Handbook of Linear Partial Differential Equations for Engineers and Scientists

HANDBOOK OF

LINEAR PARTIAL DIFFERENTIAL EQUATIONS for ENGINEERS and SCIENTISTS Andrei D. Polyanin

CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C.

Library of Congress Cataloging-in-Publication Data Polianin, A. D. (Andrei Dmitrievich) Handbook of linear partial differential equations for engineers and scientists / by Andrei D. Polyanin p. cm. Includes bibliographical references and index. ISBN 1-58488-299-9 1. Differential equations, Linear--Numerical solution--Handbooks, manuals, etc. I. Title. QA377 .P568 2001 515′.354—dc21

2001052427 CIP

This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. Apart from any fair dealing for the purpose of research or private study, or criticism or review, as permitted under the UK Copyright Designs and Patents Act, 1988, this publication may not be reproduced, stored or transmitted, in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without the prior permission in writing of the publishers, or in the case of reprographic reproduction only in accordance with the terms of the licenses issued by the Copyright Licensing Agency in the UK, or in accordance with the terms of the license issued by the appropriate Reproduction Rights Organization outside the UK. All rights reserved. Authorization to photocopy items for internal or personal use, or the personal or internal use of specific clients, may be granted by CRC Press LLC, provided that $1.50 per page photocopied is paid directly to Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923 USA. The fee code for users of the Transactional Reporting Service is ISBN 1-58488-2999/02/$0.00+$1.50. The fee is subject to change without notice. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC for such copying. Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe.

Visit the CRC Press Web site at www.crcpress.com © 2002 by Chapman & Hall/CRC No claim to original U.S. Government works International Standard Book Number 1-58488-299-9 Library of Congress Card Number 2001052427 Printed in the United States of America 1 2 3 4 5 6 7 8 9 0

FOREWORD Linear partial differential equations arise in various fields of science and numerous applications, e.g., heat and mass transfer theory, wave theory, hydrodynamics, aerodynamics, elasticity, acoustics, electrostatics, electrodynamics, electrical engineering, diffraction theory, quantum mechanics, control theory, chemical engineering sciences, and biomechanics. This book presents brief statements and exact solutions of more than 2000 linear equations and problems of mathematical physics. Nonstationary and stationary equations with constant and variable coefficients of parabolic, hyperbolic, and elliptic types are considered. A number of new solutions to linear equations and boundary value problems are described. Special attention is paid to equations and problems of general form that depend on arbitrary functions. Formulas for the effective construction of solutions to nonhomogeneous boundary value problems of various types are given. We consider second-order and higher-order equations as well as the corresponding boundary value problems. All in all, the handbook presents more equations and problems of mathematical physics than any other book currently available. For the reader’s convenience, the introduction outlines some definitions and basic equations, problems, and methods of mathematical physics. It also gives useful formulas that enable one to express solutions to stationary and nonstationary boundary value problems of general form in terms of the Green’s function. Two supplements are given at the end of the book. Supplement A lists properties of the most common special functions (the gamma function, Bessel functions, degenerate hypergeometric functions, Mathieu functions, etc.). Supplement B describes the methods of generalized and functional separation of variables for nonlinear partial differential equations. We give specific examples and an overview application of these methods to construct exact solutions for various classes of second-, third-, fourth-, and higher-order equations (in total, about 150 nonlinear equations with solutions are described). Special attention is paid to equations of heat and mass transfer theory, wave theory, and hydrodynamics as well as to mathematical physics equations of general form that involve arbitrary functions. The equations in all chapters are in ascending order of complexity. Many sections can be read independently, which facilitates working with the material. An extended table of contents will help the reader find the desired equations and boundary value problems. We refer to specific equations using notation like “1.8.5.2,” which means “Equation 2 in Subsection 1.8.5.” To extend the range of potential readers with diverse mathematical backgrounds, the author strove to avoid the use of special terminology wherever possible. For this reason, some results are presented schematically, in a simplified manner (without details), which is however quite sufficient in most applications. Separate sections of the book can serve as a basis for practical courses and lectures on equations of mathematical physics. The author thanks Alexei Zhurov for useful remarks on the manuscript. The author hopes that the handbook will be useful for a wide range of scientists, university teachers, engineers, and students in various areas of mathematics, physics, mechanics, control, and engineering sciences. Andrei D. Polyanin

© 2002 by Chapman & Hall/CRC Page iii

BASIC NOTATION Latin Characters Im[
 ]  

Re[
] , ,  ,  ,   

 , , , 

1  ,   x

|x| y

 



%

2

Laplace operator  two-dimensional Laplace operator,

3

three-dimensional Laplace operator,



( ) 



2 1

magnitude (length) of -dimensional vector, |x| =    -dimensional vector, y = { 1 ,   ,   }

2 2

+

+  +  2

Greek Characters 



fundamental solution imaginary part of a complex quantity
Green’s function     -dimensional Euclidean space, = {−  <   <  ; = 1,   , } real part of a complex quantity
cylindrical coordinates, =   2 +  2 and  = cos ,  = sin spherical coordinates, =   2 +  2 +  2 and  = sin  cos ,  = sin  sin ,  = cos   time ( ≥ 0) unknown function (dependent variable) space (Cartesian) coordinates  Cartesian coordinates in -dimensional space  -dimensional vector, x = { 1 ,   ,   }

 "

( )

& '   ) = 



-dimensional Laplace operator,





 

3

 =









2

= 

2



2

  2  2  

=1



2

 

2

= 

2

+ 

2

+ 

 

2 2

+ 

 

2 2



Dirac delta function;   ( ) ( −  )  = ( ), where ( ) is any continuous function, − ! >0    Kronecker delta,  " = # 1 if  = $ , 0 if ≠ $ % 1 if  ≥ 0, Heaviside unit step function, ( ) = # 0 if  < 0

Brief Notation for Derivatives

&  = &  , 

& 

 *) ) =  

, 



&  = & ,  2 2



,

& (' '   *) )*) =   

&

2

= &  3



3

,

2

,  ( 

&   )

=

&



= &

  



2



(partial derivatives)

2

(derivatives for





= ( ))

Special Functions (See Also Supplement A) 1 Ai( ) = +  , cos 0 

Ce2  +4 ( , 5 ) = , 

=0




1 3 3 2  +4 2  +4

+

/.



cosh[(2 +6 ) ]

Airy function; Ai( ) = 0 1 1

1 3

2

2 33 2 . 13 3 - 3 

even modified Mathieu functions, where 6 = 0, 1; Ce2  +4 ( , 5 ) = ce2  +4 (78 , 5 )

© 2002 by Chapman & Hall/CRC Page v



ce2  ( , 5 ) = , 

ce2  9 :

+1 ( 

=

2 2




=0



,5 ) = , 

9 :

2  +1 2  +1




=0

even + -periodic Mathieu functions; these satisfy the )*) equation  + ( ! − 2 5 cos 2 ) = 0, where ! = ! 2  ( 5 ) are eigenvalues

cos 2 

cos[(2 +1) ]

( )

2 .  erf  = < +  exp - −= 2 0

error function =

2 . erfc  = < +  , exp - −= 2 



 ? 2  ( ) = (−1)    : : > : (1) ( ) = @ ( ) + 7BA : : > : (2) ( ) = @ ( ) − B7 A  C ! ( ,D ,E ; ) = 1+ ,  =1: >

F:

@ 2 J M

−

2



.

2

" ) 3



=0



se2  ( , 5 ) = ,

2  +4 2  +4



se2 

:

+1 ( 

( ) =

=0

2 2

N



,5 ) = , 

@

:

=0

0

H (P ) =  , 0

sinh[(2 +6 ) ]

? −Q = R

−1



(! , D ;  ) = 1 + , 

−1

=1

 hypergeometric function, ( ! )  = ! ( ! + 1)   ( ! + − 1)

modified Bessel function of first kind Bessel function of first kind modified Bessel function of second kind generalized Laguerre polynomial Legendre polynomial associated Legendre functions odd modified Mathieu functions, where 6 = 0, 1; Se2  +4 ( , 5 ) = −7 se2  +4 (78 , 5 ) odd + -periodic Mathieu functions; these satisfy the )*) equation  + ( ! − 2 5 cos 2 ) = 0, where ! = D 2  ( 5 ) are eigenvalues

sin[(2 +1) ]

( ) cos(+ ; ) − @ sin(+ ; )

O ( P ,  ) =   ? −Q = R

 ( )

sin 2 

2  +1 2  +1

N

M

" 

N

Hankel function of first kind, 7 2 = −1 Hankel function of second kind, 7 2 = −1



2



complementary error function Hermite polynomial

( G 2) +2 ( ) = ,    =0 ! H ( ; + +: 1)    : (−1) ( G 2) +2 , ( ) =    =0 ! H ( ; + + 1) F : FI: + : ( ) − ( ) − ( ) = + 2 sin(  ; )  +K ? − . K 1 −K ?  ( ) =      -L !    1 2  ( ) =   ( − 1)  !2  "

Se2  +4 ( , 5 ) = ,

S

?

=

( ) ( )  (! ) ( D )   (E ) !



M "  ( ) = (1 − 

A

-

even 2+ -periodic Mathieu functions; these satisfy the )*) equation  + ( ! − 2 5 cos 2 ) = 0, where ! = ! 2  +1 ( 5 ) are eigenvalues parabolic cylinder function (see Paragraph 7.3.4-1); it )*) . satisfies the equation  + -8; + 12 − 14  2  = 0

−

=

= !( )     (D ) !

:

( )

odd 2+ -periodic Mathieu functions; these satisfy the )*) equation  + ( ! − 2 5 cos 2 ) = 0, where ! = D 2  +1 ( 5 ) are eigenvalues Bessel function of second kind incomplete gamma function gamma function degenerate hypergeometric function,  ( ! )  = ! ( ! + 1)   ( ! + − 1)

© 2002 by Chapman & Hall/CRC Page vi

AUTHOR Andrei D. Polyanin, D.Sc., Ph.D., is a noted scientist of broad interests, who works in various areas of mathematics, mechanics, and chemical engineering sciences. A. D. Polyanin graduated from the Department of Mechanics and Mathematics of the Moscow State University in 1974. He received his Ph.D. degree in 1981 and D.Sc. degree in 1986 at the Institute for Problems in Mechanics of the Russian (former USSR) Academy of Sciences. Since 1975, A. D. Polyanin has been a member of the staff of the Institute for Problems in Mechanics of the Russian Academy of Sciences. Professor Polyanin has made important contributions to developing new exact and approximate analytical methods of the theory of differential equations, mathematical physics, integral equations, engineering mathematics, nonlinear mechanics, theory of heat and mass transfer, and chemical hydrodynamics. He obtained exact solutions for several thousand ordinary differential, partial differential, mathematical physics, and integral equations. Professor Polyanin is an author of 27 books in English, Russian, German, and Bulgarian, as well as over 120 research papers and three patents. He has written a number of fundamental handbooks, including A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, CRC Press, 1995; A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, CRC Press, 1998; and A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Gordon and Breach, 2001. In 1991, A. D. Polyanin was awarded a Chaplygin Prize of the USSR Academy of Sciences for his research in mechanics. Address: Institute for Problems in Mechanics, RAS, 101 Vernadsky Avenue, Building 1, 117526 Moscow, Russia E-mail: [email protected]

© 2002 by Chapman & Hall/CRC Page vii

CONTENTS Foreword Basic Notation and Remarks Author Introduction. Some Definitions, Formulas, Methods, and Solutions 0.1. Classification of Second-Order Partial Differential Equations 0.1.1. Equations with Two Independent Variables 0.1.2. Equations with Many Independent Variables 0.2. Basic Problems of Mathematical Physics 0.2.1. Initial and Boundary Conditions. Cauchy Problem. Boundary Value Problems 0.2.2. First, Second, Third, and Mixed Boundary Value Problems 0.3. Properties and Particular Solutions of Linear Equations 0.3.1. Homogeneous Linear Equations 0.3.2. Nonhomogeneous Linear Equations 0.4. Separation of Variables Method 0.4.1. General Description of the Separation of Variables Method 0.4.2. Solution of Boundary Value Problems for Parabolic and Hyperbolic Equations 0.5. Integral Transforms Method 0.5.1. Main Integral Transforms 0.5.2. Laplace Transform and Its Application in Mathematical Physics 0.5.3. Fourier Transform and Its Application in Mathematical Physics 0.6. Representation of the Solution of the Cauchy Problem via the Fundamental Solution 0.6.1. Cauchy Problem for Parabolic Equations 0.6.2. Cauchy Problem for Hyperbolic Equations 0.7. Nonhomogeneohus Boundary Value Problems with One Space Variable. Representation of Solutions via the Green’s Function 0.7.1. Problems for Parabolic Equations 0.7.2. Problems for Hyperbolic Equations 0.8. Nonhomogeneous Boundary Value Problems with Many Space Variables. Representation of Solutions via the Green’s Function 0.8.1. Problems for Parabolic Equations 0.8.2. Problems for Hyperbolic Equations 0.8.3. Problems for Elliptic Equations 0.8.4. Comparison of the Solution Structures for Boundary Value Problems for Equations of Various Types 0.9. Construction of the Green’s Functions. General Formulas and Relations 0.9.1. Green’s Functions of Boundary Value Problems for Equations of Various Types in Bounded Domains 0.9.2. Green’s Functions Admitting Incomplete Separation of Variables 0.9.3. Construction of Green’s Functions via Fundamental Solutions

© 2002 by Chapman & Hall/CRC Page ix

0.10. Duhamel’s Principles in Nonstationary Problems 0.10.1. Problems for Homogeneous Linear Equations 0.10.2. Problems for Nonhomogeneous Linear Equations 0.11. Transformations Simplifying Initial and Boundary Conditions 0.11.1. Transformations That Lead to Homogeneous Boundary Conditions 0.11.2. Transformations That Lead to Homogeneous Initial and Boundary Conditions 1. Parabolic Equations with One Space Variable 1.1. Constant Coefficient Equations 2 ' 1.1.1. Heat Equation  T = !  T 2 

 ' 2 = !  T2  '  2 1.1.3. Equation of the Form  T = !  T 2  '  2 1.1.4. Equation of the Form  T = !  T 2  '  2 1.1.5. Equation of the Form  T = !  T 2   

1.1.2. Equation of the Form  T

S



+ ( , )

S  + D  + ( , ) S



+ D  T + ( , )   +D  T +E   

S



+ ( , ) 1.2. Heat Equation with Axial or Central Symmetry and Related Equations 1.2.1. Equation of the Form  T 1.2.2. Equation of the Form 1.2.3. Equation of the Form 1.2.4. Equation of the Form 1.2.5. Equation of the Form

'

 '  T  '  T  '  T  '  T  '  T 

2 = ! -  T2 +

=! =! =!

= 

.

1

 T U  U 1 -  T2 +  T  2U U  U -  T2 + 2  T U  U  2U -  T2 + 2  T U  U 2  U T 2 + 1−2V  T    2 T 2 + 1−2V  T      2U



.

S



S



+ ( , ) . .

+ ( , ) S



1.2.6. Equation of the Form =  + ( , )  1.3. Equations Containing Power Functions and Arbitrary Parameters  2  ' 1.3.1. Equations of the Form  T = !  T 2 + ( , )

 '  2   1.3.2. Equations of the Form  T = !  T 2 + ( , )  T

 '  2       1.3.3. Equations of the Form  T = !  T 2 + ( , )  T + W ( , ) + X ( , )  '      2  1.3.4. Equations of the Form  T = ( !  + D )  T 2 + ( , )  T + W ( , )  '

1.3.5. Equations of the Form  T = ( ! 

2

      2  + DY + E )  T 2 + ( , )  T + W ( , )

 '      2 1.3.6. Equations of the Form  T = ( )  T 2 + W ( , )  T + X ( , )

 '    2    1.3.7. Equations of the Form  T = ( , )  T 2 + W ( , )  T + X ( , )      2 1.3.8. Liquid-Film Mass Transfer Equation (1 −  2 )  T = !  T 2 

 

2

 

1.3.9. Equations of the Form ( ,  )  T + W ( ,  )  T =  T 2 + X ( ,  )       1.4. Equations Containing Exponential Functions and Arbitrary Parameters  2  ' 1.4.1. Equations of the Form  T = !  T 2 + ( , )

 '  2   1.4.2. Equations of the Form  T = !  T 2 + ( , )  T

 '  2      1.4.3. Equations of the Form  T = !  T 2 + ( , )  T + W ( , )

 '     2   1.4.4. Equations of the Form  T = !   T 2 + ( , )  T + W ( , )

 '  2 ?      1.4.5. Equations of the Form  T = ! V   T 2 + ( , )  T + W ( , )      1.4.6. Other Equations

1.5. Equations Containing Hyperbolic Functions and Arbitrary Parameters 1.5.1. Equations Containing a Hyperbolic Cosine 1.5.2. Equations Containing a Hyperbolic Sine 1.5.3. Equations Containing a Hyperbolic Tangent 1.5.4. Equations Containing a Hyperbolic Cotangent

© 2002 by Chapman & Hall/CRC Page x

1.6. Equations Containing Logarithmic Functions and Arbitrary Parameters '







1.6.1. Equations of the Form  T = !  T 2 + ( , )  T + W ( , )  '     2   1.6.2. Equations of the Form  T = !   T 2 + ( , )  T + W ( , )      1.7. Equations Containing Trigonometric Functions and Arbitrary Parameters 1.7.1. Equations Containing a Cosine 1.7.2. Equations Containing a Sine 1.7.3. Equations Containing a Tangent 1.7.4. Equations Containing a Cotangent 1.8. Equations Containing Arbitrary Functions 2

 2 ' 1.8.1. Equations of the Form  T = !  T 2 + (  '    2 1.8.2. Equations of the Form  T = !  T 2 + (

1.8.3. Equations of the Form 1.8.4. Equations of the Form 1.8.5. Equations of the Form 1.8.6. Equations of the Form

 '  T  '  T  '  T  '  T  '  T  '  T  Z (

 , )

 , ) T   2     = !  T 2 + ( , )  T + W ( , )     2   = !   T 2 + ( , )  T + W ( , )  2 ?      = ! V   T 2 + ( , )  T + W ( , )    2    = ( )  T 2 + W ( , )  T + X ( , )    2      = ( )  T 2 + W ( , )  T + X ( , )     2   = ( , )  T 2 + W ( , )  T + X ( , ) S      ' )  T =  6 ( )  T − 5 ( ) + ( , )    [   \

1.8.7. Equations of the Form 1.8.8. Equations of the Form 1.8.9. Equations of the Form 1.9. Equations of Special Form 1.9.1. Equations of the Diffusion (Thermal) Boundary Layer ` 2 2 ' 1.9.2. One-Dimensional Schro¨ dinger Equation 7^]X  T = − 2_ "  T 2 + a ( ) 

2. Parabolic Equations with Two Space Variables  ' 2.1. Heat Equation  T = ! 2   2.1.1. Boundary Value Problems in Cartesian Coordinates 2.1.2. Problems in Polar Coordinates 2.1.3. Axisymmetric Problems  S  2.2. Heat Equation with a Source  T ' = ! 2  + ( ,  , )  2.2.1. Problems in Cartesian Coordinates 2.2.2. Problems in Polar Coordinates 2.2.3. Axisymmetric Problems 2.3. Other Equations 2.3.1. Equations Containing Arbitrary Parameters 2.3.2. Equations Containing Arbitrary Functions

 

3. Parabolic Equations with Three or More Space Variables  ' 3.1. Heat Equation  T = ! 3  3.1.1. Problems in Cartesian Coordinates 3.1.2. Problems in Cylindrical Coordinates 3.1.3. Problems in Spherical Coordinates  S  ' 3.2. Heat Equation with Source  T = ! 3  + ( ,  ,  , ) 3.2.1. Problems in Cartesian Coordinates 3.2.2. Problems in Cylindrical Coordinates 3.2.3. Problems in Spherical Coordinates 3.3. Other Equations with Three Space Variables 3.3.1. Equations Containing Arbitrary Parameters 3.3.2. Equations Containing Arbitrary Functions S  ' 3.3.3. Equations of the Form b ( ,  ,  )  T = div[ ! ( ,  ,  )∇ ]− 5 ( ,  ,  ) + ( ,  ,  , ) 

© 2002 by Chapman & Hall/CRC Page xi



3.4. Equations with Space Variables  S  ' 3.4.1. Equations of the Form  T = !   + ( 1 ,   ,   , )  3.4.2. Other Equations Containing Arbitrary Parameters 3.4.3. Equations Containing Arbitrary Functions 4. Hyperbolic Equations with One Space Variable 4.1. Constant Coefficient Equations 2' 4.1.1. Wave Equation  T 2 = !

4.1.2. 4.1.3. 4.1.4.

2

2

 T2  2'   2 Equations of the Form  T 2 = ! 2  T 2 + 2'  2  Equation of the Form  T 2 = ! 2  T 2 − D  2'  2 Equation of the Form  T 2 = ! 2  T 2 − D  2'  2 Equation of the Form  T 2 = ! 2  T 2 + D   

S



( , ) S



S   T + ( , )   S   T + E  + ( , )  

4.1.5. 4.2. Wave Equation with Axial or Central Symmetry 2' 4.2.1. Equations of the Form  T 2 = !

2

2 '

2

 2'

-

2

 2'

-

2

-

2

-

4.2.2. Equation of the Form  T 2 = ! 4.2.3. Equation of the Form  T 2 = !

4.2.4. Equation of the Form  T 2 = !  2' 4.2.5. Equation of the Form  T 2 = !  2'



+ ( , )

2

-  T2 + 1 U 2 U  T2 + 1  U   2U  T2 + 2   2U U   T2 + 2  U   2U  T2 + 1  U   2U  T2 + 2   U U 

.

 T  U T U T U T U T U T U

.

S



S



+ ( , ) . .

+ ( , )

S  − D  + ( , )

.

S

.



4.2.6. Equation of the Form  T 2 = ! 2 − D  + ( , )  4.3. Equations Containing Power Functions and Arbitrary Parameters

2' 2 S  4.3.1. Equations of the Form  T 2 = ( !  + D )  T 2 + E  T +  + ( , )

 2'

4.3.2. Equations of the Form  T 2 = ( !   4.3.3. Other Equations

2

   2 S  + D )  T 2 + Ec  T +  + ( , )  

 

4.4. Equations Containing the First Time Derivative 2' 2 S  ' 4.4.1. Equations of the Form  T 2 +  T = ! 2  T 2 + D  T + E  + ( , )

 2'  '      2 S  4.4.2. Equations of the Form  T 2 +  T = ( )  T 2 + W ( )  T + X ( ) + ( , )       4.4.3. Other Equations 4.5. Equations Containing Arbitrary Functions

2' S  6 ( )  T − 5 ( ) + ( , ) \    !   ' [ S   + ( )  T = D ( ) d  6 ( )  T − 5 ( ) e + ( , )    [   \

4.5.1. Equations of the Form Z ( )  T 2 =  '

2

4.5.2. Equations of the Form  T 2  4.5.3. Other Equations

5. Hyperbolic Equations with Two Space Variables 

'

5.1. Wave Equation  T 2 = ! 2 2   5.1.1. Problems in Cartesian Coordinates 5.1.2. Problems in Polar Coordinates 5.1.3. Axisymmetric Problems 2

'

5.2. Nonhomogeneous Wave Equation  T 2 = ! 2  5.2.1. Problems in Cartesian Coordinates 5.2.2. Problems in Polar Coordinates 5.2.3. Axisymmetric Problems 2

'





S

2



S



+ ( ,  , )



5.3. Equations of the Form  T 2 = ! 2 2  − D  + ( ,  , )  5.3.1. Problems in Cartesian Coordinates 5.3.2. Problems in Polar Coordinates 5.3.3. Axisymmetric Problems 2

© 2002 by Chapman & Hall/CRC Page xii

 2' S  ' 5.4. Telegraph Equation  T 2 +  T = ! 2 2  − D  + ( ,  , )   5.4.1. Problems in Cartesian Coordinates 5.4.2. Problems in Polar Coordinates 5.4.3. Axisymmetric Problems 5.5. Other Equations with Two Space Variables

6. Hyperbolic Equations with Three or More Space Variables 

'

6.1. Wave Equation  T 2 = ! 2 3   6.1.1. Problems in Cartesian Coordinates 6.1.2. Problems in Cylindrical Coordinates 6.1.3. Problems in Spherical Coordinates 2

'

6.2. Nonhomogeneous Wave Equation  T 2 = ! 2  6.2.1. Problems in Cartesian Coordinates 6.2.2. Problems in Cylindrical Coordinates 6.2.3. Problems in Spherical Coordinates 2



'



3

S





+ ( ,  ,  , )

S



6.3. Equations of the Form  T 2 = ! 2 3  − D  + ( ,  ,  , )  6.3.1. Problems in Cartesian Coordinates 6.3.2. Problems in Cylindrical Coordinates 6.3.3. Problems in Spherical Coordinates 2

'

'



S



6.4. Telegraph Equation  T 2 +  T = ! 2 3  − D  + ( ,  ,  , )   6.4.1. Problems in Cartesian Coordinates 6.4.2. Problems in Cylindrical Coordinates 6.4.3. Problems in Spherical Coordinates 6.5. Other Equations with Three Space Variables 6.5.1. Equations Containing Arbitrary Parameters 2

2' 6.5.2. Equation of the Form b ( ,  ,  )  T 2 = div ! ( ,  ,  )∇



6.6. Equations with Space Variables  2' 6.6.1. Wave Equation  T 2 = ! 2  



[

\

S  − 5 ( ,  ,  ) + ( ,  ,  , )

  2' S  6.6.2. Nonhomogeneous Wave Equation  T 2 = ! 2   + ( 1 ,   ,   , )   2' S  6.6.3. Equations of the Form  T 2 = ! 2   − D  + ( 1 ,   ,   , )  6.6.4. Equations Containing the First Time Derivative

7. Elliptic Equations with Two Space Variables 

7.1. Laplace Equation 2  = 0 7.1.1. Problems in Cartesian Coordinate System 7.1.2. Problems in Polar Coordinate System 7.1.3. Other Coordinate Systems. Conformal Mappings Method 

S

7.2. Poisson Equation 2  = − (x) 7.2.1. Preliminary Remarks. Solution Structure 7.2.2. Problems in Cartesian Coordinate System 7.2.3. Problems in Polar Coordinate System 7.2.4. Arbitrary Shape Domain. Conformal Mappings Method  S 7.3. Helmholtz Equation 2  + f  = − (x) 7.3.1. General Remarks, Results, and Formulas 7.3.2. Problems in Cartesian Coordinate System 7.3.3. Problems in Polar Coordinate System 7.3.4. Other Orthogonal Coordinate Systems. Elliptic Domain

© 2002 by Chapman & Hall/CRC Page xiii

7.4. Other Equations   7.4.1. Stationary Schro¨ dinger Equation 2  = ( ,  ) 7.4.2. Convective Heat and Mass Transfer Equations 7.4.3. Equations of Heat and Mass Transfer in Anisotropic Media 7.4.4. Other Equations Arising in Applications 2 2 S 7.4.5. Equations of the Form ! ( )  T 2 +  T 2 + D ( )  T + E ( ) = − ( ,  )  

 

 

8. Elliptic Equations with Three or More Space Variables  8.1. Laplace Equation 3  = 0 8.1.1. Problems in Cartesian Coordinates 8.1.2. Problems in Cylindrical Coordinates 8.1.3. Problems in Spherical Coordinates 8.1.4. Other Orthogonal Curvilinear Systems of Coordinates  S 8.2. Poisson Equation 3  + (x) = 0 8.2.1. Preliminary Remarks. Solution Structure 8.2.2. Problems in Cartesian Coordinates 8.2.3. Problems in Cylindrical Coordinates 8.2.4. Problems in Spherical Coordinates  S 8.3. Helmholtz Equation 3  + f  = − (x) 8.3.1. General Remarks, Results, and Formulas 8.3.2. Problems in Cartesian Coordinates 8.3.3. Problems in Cylindrical Coordinates 8.3.4. Problems in Spherical Coordinates 8.3.5. Other Orthogonal Curvilinear Coordinates 8.4. Other Equations with Three Space Variables 8.4.1. Equations Containing Arbitrary Functions S 8.4.2. Equations of the Form div [ ! ( ,  ,  )∇ ] − 5 ( ,  ,  ) = − ( ,  ,  )  8.5. Equations with Space Variables  8.5.1. Laplace Equation   = 0 8.5.2. Other Equations 9. Higher-Order Partial Differential Equations 9.1. Third-Order Partial Differential Equations 9.2. Fourth-Order One-Dimensional Nonstationary Equations 4 S  ' 9.2.1. Equations of the Form  T + ! 2  T 4 = ( , )  2'

9.2.2. Equations of the Form  T 2 + !  2'

9.2.3. Equations of the Form  T 2 + !  2'

 4  T4 = 0   S  2 4  T 4 = ( , )    = S ( ,  ) 2  4T 4 +   2

9.2.4. Equations of the Form  T 2 + !  9.2.5. Other Equations 9.3. Two-Dimensional Nonstationary Fourth-Order Equations 4 4 . S  ' 9.3.1. Equations of the Form  T + ! 2 -  T 4 +  T 4 = ( ,  , ) 

 

 

'

9.3.2. Two-Dimensional Equations of the Form  T 2 + ! 

2



2

   '

2

+!

9.3.3. Three- and -Dimensional Equations of the Form  T 2      2' S 9.3.4. Equations of the Form  T 2 + ! 2 +  = ( ,  , )

=0 2

 

=0

 2' 4 4 S .  9.3.5. Equations of the Form  T 2 + ! 2 -  T 4 +  T 4 +  = ( ,  , )      9.4. Fourth-Order Stationary Equations    9.4.1. Biharmonic Equation   = 0  = S ( ,  ) 9.4.2. Equations of the Form

© 2002 by Chapman & Hall/CRC Page xiv

 

S

 − f  = ( ,  ) 9.4.3. Equations of the Form 4 4 S 9.4.4. Equations of the Form  T 4 +  T 4 = ( ,  )  4

 4

S

9.4.5. Equations of the Form  T 4 +  T 4 +  = ( ,  )     9.4.6. Stokes Equation (Axisymmetric Flows of Viscous Fluids) 9.5. Higher-Order Linear Equations with Constant Coefficients 9.5.1. Fundamental Solutions. Cauchy Problem 9.5.2. Elliptic Equations 9.5.3. Hyperbolic Equations 9.5.4. Regular Equations. Number of Initial Conditions in the Cauchy Problem 9.5.5. Some Special-Type Equations 9.6. Higher-Order Linear Equations with Variable Coefficients 9.6.1. Equations Containing the First Time Derivative 9.6.2. Equations Containing the Second Time Derivative 9.6.3. Nonstationary Problems with Many Space Variables 9.6.4. Some Special-Type Equations Supplement A. Special Functions and Their Properties A.1. Some Symbols and Coefficients A.1.1. Factorials A.1.2. Binomial Coefficients A.1.3. Pochhammer Symbol A.1.4. Bernoulli Numbers A.2. Error Functions and Exponential Integral A.2.1. Error Function and Complementary Error Function A.2.2. Exponential Integral A.2.3. Logarithmic Integral A.3. Sine Integral and Cosine Integral. Fresnel Integrals A.3.1. Sine Integral A.3.2. Cosine Integral A.3.3. Fresnel Integrals A.4. Gamma and Beta Functions A.4.1. Gamma Function A.4.2. Beta Function A.5. Incomplete Gamma and Beta Functions A.5.1. Incomplete Gamma Function A.5.2. Incomplete Beta Function A.6. Bessel Functions A.6.1. Definitions and Basic Formulas A.6.2. Integral Representations and Asymptotic Expansions A.6.3. Zeros and Orthogonality Properties of Bessel Functions A.6.4. Hankel Functions (Bessel Functions of the Third Kind) A.7. Modified Bessel Functions A.7.1. Definitions. Basic Formulas A.7.2. Integral Representations and Asymptotic Expansions A.8. Airy Functions A.8.1. Definition and Basic Formulas A.8.2. Power Series and Asymptotic Expansions

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A.9. Degenerate Hypergeometric Functions A.9.1. Definitions and Basic Formulas A.9.2. Integral Representations and Asymptotic Expansions A.10. Hypergeometric Functions A.10.1. Definition and Some Formulas A.10.2. Basic Properties and Integral Representations A.11. Whittaker Functions A.12. Legendre Polynomials and Legendre Functions A.12.1. Definitions. Basic Formulas A.12.2. Zeros of Legendre Polynomials and the Generating Function A.12.3. Associated Legendre Functions A.13. Parabolic Cylinder Functions A.13.1. Definitions. Basic Formulas A.13.2. Integral Representations and Asymptotic Expansions A.14. Mathieu Functions A.14.1. Definitions and Basic Formulas A.15. Modified Mathieu Functions A.16. Orthogonal Polynomials A.16.1. Laguerre Polynomials and Generalized Laguerre Polynomials A.16.2. Chebyshev Polynomials and Functions A.16.3. Hermite Polynomial A.16.4. Jacobi Polynomials Supplement B. Methods of Generalized and Functional Separation of Variables in Nonlinear Equations of Mathematical Physics B.1. Introduction B.1.1. Preliminary Remarks B.1.2. Simple Cases of Variable Separation in Nonlinear Equations B.1.3. Examples of Nontrivial Variable Separation in Nonlinear Equations B.2. Methods of Generalized Separation of Variables B.2.1. Structure of Generalized Separable Solutions B.2.2. Solution of Functional Differential Equations by Differentiation B.2.3. Solution of Functional Differential Equations by Splitting B.2.4. Simplified Scheme for Constructing Exact Solutions of Equations with Quadratic Nonlinearities B.3. Methods of Functional Separation of Variables B.3.1. Structure of Functional Separable Solutions B.3.2. Special Functional Separable Solutions B.3.3. Differentiation Method B.3.4. Splitting Method. Reduction to a Functional Equation with Two Variables B.3.5. Some Functional Equations and Their Solutions. Exact Solutions of Heat and Wave Equations B.4. First-Order Nonlinear Equations B.4.1. Preliminary Remarks B.4.2. Individual Equations B.5. Second-Order Nonlinear Equations B.5.1. Parabolic Equations B.5.2. Hyperbolic Equations B.5.3. Elliptic Equations B.5.4. Equations Containing Mixed Derivatives

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B.5.5. General Form Equations B.6. Third-Order Nonlinear Equations B.6.1. Stationary Hydrodynamic Boundary Layer Equations B.6.2. Nonstationary Hydrodynamic Boundary Layer Equations B.7. Fourth-Order Nonlinear Equations B.7.1. Stationary Hydrodynamic Equations (Navier–Stokes Equations) B.7.2. Nonstationary Hydrodynamic Equations B.8. Higher-Order Nonlinear Equations C .  ' B.8.1. Equations of the Form  T = -L , ,  ,  T ,   ,  g T  2'   g . C     B.8.2. Equations of the Form  T 2 = -L , , ,  T ,   ,  g T     g  B.8.3. Other Equations References

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Introduction

Some Definitions, Formulas, Methods, and Solutions 0.1. Classification of Second-Order Partial Differential Equations 0.1.1. Equations with Two Independent Variables 0.1.1-1. Examples of equations encountered in applications. Three basic types of partial differential equations are distinguished—parabolic, hyperbolic, and elliptic. The solutions of the equations pertaining to each of the types have their own characteristic qualitative differences. The simplest example of a parabolic equation is the heat equation


− 



2 




2

= 0,

(1)



where the variables and play the role of time and the spatial coordinate, respectively. Note that equation (1) contains only one highest derivative term. The simplest example of a hyperbolic equation is the wave equation


2  

2




−

2 

2

= 0,

(2)



where the variables and play the role of time and the spatial coordinate, respectively. Note that the highest derivative terms in equation (2) differ in sign. The simplest example of an elliptic equation is the Laplace equation 2
 

2

2


+

2

= 0,

(3)



where and play the role of the spatial coordinates. Note that the highest derivative terms in equation (3) have like signs. Any linear partial differential equation of the second-order with two independent variables can be reduced, by appropriate manipulations, to a simpler equation which has one of the three highest derivative combinations specified above in examples (1), (2), and (3). 0.1.1-2. Types of equations. Characteristic equations. Consider a second-order partial differential equation with two independent variables which has the general form 





( , )

2 

© 2002 by Chapman & Hall/CRC




2



+ 2 ( , ) 

2








+ ( , )

2 




2

= 








, , ,







, 



,

(4)





where  ,  ,  are some functions of and that have continuous derivatives up to the second-order inclusive.*   Given a point ( , ), equation (4) is said to be parabolic

if

hyperbolic



if

elliptic



if

2

−

2

−



−





2



= 0, 

> 0, 

0( 1 ): ; [ ] + > =1 operator : ; depends on

operator : F

(







(x, ) = exp( 1 1 , &'&'/ & ,




[ ] = : ; [ -] + : 1 [ ] + I'I'I + :  - [ ], operator : ; depends on only , operator : depends on only

6

F




(  (



(x, ) = Z [ ; " (x),   is an arbitrary constant, x = { 1 , &'&'& ,

[ ] = : ; [ ] + : x [ ],  operator : ; depends on only , operator : x depends on only x

5

8

/

Y

3





(x, ) =( G exp( Y + 1 1 + I'I'I + ), , 1 , &'&'& , are related by / an algebraic / equation

b

"

+:

a b Y

−

/ \]Y > 0 ( ^ 1 )

1 (^ 1 , b

)"

-

+

0

2 (^ 2 ,

>

=2 / b

) &'&'&e"

( ^ 1 )_ (

(^

(

" 1

=0

, ), b

] = Y ( )" , E = 1, &'&'& , % , b ( ( ) + Y ( )=0 1 2 ( ) + I'I'I + Y b

,; ["

b

b

0.3.1-3. Separable solutions. Many homogeneous linear partial differential equations have solutions that can be represented as the product of functions depending on different arguments. Such solutions are referred to as separable solutions. Table 2 presents the most commonly encountered types of homogeneous linear differential equations with many independent variables that admit exact separable solutions. Linear combinations of ( particular solutions that correspond to different values of the separation parameters, Y , 1 , &'&'& , , are also solutions of the equations in question. For brevity, the word “operator” is used to denote / / “linear differential operator.” For a constant coefficient equation (see the first row in Table 2), the separation parameters must satisfy the algebraic equation f ( ( Y , 1 , &'&'& , ) = 0, (4) /

/

which results from substituting the solution into the equation (1). In physical applications, equation (4) is usually referred to as a variance equation. Any % of the % + 1 separation parameters in (4) can be treated as arbitrary.

© 2002 by Chapman & Hall/CRC Page 9

Note that constant coefficient equations also admit more sophisticated solutions; see the second and third rows, the last column. The eighth row of Table 2 presents the case of incomplete separation of variables where the ( solution is separated with respect to the space variables ^ 1 , &'&'& , ^ but is not separated with respect to time .   b    ! For stationary equations, which do not depend on , one should set Y = 0, : ; [` ] ≡ 0, b and  ( ) ≡ 1 in rows 1, 6, and 7 of Table 2. b

0.3.1-4. Solutions in the form of infinite series in . b

1 H . The equation a

`

=

[` ], g

a b where g is an arbitrary linear differential operator of the second (or any) order that only depends on the space variables, has the formal series - solution `

(x, ) = > (x) + b

).J

b

=1

E

[ > (x)], g

!

[> ] = g

g

h5g

−1

[ > ]i ,

where > (x) is an arbitrary infinitely differentiable function. This solution satisfies the initial condition ` (x, 0) = > (x). 2 H . The equation

2 a

`

2

=

[` ], g

a b as in Item 1 , has a formal solution represented by the where g is a linear differential operator, just H sum of two series as `

(x, ) = b

)-

2 J

b

=0

(2 E )!

g

[ > (x)] +

)-

2 +1 J

b

=0

(2 E + 1)!

g

[@ (x)],

where > (x) and @ (x) are arbitrary infinitely differentiable functions. This solution satisfies the initial conditions ` (x, 0) = > (x) and ; ` (x, 0) = @ (x). a

0.3.2. Nonhomogeneous Linear Equations 0.3.2-1. Simplest properties of nonhomogeneous linear equations. For brevity, we write a nonhomogeneous linear partial differential equation in the form F [` ] = < (x, ), (5) b F where the linear differential operator is defined above, see the beginning of Paragraph 0.3.1-1. Below are the simplest properties of particular solutions of the nonhomogeneous equation (5). 1 H . If ` K j (x, ) is a particular solution of the nonhomogeneous equation (5) and ` K 0 (x, ) is a particular b b solution of the corresponding homogeneous equation (1), then the sum G ` K 0 (x, ) + ` K j (x, ), b b where G is an arbitrary constant, is also a solution of the nonhomogeneous equation (5). The following, more general statement holds: The general solution of the nonhomogeneous equation (5) is the sum of the general solution of the corresponding homogeneous equation (1) and any particular solution of the nonhomogeneous equation (5). 2 H . Suppose ` 1 and ` 2 are solutions of nonhomogeneous linear equations with the same left-hand side and different right-hand sides, i.e., F F [` 1 ] = < 1 (x, ), [` 2 ] = < 2 (x, ). b b Then the function ` = ` 1 + ` 2 is a solution of the equation F [` ] = < 1 (x, ) + < 2 (x, ). b

b

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0.3.2-2. Fundamental and particular solutions of stationary equations. Consider the second-order linear stationary (time-independent) nonhomogeneous equation : x [`

] = < (x).

(6)

Here, : x is a linear differential operator of( the second (or any) order of general form whose 6 coefficients are dependent on x, where x 8 . A distribution k = k (x, y) that satisfies the equation with a special right-hand side : x[ k

] = A (x − y)

(7)

is called a fundamental solution corresponding to the operator : x . In (7), A (x) is an l -dimensional Dirac delta function and the vector quantity y = {m 1 , n'n'n , m o } appears in equation (7) as 6 o . an l -dimensional free parameter. It is assumed that y 8 The l -dimensional Dirac delta function possesses the following basic properties: 1. 2.

A

-

(x) = A (^ 1 ) A (^ 2 ) n'n'npA (^

P q r


0. b

0.4.1-2. Search for particular solutions. Derivation of equations and boundary conditions. The approach is based on searching for particular solutions of equation (1) in the product form `

(^ , ) =  (^ ) Ž ( ). b

(5)

b

After separation of the variables and elementary manipulations, one arrives at the following linear ordinary differential equations for the functions  =  (^ ) and Ž = Ž ( ): b

‚

(^ )  ‹

+ ƒ (^ ) 



( )Ž

‡ ‡

b

 Š’Š



+  ( )Ž b

‡ Š



+ [ ‘ + „ (^ )]  = 0,

(6)

+ [ ‘ − … ( )]Ž = 0.

(7)

b

These equations contain a free parameter ‘ called the separation constant. With the notation adopted in Fig. 1, equations (6) and (7) can be rewritten as follows:  “ 1 (^ ,  ,   ,   ) + ‘  = 0 ‡ ‡ ‡ and Ž “ 2 ( , Ž , Ž  , Ž  ) + ‘ Ž = 0. b Š Š’Š Substituting (5) into (2) yields the boundary conditions for  =  (^ ): †

1  †

‡

2  ‡

+ˆ +ˆ

1 2

= 0 at = 0 at ^

^

= =

^ 1, ^ 2.

(8)

The homogeneous linear ordinary differential equation (6) in conjunction with the homogeneous linear boundary conditions (8) make up an eigenvalue problem.

© 2002 by Chapman & Hall/CRC Page 12

Figure 1. Scheme of solving linear boundary value problems by separation of variables (for parabolic equations, the function ” 2 does not depend on • ˜™–—– ˜ , and š › = 0).

© 2002 by Chapman & Hall/CRC Page 13

0.4.1-3. Solution of eigenvalue problems. Orthogonality of eigenfunctions. Suppose œ 1 = œ 1 ( , ‘ ) and œ 2 = œ 2 ( , ‘ ) are linearly independent particular solutions of equation (6). Then the general solution of this equation can be represented as the linear combination = 

,‘ )+ž

ž 1 œ 1 ( 

2 œ 2 ( 

, ‘ ),

(9)

where ž 1 and ž 2 are arbitrary constants. Substituting solution (9) into the boundary conditions (8) yields the following homogeneous linear algebraic system of equations for ž 1 and ž 2 : Ÿ

11 ( ‘ Ÿ

where Ÿ  ¢¡ ( ‘ ) = h †   ( œ ¡ )  + ˆ must be zero; we have ‡

 

¡ œ

i

= ‡ Ÿ

‡ £

)ž

21 ( ‘

1

)ž

+Ÿ

12 ( ‘

)ž

2

= 0,

Ÿ

22 ( ‘

)ž

2

= 0,

+

1

(10)

. For system (10) to have nontrivial solutions, its determinant )Ÿ

11 ( ‘

)−Ÿ

22 ( ‘

12 ( ‘

)Ÿ

21 ( ‘

) = 0.

(11)

Solving the transcendental equation (11) for ‘ , one obtains the eigenvalues ‘ = ‘ For these values of ‘ , there are nontrivial solutions of equation (6), ( ) = Ÿ

 o

12 ( ‘ o

) œ

1 (

,‘

)−Ÿ o

11 ( ‘ o

) œ

2 (

,‘

o

, where l = 1, 2,

), o

n'n'n

(12)

which are called eigenfunctions (these functions are defined up to a constant multiplier). To facilitate the further analysis, we represent equation (6) in the form [¤ ( )   ‡

] + [‘ ‡

¥

( ) − ¦ ( )]  = 0,

(13)

where ¤

( ) = exp §©¨ ‚

ª

( ) ( ) «

 ¬

,

(­ ) exp §¯¨ (­ )

(­ ) = − ® ¦

‚

‚

(­ ) (­ ) ª

«

­ ¬

, ¥

(­ ) = ‚

1 exp §©¨ (­ ) ‚

ª

(­ ) (­ ) «

­ ¬

. (14)

It follows from the adopted assumptions (see the end of Paragraph 0.4.1-1) that ¤ (­ ), ¤ °± (­ ), ¦ (­ ), and ¥ (­ ) are continuous functions, with ¤ (­ ) > 0 and ¥ (­ ) > 0. The eigenvalue problem (13), (8) is known to possess the following properties: 1. All eigenvalues ² 1 , ² 2 , ³'³'³ are real, and ² ´ µ ¶ as · µ ¶ ; consequently, the number of negative eigenvalues is finite. 2. The system of eigenfunctions ¸ 1 (­ ), ¸ 2 (­ ), ³'³'³ is orthogonal on the interval ­ 1 ≤ ­ ≤ ­ 2 with weight ¹ (­ ), i.e., ±

¨

3. If

±

2

(­ ) ¸ ¹

´

(­ ) ¸ º

(­ )

1

«

­

= 0 for ·

≠» .

(15)

¼

(­ ) ≥ 0, ¼

½ 1¾ 1

≤ 0,

½ 2¾ 2

≥ 0,

(16)

there are no negative eigenvalues. If ≡ 0 and ¾ 1 = ¾ 2 = 0, the least eigenvalue is ² 1 = 0 and the corresponding eigenfunction is ¸ 1 = const. Otherwise, all eigenvalues are positive, provided that conditions (16) are satisfied; the first inequality in (16) is satisfied if (­ ) ≤ 0. ® Subsection 1.8.9 presents some estimates for the eigenvalues ² ´ and eigenfunctions ¸ ´ (­ ).

© 2002 by Chapman & Hall/CRC Page 14

0.4.2. Solution of Boundary Value Problems for Parabolic and Hyperbolic Equations 0.4.2-1. Solution of boundary value problems for parabolic equations. For parabolic equations, one should set ¿ (À ) ≡ 0 in (1) and (7). In addition, we assume that  (À ) > 0 and Á (À ) < min ² ´ . First we search for the solutions of equation (7) corresponding to the eigenvalues ² = ² ´ and satisfying the normalizing conditions  ´ (0) = 1 to obtain (À ) = exp é¨

 ´

0

(Å ) − ²  (Å ) Á

Ä

´ «

Å ¬

.

(17)

Then the solution of the original nonstationary boundary value problem (1)–(3) for the parabolic equation is sought in the form Æ (­ , À ) =

È

(­ Æ ) Â

´ ¸ ´

Ç ´ =1 É

(À ), ´

(18)

where the ´ are arbitrary constants and the functions ´ (­ , À ) = ¸ ´ (­ ) Â ´ (À ) are particular soluÉ tions (5) satisfying the boundary conditions (2). By the principle of linear superposition, series (18) is also a solution of the original partial differential equation which satisfies the boundary conditions. To determine the coefficients ´ , we substitute series (18) into the initial condition (3), thus É obtaining È

(­ ) =

´ ¸ ´ Ç

Ê 0 (­

).

´ =1 É

Multiplying this equation by ¹ (­ ) ¸ ´ (­ ), integrating the resulting relation with respect to ­ over the interval ­ 1 ≤ ­ ≤ ­ 2 , and taking into account the properties (15), we find ´ É

= Ë

1 ¸ ´

± Ë

2

¨ ±

2

¹

(­ ) ¸ ´

(­ ) Ê 0 (­ )

1

«

­

Ë

,

¸ ´

Ë 2

±

= ¨

±

2

¹

(­ ) ¸

2 ´ (­

1

) «

­

.

(19)

The weight function ¹ (­ ) is defined in (14). Relations (18), (12), (17), and (19) give a formal solution of the nonstationary boundary value problem (1)–(3) if ¿ (À ) ≡ 0. Example 1. Let Ì (Í ) = 1 and Î ( Í ) = 0. Substituting these values into (17) yields Ï Ð

( Í ) = exp(− Ñ

Ð

Í ).

(20)

If the function Ò 0 ( Ó ) is twice continuously differentiable and the compatibility conditions (see Paragraph 0.4.2-3) are satisfied, then series (18) is convergent and admits termwise differentiation, once with respect to Í and twice with respect to Ó . In this case, relations (18), (12), (19), and (20) give the classical smooth solution of problem (1)–(3). [If Ò 0 ( Ó ) is not as smooth as indicated or if the compatibility conditions are not met, then series (18) may converge to a discontinuous function, thus giving only a generalized solution.]

0.4.2-2. Solution of boundary value problems for hyperbolic equations. Æ For hyperbolic equations, the solution of the boundary value problem (1)–(4) is sought in the form

(­ , À ) =

È ¸ ´ Ç ´ =1

(­ )

´ Â ´ 1 (À Ô

)+Õ

´ Â ´ 2 ( À )Ö

.

(21)

É

Here, ´ and Õ ´ are arbitrary constants. The functions  ´ 1 (À ) and  ´ 2 (À ) are particular solutions of the É linear equation (7) for  (with ² = ² ´ ) which satisfy the conditions  ´ 1 (0)

= 1,

 ´ ° 1 (0)

= 0;

 ´ 2 (0)

= 0,

 ´ ° 2 (0)

= 1.

(22)

© 2002 by Chapman & Hall/CRC Page 15

Substituting solution (21) into the initial conditions (3)–(4) yields È

(­ ) =

´ ¸ ´ Ç

Ê 0 (­

È

),

Ç

´ =1 É

Õ

´ ¸ ´

(­ ) =

Ê 1 (­

).

´ =1

Multiplying these equations by ¹ (­ ) ¸ ´ (­ ), integrating the resulting relations with respect to ­ on the interval ­ 1 ≤ ­ ≤ ­ 2 , and taking into account the properties (15), we obtain the coefficients of series (21) in the form = ´ É Ë

Ë

1 ¸ ´

± Ë

2

¨ ±

2

¹

(­ ) ¸

(­ ) Ê 0 (­ ) ´

1

, ­

«

Õ

´

=

1 Ë

¸ ´

± Ë

¨

2

2

¹ ±

(­ ) ¸

1

´

(­ ) Ê 1 (­ ) «

­

.

(23)

Ë

The quantity ¸ ´ is defined in (19). Relations (21), (12), and (23) give a formal solution of the nonstationary boundary value problem (1)–(4) for ¿ (À ) > 0. Ð

Example 2. Let × (Í ) = 1, Ì ( Í ) = Î (Í ) = 0, and Ñ Ï Ð

1 (Í

Ð

) = cos ØeÙ Ñ

> 0. The solutions of (7) satisfying conditions (22) are expressed as Ï Ð

Í]Ú ,

2 (Í

)= Û

1 Ñ

Ð

Ð

sin Ø Ù Ñ

Í]Ú .

(24)

If Ò 0 ( Ó ) and Ò 1 ( Ó ) have three and two continuous derivatives, respectively, and the compatibility conditions are met (see Paragraph 0.4.2-3), then series (21) is convergent and admits double termwise differentiation. In this case, formulas (21), (12), (23), and (24) give the classical smooth solution of problem (1)–(4). Æ

0.4.2-3. Conditions of compatibility of initial and boundary conditions. Parabolic equations, ¿ (À ) ≡ 0. Suppose the function has a continuous derivative with respect to À and two continuous derivatives with respect to ­ and is a solution of problem (1)–(3). Then the boundary conditions (2) and the initial condition (3) must be consistent; namely, the following compatibility conditions must hold: [½ If ½

1

= 0 or ½

2

1 Ê 0°

+¾

1 Ê 0 ]± =±

= 0,

1

[½

2 Ê 0°

+¾

2 Ê 0 ]± =±

2

= 0.

(25)

= 0, then the additional compatibility conditions [ ‚ (­ ) Ê [ ‚ (­ ) Ê

0°° 0°° Æ

+ (­ ) Ê 0° ] ± ª + (­ ) Ê 0° ] ± ª

=±

1

=±

2

= 0 if = 0 if

½ 1 ½ 2

= 0, =0

(26)

must also hold; the primes denote the derivatives with respect to ­ . Hyperbolic equations. Suppose is a twice continuously differentiable solution of problem (1)–(4). Then conditions (25) and (26) must hold. In addition, the following conditions of compatibility of the boundary conditions (2) and initial condition (4) must be satisfied: [½

1 Ê 1°

+¾

1 Ê 1 ]± =±

= 0,

1

[½

2 Ê 1°

+¾

2 Ê 1 ]± =±

2

= 0.

0.4.2-4. Linear nonhomogeneous equations with nonhomogeneous boundary conditions. Parabolic equations, ¿ (À ) ≡ 0. The solution of the boundary value problem for the parabolic linear homogeneous equation (1) subject to the homogeneous linear boundary conditions (2) and nonhomogeneous initial condition (3) is given by relations (18), (12), (17), and (19). This solution Æ can be rewritten in the form ±

(­ , À ) = ¨

±

1

2

(­ , Ý , À , 0) Ê 0(Ý ) Ü

«

Ý

.

© 2002 by Chapman & Hall/CRC Page 16

Here,

(­ , Ý , À , Þ ) is the Green’s function, which is expressed as Ü

(­ , Ý , À , Þ ) = ¹ (Ý )

È

¸ ´ Ç

Ü

(­ ) ¸ Ë

´ =1

¸ ´

Ë

(Ý ) ´

2

(À , Þ ),

 ´

where  ´ =  ´ (À , Þ ) is the solution of equation (7) with ¿ (À ) ≡ 0 and initial condition  ´ = 1 at À = Þ .

(27) = ²

which satisfies the

² ´

The function  ´ (À , Þ ) can be calculated by formula (17) with the lower limit of integration equal to Þ (rather than zero). The simplest way to obtain the solutions of more general boundary value problems for the corresponding nonhomogeneous linear equations with nonhomogeneous boundary and initial conditions is to take advantage of formula (6) from Subsection 0.7.1 and use the Green’s function (27). Hyperbolic equations. The solution of the boundary value problem for the hyperbolic linear homogeneous equation (1) subject to the homogeneous linear boundary conditions (2) and semihomogeneous initial conditions (3)–(4) with Ê 0 (­ ) ≡ 0 is given by relations (21), (12), and (23) with Æ ´ = 0 ( · = 1, 2, ³'³'³ ). This solution can be rewritten in the form ±

É

(­ , À ) = ¨

±

2

(­ , Ý , À , 0) Ê 1(Ý ) Ü

1

«

. Ý

Here, (­ , Ý , À , Þ ) is the Green’s function defined by relation (27), where Â Ü of equation (7) for  with ² = ² ´ which satisfies the initial conditions  ´

= 0 at À

=Þ ,

 ´°

= 1 at À

´

=Â

(À , Þ ) is the solution ´

=Þ .

−1 ß 2 ² ´

1ß 2

In the special case ¿ (À ) = 1,  (À ) = Á (À ) = 0, we have  ´ (À , Þ ) = sin Ôಠ´ (À − Þ )Ö . The simplest way to obtain the solutions of more general boundary value problems for the corresponding nonhomogeneous linear equations with nonhomogeneous boundary and initial conditions is to take advantage of formula (14) from Subsection 0.7.2 and use the Green’s function (27).

á5â

References for Section 0.4: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), E. Butkov (1968), E. C. Zachmanoglou and D. W. Thoe (1986), T. U.-Myint and L. Debnath (1987), A. N. Tikhonov and A. A. Samarskii (1990), R. B. Guenther and J. W. Lee (1996), D. Zwillinger (1998), I. Stakgold (2000), A. D. Polyanin (2001a).

0.5. Integral Transforms Method 0.5.1. Main Integral Transforms Various integral transforms are widely used to solve linear problems of mathematical physics. An integral transform is defined as ã ã

Ê

(² ) = ¨

å

ä ¸

(­ , ² ) Ê (­ )

. ­

«

ã

The function Ê ( ² ) is called the transform of the function Ê (­ ) and ¸ (­ , ² ) is called the kernel of the integral transform. The function Ê (­ ) is called the inverse transform of Ê ( ² ). The limits of integration ‚ and are real numbers (usually, ‚ = 0, = ¶ or ‚ = − ¶ , = ¶ ). ª ª ª ã Corresponding inversion formulas, which have the form ãÊ

(­ ) =

¨ æ

Â

(­ , ² ) Ê ( ² ) «

²

make it possible to recover Ê (­ ) if Ê ( ² ) is given. The integration path ç can lie either on the real axis or in the complex plane. The most commonly used integral transforms are listed in Table 3 (for the constraints imposed on the functions and parameters occurring in the integrand, see the references given at the end of Section 0.5).

© 2002 by Chapman & Hall/CRC

Page 17

TABLE 3 Main integral transforms Integral Transform

Definition

Laplace transform

Ò (é ) = ê ë è 0

Fourier transform

Ò (õ ) = Û è 2ñ

Fourier sine transform

Ò s( õ ) = ø è

Fourier cosine transform

Ò c( õ ) = ø è

−íïî

ê

2 ñ

2 ñ

Mellin transform

Ò (ú ) = ê ë ù 0

Hankel transform

Ò ü (õ ) = ê ë ù 0

Meijer transform

− ô5ö'÷

ë ì

−

ë ê

ë

0

ê

0

ë

sin( Ó õ ) Ò (Ó ) ðïÓ

Ò (Ó ) = ø

cos( Ó õ ) Ò (Ó ) ðïÓ

Ò (Ó ) = ø

ñ

ê

0

ë

úMÓ þ

Ò (Ó ) =

1 ê 2ñ ò

Ò (Ó ) = ê

ë

0

ë

−

ë ì

ë

ë

sin( Ó õ ) Ò s( õ ) ðïõ ë

cos( Ó õ ) Ò c( õ ) ðïõ

è

0

ê

0

è

+ô

ó

íïî Ò (é ) ðMé è ô5ö'÷ Ò ( õ ) ðïõ è

ì

ë ê

ñ

−ô

Ó − û Ò ( ú ) ðïú ù ë

ó ë õ ý ü ( Ó õ ) Ò ü ( õ ) ðïõ ù

1

Ò (Ó ) = Û ò 2ñ

ü ( úMÓ ) Ò ( Ó ) ðïÓ

ê

2 2

−ô

ó

2ñ

ñ

+ô

ó

1

Ò (Ó ) = Û

Ò ( Ó ) ðïÓ

Ó ý ü ( Ó õ ) Ò ( Ó ) ðïÓ Û

1 ê 2ñ ò

Ò (Í ) =

Ó û −1 Ò ( Ó ) ðïÓ

2

Ò (ú ) = ø ù

Ò ( Í ) ðïÍ

ì

1

Inversion Formula

ê ó

ó



−ô

+ô

ë

Û

úMÓ ÿpü ( úMÓ ) Ò ( ú ) ðïú ù

ë

Notation: ò 2 = −1, ý
( Ó ) and ( Ó ) are the Bessel functions of the first and the second kind, respectively; ÿ ( Ó ) and þ are the modified Bessel functions of the first and the second kind.

(Ó )

The Laplace transform and the Fourier transform are in most common use. These integral transforms are briefly described below.

0.5.2. Laplace Transform and Its Application in Mathematical Physics 0.5.2-1. The Laplace transform. The inverse Laplace transform. The Laplace transform of an arbitrary ã (complex-valued) function Ê (À ) of a real variable À (À ≥ 0) is defined by  Ê

( ) =

0

Ç

−

Ê Ä



(À )  À ,

(1)

where  = ½ +  is a complex variable, 2 = −1. ã The Laplace transform exists for any continuous or piecewise-continuous function satisfying 0 the condition | Ê (À )| < with some > 0 and 0 ≥ 0. In what follows, 0 often means the Ä  ã possible greatest lower bound of the values of 0 in this condition. For any Ê (À ), the transform Ê ( ) is defined in the half-plane Re  > 0 and is analytic there. Given the transform Ê ( ), the function Ê (À ) can ã be found by means of the inverse Laplace transform   + 1  
 , Ê (À ) = (2) Ç Ê ( )  2  Ä −   ã where the integration path is parallel to the imaginary axis and lies to the right of all singularities Ç of Ê ( ), which corresponds to  > 0 . The integral in (2) is understoodã in the sense of the Cauchy ã principal value: 

 +  −

Ç

Ê

( )  
 =  lim  Ä 

In the domain À < 0, formula (2)Ç gives Ê (À ) ≡ 0.

Ç



 + 

 − 

Ê

( )  
 . 

Ä

© 2002 by Chapman & Hall/CRC Page 18

TABLE 4 Main properties of the Laplace transform No

ã ã Laplace transform

Function

1

‚ Ê 1 (À

2 Ê

3

 Ê

)+ 

( ‚ ),

å



5 6 Ê

8

Ê

 

9

0

Ê

>0

0 Ê 1 (Þ

(Þ ) 

Scaling

( − ‚ )

Differentiation of the transform Shift in the complex plane

( ) − Ê (+0)

Differentiation

ã Ê ã

  Ê

≥

 Ê



( ) −

ã

−

 ã

Ê 1 (

Ê

−1)

( (

( )Ö  !

Ô  ã Ê

1

Þ

 

=1

(−1) !

Þ

) Ê 2 ( − Þ ) 

Linearity

)

( ) (−1) ã  Ê   ( )

( )

 (  ) ( ), "  

Ê 2 (

(㠂  )

‚ Ê

 (  ) ( )

Ê

ã

)+

‚ Ê 1 (

  ( ) Ê

!

‚

)

( );  = 1, 2, 



4

7

Ê 2 (

Operation

(+0)

)

Differentiation Differentiation

( )

Integration

) Ê 2 ( )

Convolution

Ê

ã

Formula (2) holds for continuous functions. If Ê ( ) has a (finite) jump discontinuity at a point 1 0 > 0, then the left-hand side of (2) is equal to 2 [ Ê (  0 − 0) + Ê (  0 + 0)] at this point (for  0 = 0, the first term in the square brackets must be omitted). ã ã We will briefly denote the Laplace transform (1) and the inverse Laplace transform (2) as  =

Ê

( ) = ç { Ê ( )}, Ê

( ) = ç

−1

{ Ê ( )}.

0.5.2-2. Main properties of the Laplace transform. The main properties of the correspondence between functions and their Laplace transforms are gathered in Table 4. The Laplace transforms of some functions are listed in Table 5. There are a number of books that contain detailed tables of direct and inverse Laplace transforms elsewhere; see the references at the end of Section 0.5. Such tables are convenient to use in solving linear differential equations. Note the important case in which the transform is a rational function of the form ã Ê

( ) = #$

( ) , ( )

$

$

where ( ) and ( ) are polynomials in the variable  and$ the degree of ( ) exceeds that of ( ). # # Assume that the zeros of the denominator are simple, i.e., ( ) ≡ const ( − % 1 )( − % 2) 

 ( − % ).  Then the inverse transform can be determined by the formula &

( ) =

' 

$# =1

(% ) exp( %  (% )

 ),

where the primes denote the derivatives.

© 2002 by Chapman & Hall/CRC Page 19

TABLE 5 The Laplace transforms of some functions &

No

Function, ( )

1

1

2



3

+* - −*

4

&

Laplace transform, ( () ) 1 )  !

6



, ( ‚ + 1)() +  )− *

cosh( ‚  )

8

ln 

9

sin( ‚  )

10

cos( ‚  )

> −1

−1 ‚

> −1

‚

−‚

2

)

2

)

−‚

2

)

2

1 − (ln ) + / )

/ = 0.5772 

 is the Euler constant

)

‚

)

2

)

2

+‚

2

)

1 ‚

21 2 3 )

+‚

2

exp 4 − ‚ 1 ) 5 7

6 0(‚ 2 )

12

‚

*

() + ‚ )−1

7

erfc 0

− −1

, ( ‚ + 1))

sinh( ‚  )

11

 = 1, 2, 



+1

) 

 +* - − .

5

Remarks

‚

)

1 +‚

2

≥0

6 0 (8 ) is the Bessel function

2

0.5.2-3. Solving linear problems of mathematical physics by the Laplace transform. Figure 2 shows schematically how one can utilize the Laplace transforms to solve boundary value problems for linear parabolic or hyperbolic equations with two independent variables in the case where the equation coefficients are independent of 2 . It is significant that with the Laplace transform, the original problems for a partial differential equation is reduced to a simpler problem for an ordinary differential equation with parameter ) ; the derivatives with respect to 2 are replaced by appropriate algebraic expressions taking into account the initial conditions (see property 5 or 6 in Table 4). Example 1. Consider the following problem for the heat equation:

9
:= ;

@ =0

(initial condition),

0

at

(boundary condition),

0

at

? =0 A B ?

We apply the Laplace transform with respect to @ . Setting

C {

9
: 0, @ > 0),

,

at

=0

;

=

(boundary condition).

;

;

è

= C { } and taking into account the relations

(used are property 5 of Table 4 and the initial condition), (used are property 1 of Table 4 and the relation C {1} = 1 D é ),

© 2002 by Chapman & Hall/CRC Page 20

Figure 2. Scheme for solving linear boundary value problems by the Laplace transform. we arrive at the following problem for a second-order linear ordinary differential equation with parameter : 




= 

0





at

= 0,

=0

at

0




−

 



(boundary condition),

(boundary condition).








Integrating the equation yields the general solution =  1 ( ) − +  determine the constants,  1 ( ) =  0 and  2 ( ) = 0. Thus, we have 


=



0

− 



2( 



)



. Using the boundary conditions, we

.

Let us apply the inverse Laplace transform to both sides of this relation. We refer to Table 5, row 11 with  = inverse transform of the right-hand side. Finally, we obtain the solution of the original problem in the form 

to find the



= 





0 erfc

2 

. 

0.5.3. Fourier Transform and Its Application in Mathematical Physics 0.5.3-1. The Fourier transform and its properties. The Fourier transform is defined as follows: 



( ) = 

1 2

 

−

( ) 

− 

!

" 

,

2 #

= −1.

(3)

 

This relation is meaningful for any function ( ) absolutely integrable on the interval ( $ , + $ ). We    will briefly write ( ) = % { ( )} to denote the Fourier transform (3).   Given ( ), the function ( ) can be found by means of the inverse Fourier transform 

( ) = 

1 2 



−



( ) 

 

!

" 

,

(4)



© 2002 by Chapman & Hall/CRC Page 21

TABLE 6 Main properties of the Fourier transform No

Function 

1



2 

3 

)

( (

), &

( );

./. !



5 

6 

−

1 (1



)

>0 &









#-)

−





(#0 ) )



) 2 ( − 1 ) "

 



1 (

1

Linearity

)

( )

Differentiation of the transform

( )

Differentiation

)

2

2 (

Scaling



( ) 



)

(&

&

( )

)



)+'

1 (

&

( ) !

( ) !

2 (



= 1, 2, +,+,+ *



4



)+'

1 (

&

Operation

Fourier transform



Differentiation

( )



Convolution

) 2 ( )



where the integral is understood in the sense of the Cauchy principal value. We will briefly write    ( ) = % −1 { ( )} to denote the inverse Fourier transform (4).  ( ) has a (finite) jump discontinuity The inversion formula (4) holds for continuous functions. If  at a point  =  0 , then the left-hand side of (4) is equal to 12 2 ( 0 − 0) + ( 0 + 0)3 at this point. The main properties of the correspondence between functions and their Fourier transforms are gathered in Table 6. 0.5.3-2. Solving linear problems of mathematical physics by the Fourier transform. The Fourier transform is usually employed to solve boundary value problems for linear partial differential equations whose coefficients are independent of the space variable  , − $ <  < $ . The scheme for solving linear boundary value problems with the help of the Fourier transform is similar to that used in solving problems with help of the Laplace transform. With the Fourier transform, the derivatives with respect to  in the equation are replaced by appropriate algebraic expressions; see property 4 or 5 in Table 6. In the case of two independent variables, the problem for a partial differential equation is reduced to a simpler problem for an ordinary differential equation with parameter  . On solving the latter problem, one determines the transform. After that, by applying the inverse Fourier transform, one obtains the solution of the original boundary value problem. Example 2. Consider the following Cauchy problem for the heat equation: 4,5 

4

=





= 6 (  ) at 



(− <  < ), = 0 (initial condition).


We 4 apply the Fourier transform with respect to the space variable  . Setting  = 7 {  } and taking into account the relation
7 {   } = − 8 2  (see property 4 of Table 6), we arrive at the following problem for a first-order linear ordinary differential equation with parameter 8 : 


5


+8

2


= 0,

= 6 ( 8 ) at 

= 0, 







where 6 ( 8 ) is defined by (3). On solving this problem for the transform  , we find 







−9 2

= 6 (8 )

5

.

Let us apply the inversion formula to both sides of this equation. After some calculations, we obtain the solution of the original problem in the form 

= =

1 

1 2:




2:

−
W ≥ 0, the homogeneous linear equation Q Q

X S

−T

x, U

[X ] = 0

(2)

with the nonhomogeneous initial condition of special form X

= Y (x − y) at

S

=W .

(3)

The quantities W and y appear in problem (2)–(3) as free parameters, and Y (x) = Y ( 1 ) +,+,+ZY ( ) is ) the * -dimensional Dirac delta function. S [ \^] _ `>a b c If the coefficients of the differential operator T x,U in (2) are independent of time , S then theS fundamental solution of the Cauchy problem depends on only three arguments, X (x, y, , W ) = X (x, y, − W ). [ \^] _ `>a d c If the differential operator T x,U has constant coefficients, then the fundamental S S solution of the Cauchy problem depends on only two arguments, X (x, y, , W ) = X (x − y, − W ).

© 2002 by Chapman & Hall/CRC Page 23

0.6.1-2. The fundamental solution allowing incomplete separation of variables. Consider the special case where the differential operator T the sum R R T x, U [ ] = T 1,U [ ] + J,J,J + T

in equation (1) can be represented as

x, U

R

[ ],

,U )

(4)

where each term depends on a single space coordinate and time, Q R

T

,U e

[ ]≡

( e

&

2

S

e

, )

R

Q

Q

+ 'e (

2 

R

S

, ) e



e

S

+ f,e (

Q e

R

, ) , e

= 1, g

+,+,+

,* .

Equations of this form are often encountered in applications. The fundamental solution of the Cauchy problem for the * -dimensional equation (1) with operator (4) can be represented in the product form h )

S

(x, y, , W ) =

X

X

X e

=

X

S

( e

e

, P e , , W ), e

(5)

=1 e

where

S

( e

, P e , , W ) are the fundamental solutions satisfying the one-dimensional equations Q X Q

e S

−T e

X

= Y (

with the initial conditions e

,U

[X

]=0 e

( g = 1, S

− P e ) at e

,* )

+,+,+

=W .

In this case, the fundamental solution of the Cauchy problem (5) admits incomplete separation of S variables; the fundamental solution is separated in the space variables  1 , +,+,+ ,  but not in time . )

0.6.2. Cauchy Problem for Hyperbolic Equations Consider a nonhomogeneous linear equation of the hyperbolic type with an arbitrary right-hand side, Q

2 Q

S

R

Q

R

S

+ i (x, )

2

Q

R

−T S

x, U

S

[ ] = V (x, ),

(6)

where the second-order linear differential operator T x,U is defined by relation (2) from Subsection 0.2.1, with x K L ) . The solution of the Cauchy problem for equation (6) with general initial conditions, R Q

R U

=



0 (x)

=



1 (x) at

S

at

= 0, S

= 0,

can be represented as the sum R

Q U

S

(x, ) =

S

V

0 

H 



H I

2

"

1 (y) +



− W

I 

+

(y, W ) X (x, y, , W ) y "





(y, 0)3

0 (y) i

H

0 (y)

S

S

S

(x, y, , W )k

X W

I

(x, y, , 0) " y,

X

Q j

"

y= "

P

1

+,+,+

" P

)

" l

=0

y

. S

Here, X = X (x, y, , W ) is the fundamental solution of the Cauchy problem that satisfies, for > W ≥ 0, the homogeneous linear equation Q

2 Q

S

Q S

X

+ i (x, )

2

Q

X S

−T

[X ] = 0

x, U

(7)

with the semihomogeneous initial conditions of special form X Q U

X

=0 at = Y (x − y) at S

S

=W , =W .

(8)

© 2002 by Chapman & Hall/CRC Page 24

The quantities W and y appear in problem (7)–(8) as free parameters (y K L ) ). S [ \^] _ `>a b c If the coefficients of the differential operator T x,U in (7) are independent of time , S then theS fundamental solution ofS the Cauchy problem depends on only three arguments, X (x, y, , W ) = S l l X (x, y, − W ). Here, m X (x, y, , W ) n =0 = − m U X (x, y, ). [ \^] _ `>a d c n If them differential operatorm T x,U has constant coefficients, then the fundamental S S solution of the Cauchy problem depends on only two arguments, X (x, y, , W ) = X (x − y, − W ).

MON

References for Section 0.6: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), G. E. Shilov (1965), A. D. Polyanin (2000a, 2000b, 2000c, 2001a).

0.7. Nonhomogeneous Boundary Value Problems with One Space Variable. Representation of Solutions via the Green’s Function 0.7.1. Problems for Parabolic Equations S

0.7.1-1. Statement of the problem ( ≥ 0,

1 

≤

≤  2 ). 

In general, a nonhomogeneous linear differential equation of the parabolic type with variable coefficients in one dimension can be written as Q

R Q

where Q R

T !

,U

2

S

[ ] ≡ & ( , ) Q



R

−T S

S

[ ] = V ( , ),

,U !

R

Q

R

S

+ ' ( , )

2

(1)

S

R

S

+ f ( , ) ,

Q 

&

( , ) > 0.

(2)

Consider the nonstationary boundary value problem for equation (1) with an initial condition of general form, R S  = ( ) at = 0, (3) and arbitrary nonhomogeneous linear boundary conditions, Q

R S

o

R Q

o

S

2

R

S

+ g 1( )

Q

1 Q



+ g 2( )

= p 1 ( ) at R



= 

1,



= 

2.

S

= p 2 ( ) at

(4) (5) S

S

By appropriately choosing the coefficients o 1 , o 2 and the functions g 1 = g 1 ( ), g 2 = g 2 ( ) in (4) and (5), we obtain the first, second, third, and mixed boundary value problems for equation (1). 0.7.1-2. Representation of the problem solution in terms of the Green’s function. The solution of the nonhomogeneous linear boundary value problem (1)–(5) can be represented as R

U

S

( , ) =

! 2

S

V

0 



(P , W ) q ( , P , , W ) "

p 

0

"

1 (W

! 2

+ W



! 1

U

+

P

) & ( 1 , W ) r

1 (

S

, ,W )" W



S

(P ) q ( , P , , 0) "

U

+

p 

0

S

P

! 1

2 (W

) & ( 2 , W ) r

2 (

S

, ,W )" W

.

(6)

S

Here, q ( , P , , W ) is the Green’s function that satisfies, for > W ≥ 0, the homogeneous equation Q Q

q S

−T !

,U

[q ] = 0

(7)

© 2002 by Chapman & Hall/CRC Page 25

TABLE 7 S S Expressions of the functions r 1 ( , , W ) and r 2 ( , , W ) involved in the integrands of the last two terms in solution (6) Type of problem

Form of boundary conditions R

First boundary value problem ( o 1 = o 2 = 0, g 1 = g 2 = 1)

S

R



=

1

r

Q

= p 2( ) at 

=

2 r

Second boundary value problem ( o 1 = o 2 = 1, g 1 = g 2 = 0)

R !

S

Q

R

Q

R

Third boundary value problem ( o 1 = o 2 = 1, g 1 < 0, g 2 > 0)

! Q

+g +g R

!

Q

Mixed boundary value problem ( o 1 = g 2 = 1, o 2 = g 1 = 0)

R

t

S

=!

=! n

S

= p 2( ) at 

=

2 r

2( 

, , W ) = q (  ,  2, , W )

1 r

1( 

2 r

S

R



= = 

S

S

S

S

, , W ) = − q (  ,  1, , W ) S S (  2 , , W ) = q (  ,  2, , W ) S

, ,W )=

1( 

Q

t

S



=

1 r

= p 2( ) at 

=

2 r

= p 1( ) at 

=

1 r

1( 

, , W ) = − q (  ,  1, , W )

= p 2( ) at 

=

2 r

2( 

, ,W )=−

S

t

( , P , , W ) n

= p 1( ) at S

2

S

, , W ) = − q (  ,  1, , W )

= p 1( ) at S = p 2( ) at

1

t

( , P , , W ) n q

1( 

S

R !

Q

r

2

R

S

2(  , , W ) = −

t n

1

R

!

S

( , P , , W ) n q

=

S

Q

t

( , , W )



1

R

Mixed boundary value problem ( o 1 = g 2 = 0, o 2 = g 1 = 1)

Q

s

= p 1( ) at S

!

S

, ,W )=

1( 

= p 1( ) at S

S

Functions r

q

S

=! n

S

1

2(  , , W ) = q (  ,  2, , W ) S

S

S

Q

t

S

q

t

( , P , , W ) n n

=!

2

with the nonhomogeneous initial condition of special form S

= Y ( − P ) at q

=

(8) W

and the homogeneous boundary conditions Q o

S

q Q

1



+ g 1 ( )q

Q o

Q

2

S

q 

+ g 2 ( )q

= 0 at 

=  1,

(9)

= 0 at 

=  2.

(10)

The quantities P and W appear in problem (7)–(10) as free parameters ( Dirac delta function. The initial condition (8) implies the limit relation 

! 2

( ) = liml Uvu







1

≤ P

≤  2 ), and Y ( ) is the

S

(P ) q ( , P , , W ) " P

! 1



for any continuous function = ( ). S S The functions r 1 ( , , W ) and r 2 ( , , W ) involved in the integrands S of the last two terms in solution (6) can be S expressed in terms of the Green’s function q ( , P , , W ). The corresponding formulas for r s ( , , W ) are given in Table 7 for the basic types of boundary value problems. It is significant that the Green’s function q and the functions r 1 , r 2 are independent of the  functions V , , p 1 , and p 2 that characterize various nonhomogeneities of the boundary value problem. If the coefficients of equation (1)–(2) and the coefficients g 1 , g 2 in the boundary conditions (4) S and (5) are independent of time , i.e., the conditions &

= & ( ), '

= ' ( ), f

= f ( ), g

= const,

1

g

2

= const

(11)

hold, then the Green’s function depends on only three arguments, S

q

( , P , , W ) =

S

q

( , P , − W ).

© 2002 by Chapman & Hall/CRC Page 26

S

In this case, the functions r s depend on only two arguments, r s = r s ( , − W ), w = 1, 2. Formula (6) also remains valid for the problem with boundary conditions of the third kind if S S g 1 = g 1 ( ) and g 2 = g 2 ( ). Here, the relation between r s (w = 1, 2) and the Green’s function q is function itself is now different. the same as that in the case of constants g 1 and g 2 ; the Green’s x R x The condition that the solution must vanish at infinity, 0 as  $ , is often set for the first, second, and third boundary value problems that are considered on the interval  1 ≤  < $ . In this case, the solution is calculated by formula (6) with r 2 = 0 and r 1 specified in Table 7.

0.7.2. Problems for Hyperbolic Equations S

0.7.1-2. Statement of the problem ( ≥ 0,

≤

1 

≤  2 ). 

In general, a one-dimensional nonhomogeneous linear differential equation of hyperbolic type with variable coefficients is written Qas R Q R 2

Q R

S

S

+ i ( , )

2

Q

R

−T S

!

S

[ ] = V ( , ),

,U

(12)

where the operator T ! ,U [ ] is defined by (2). Consider the nonstationary boundary value problem for equation (12) with the initial conditions R S = y 0 ( ) at = 0, Q R S (13) = y (  ) at =0 1 U and arbitrary nonhomogeneous linear boundary conditions (4)–(5). 0.7.2-2. Representation of the problem solution in terms of the Green’s function. The solution of problem (12), (13), (4), (5) can be represented as the sum R

U

S

( , ) =

! 2

S

V

0 



! 1

! 2

−

y 

(P , W ) q ( , P , , W ) "

0 (P

)j

S

Q q W

(  , P , , W )k



0

1 (W

" l

) & ( 1 , W ) r

S

1 (

, ,W )"



2

y

1 (P

S

) + y 0 (P ) i (P , 0)3^q ( , P , , 0) "

p

0

 S

P

! 1

U

+ W

! 2

+ P

=0

U p

W

Q

! 1

+

" P

2 (W

) & ( 2 , W ) r

2 (

S

, ,W )" W

.

(14)

Here, the Green’s function q ( , P , , W ) is determined by solving the homogeneous equation Q

2 Q

S

Q

S

q

2

+ i ( , ) Q

q S

−T !

,U

[q ] = 0

(15)

with the semihomogeneous initial conditions S q =0 at =W , (16) Q S = Y (  − P ) at = W , (17) q U and the homogeneous boundary conditions (9) and (10). The quantities P and W appear in problem (15)–(17), (9), (10) as freeS parameters ( 1S ≤ P ≤  2 ), and Y ( ) is the Dirac delta function. The functions r 1 ( , , W ) and r 2 ( , , W ) involved in the integrands of the last two terms in S solution (14) can be expressed via the Green’s function q ( , P , , W ). The corresponding formulas S for r s ( , , W ) are given in Table 7 for the basic types of boundary value problems. It is significant that the Green’s function q and r 1 , r 2 are independent of the functions V , y 0 , y 1 , p 1 , and p 2 that characterize various nonhomogeneities of the boundary value problem. If the coefficients of equationS (12) and the coefficients g 1 , g 2 in the boundary conditions (4) and (5) S are independent of time , then the Green’s function depends on only three arguments, S S S l l q ( , P , , W ) = q ( , P , − W ). In this case, one can set m q ( , P , , W ) n =0 = − m U q ( , P , ) in solun m m tion (14). MON References for Section 0.7: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), E. Butkov (1968), A. G. Butkovskiy (1982), E. Zauderer (1989), A. D. Polyanin (2000a, 2000b, 2000c, 2001a).

© 2002 by Chapman & Hall/CRC Page 27

0.8. Nonhomogeneous Boundary Value Problems with Many Space Variables. Representation of Solutions via the Green’s Function 0.8.1. Problems for Parabolic Equations 0.8.1-1. Statement of the problem. In general, a nonhomogeneous linear differential equation of the parabolic type in * space variables Q R has the form R S Q S − T x,U [ ] = V (x, ), (1) where Q z

R

x, U T

[ ]≡

2

S

) &

(x, )

| {

Q

}

x = { 1 , +,+,+ ,

Q

+ }

|

, =1

 { }

R

Q

} )

z

},

z

|

&

| {

S

Q

(x, )1

R

+ f (x, ) , } |

=1

S

)

(x, )

' |

{

R

S

)

≥

| 1 {

(2) z )

~

1

, =1 |

| {

|

2

, ~

> 0.

=1 Q

Let  be some simply connected domain in L € with a sufficiently smooth boundary  =  . We consider the nonstationary boundary value problem for equation (1) in the domain  with an arbitrary initial condition, R S = y (x) at = 0, (3) and nonhomogeneous linear boundary conditions, R

‚

S

[ ] = p (x, ) for x

x, U

K

. 

(4)

‚

In the general case, x,U is a first-order linear differential operator in the space coordinates with S coefficients dependent on x and . 0.8.1-2. Representation of the problem solution in terms of the Green’s function. The solution of the nonhomogeneous linear boundary value problem (1)–(4) can be represented as the sum R

S

(x, ) = ƒ

+ƒ

„ ƒ

…

(y, ‡ ) ˆ (x, y, ‰ , ‡ ) Š †

0 „ ƒ

0

Œ

p

(y, ‡ )  (x, y, ‰ , ‡ ) Š

where ˆ (x, y, ‰ , ‡ ) is the Green’s function; for ‰ >





‹

‹

Š

Š

+ƒ ‡

…

y

(y) ˆ (x, y, ‰ , 0) Š 

‹

, ‡

(5)

≥ 0, it satisfies the homogeneous equation ‡

Ž ˆ Ž ‰

−

[ˆ ] = 0

x,

(6)

„

with the nonhomogeneous initial condition of special form = Y (x − y) at ˆ

=‡ ‰

(7)

and the homogeneous boundary condition ‚

[ ˆ ] = 0 for x

x,





.

(8)

„

The vector y = {‘ 1} , ’,’,’ , ‘ } appears in problem (6)–(8) as an “ -dimensional free parameter (y   ), } € and Y (x − y) = Y ( 1 − ‘ 1 ) ’,’,’ZY ( − ‘ ) is the “ -dimensional Dirac delta function. The Green’s €

€

© 2002 by Chapman & Hall/CRC Page 28

TABLE 8 The form of the function  (x, y, ‰ , ‡ ) for the basic types of nonstationary boundary value problems Type of problem

Function  (x, y, ‰ , ‡ )

Form of boundary condition (4)

1st boundary value problem

= p (x, ‰ ) for x ”

2nd boundary value problem m m

3rd boundary value problem m

™

+š ˜

m

= p (x, ‰ ) for x

˜ –

–









m m

(x, y, ‰ , ‡ )

• –

—



(x, y, ‰ , ‡ ) = ˆ (x, y, ‰ , ‡ ) 

(x, y, ‰ , ‡ ) = ˆ (x, y, ‰ , ‡ )



= p (x, ‰ ) for x ”

™

(x, y, ‰ , ‡ ) = − 



function ˆ is independent of the functions † , › , and p that characterize various nonhomogeneities of the boundary value problem. In (5), the integration is everywhere performed with respect to y, with Š  ‹ = Š ‘ 1 ’,’,’>Š ‘ . € The function  (x, y, ‰ , ‡ ) involved in the integrand of the last term in solution (5) can be expressed via the Green’s function ˆ (x, y, ‰ , ‡ ). The corresponding formulas for  (x, y, ‰ , ‡ ) are given in Table 8 for the three basic types of boundary value problems; in the third boundary value problem, the coefficient š can depend on x and ‰ . The boundary conditions of the second and third kind, as well as the solution of the first boundary value problem, involve operators of differentiation along the conormal of operator (2); these operators act as follows: Ž ˆ Ž

œ

ž

≡ 

Ž € Ÿ

Ÿ

,  =1

 

Ž ˆ

(x, ‰ ) ¢  

Ž

£

ˆ

, Ÿ

Ž

‹

¡

ž

≡

œ

Ž €

Ÿ

Ÿ  

ˆ

(y, ‡ ) ¢  

,  =1 ¡

Ž ‘

Ÿ

,

(9)

where N = { ¢ 1 , ’,’,’ , ¢ } is the unit outward normal to the surface  . In the special case where € Ÿ Ÿ (x, ‰ ) = 1 and Ÿ   (x, ‰ ) = 0 for ¤ ≠ ¥ , operator (9) coincides with the ordinary operator of differen¡ tiation along the ¡ outward normal to ¦ . If the coefficient of equation (6) and the boundary condition (8) are independent of ‰ , then the Green’s function depends on only three arguments, ˆ (x, y, ‰ , ‡ ) = ˆ (x, y, ‰ − ‡ ). [

\^§

¨

©>ª

«

Let ¦ Ÿ

(¤ = 1, ’,’,’ , ¬ ) be different portions of the surface ¦ such that ¦ =

boundary conditions of various types be set on the ¦ Ÿ , ¯

Ÿ

() x, [ ”

] = p Ÿ (x, ‰ ) for x 

Ÿ ¦

,

Ÿ

® ­

¦

Ÿ

=1

= 1, ’,’,’ , ¬ . ¤

and let (10)

„ Then formula (5) remains valid but the last term in (5) must be replaced by the sum ž ­ Ÿ

ƒ

„ ƒ

Œ

0

=1

°

p

Ÿ

(y, ‡ )  Ÿ

(x, y, ‰ , ‡ ) Š ¦

‹

Š

‡

.

(11)

0.8.2. Problems for Hyperbolic Equations 0.8.2-1. Statement of the problem. The general nonhomogeneous linear differential hyperbolic equation in written as Ž Ž 2”

Ž ‰

2

+ ± (x, ‰ )

” Ž ‰

−

“

space variables can be

[” ] = † (x, ‰ ),

x,

where the operator  x, [” ] is explicitly defined in (2). We consider the nonstationary boundary value problem for equation (12) in the domain ² „ arbitrary initial conditions, ” = › 0 (x) at ‰ = 0,

(12)

„

Ž

” = › 1 (x) at ‰ = 0, „ and the nonhomogeneous linear boundary condition (4).

with (13) (14)

© 2002 by Chapman & Hall/CRC Page 29

0.8.2-2. Representation of the problem solution in terms of the Green’s function. The solution of the nonhomogeneous linear boundary value problem (12)–(14),(4) can be represented as the sum Ž ”

(x, ‰ ) = ƒ

+ƒ

„ ƒ …

(y, ‡ ) ˆ (x, y, ‰ , ‡ ) Š †

0 …

µ ›

1 (y) + ›

²

‹

Š

−ƒ ‡

(y, 0)¶^ˆ (x, y, ‰ , 0) Š

0 (y) ±

Here, ˆ (x, y, ‰ , ‡ ) is the Green’s function; for ‰ > Ž

0 (y) ³ ›

…

²

+ƒ ‹

Ž

(x, y, ‰ , ‡ )´ ˆ

‡

„ ƒ

p Œ

0

l

Š

²

‹

=0

(y, ‡ )  (x, y, ‰ , ‡ ) Š ¦

‹

Š

. (15) ‡

≥ 0 it satisfies the homogeneous equation ‡

Ž

2 ˆ

Ž

2 ‰

ˆ

+ ± (x, ‰ ) Ž

‰

−

[ˆ ] = 0

x,

(16)

„

with the semihomogeneous initial conditions =0 at = Y (x − y) at

ˆ Ž ˆ

=‡ , =‡ , ‰

‰

„

and the homogeneous boundary condition (8). If the coefficients of equation (16) and the boundary condition (8) are independent of time ‰ , then the Green’s function depends on only three arguments, ˆ (x, y, ‰ , ‡ ) = ˆ (x, y, ‰ − ‡ ). In this case, l l one can set m ˆ (x, y, ‰ , ‡ ) n =0 = − m ˆ (x, y, ‰ ) in solution (15). n m m The function  (x, y, ‰ , ‡ ) involved in the integrand of the last term in solution (15) can be „ expressed via the Green’s function ˆ (x, y, ‰ , ‡ ). The corresponding formulas for  are given in Table 8 for the three basic types of boundary value problems; in the third boundary value problem, the coefficient š can depend on x and ‰ . [

\^§

¨

©>ª

«

Let ¦ Ÿ

(¤ = 1, ’,’,’ , ¬ ) be different portions of the surface ¦ such that ¦ = Ÿ

® ­

=1

¦

Ÿ

and let

boundary conditions of various types (10) be set on the ¦ . Then formula (15) remains valid but the last term in (15) must be replaced by the sum (11). Ÿ

0.8.3. Problems for Elliptic Equations 0.8.3-1. Statement of the problem. In general, a nonhomogeneous linear elliptic equation can be written as − where 

x [”

Ž

]≡

ž Ÿ

Ÿ

·

,  =1

 

¡

(x) Ž

] = † (x),

x [”

£ Ÿ

2” Ž

+ £  

(17) Ž

ž · Ÿ

=1

(x) Ž Ÿ

” £ Ÿ

+ ¹ (x)” .

(18)

¸

Two-dimensional problems correspond to “ = 2 and three-dimensional problems, to “ = 3. We consider equation (17)–(18) in a domain ² and assume that the equation is subject to the general linear boundary condition ¯

x [”

] = p (x) for x 

¦

.

(19)

The solution of the stationary problem (17)–(19) can be obtained by passing in (5) to the limit as » . To this end, one should start with equation (1) whose coefficients are independent of ‰ and take the homogeneous initial condition (3), with › (x) = 0, and the stationary boundary condition (4). ‰

º

© 2002 by Chapman & Hall/CRC Page 30

TABLE 9 The form of the function  (x, y) involved in the integrand of the last term in solution (20) for the basic types of stationary boundary value problems Type of problem

Function  (x, y)

Form of boundary condition (19)

1st boundary value problem

= p (x) for x ”

2nd boundary value problem m m

3rd boundary value problem m m

™

+š ˜

–





¦

m m

(x, y)

• –

—



(x, y) = ˆ

(x, y)



(x, y) = ˆ

(x, y)

¦

= p (x) for x ”

™

(x, y) = − 

¦

= p (x) for x

˜ –



0.8.3-2. Representation of the problem solution in terms of the Green’s function. The solution of the linear boundary value problem (17)–(19) can be represented as the sum (x) = ”

ƒ

…

(y) ˆ (x, y) Š †

²

‹

+ƒ Œ

(y)  (x, y) Š p

¦

‹

.

(20)

Here, the Green’s function ˆ (x, y) satisfies the nonhomogeneous equation of special form −

] = ¼ (x − y)

x[ˆ

(21)

with the homogeneous boundary condition ¯

x[ˆ

] = 0 for x 

. ¦

(22)

The vector y = {‘ 1, ’,’,’ , ‘ } appears in problem (21), (22) as an “ -dimensional free parameter (y  ² ). Note that ˆ is independent of the functions † and p characterizing various nonhomogeneities of the · original boundary value problem. The function  (x, y) involved in the integrand of the second term in solution (20) can be expressed via the Green’s function ˆ (x, y). The corresponding formulas for  are given in Table 9 for the three basic types of boundary value problems. The boundary conditions of the second and third kind, as well as the solution of the first boundary value problem, involve operators of differentiation along the conormal of operator (18); these operators are defined by (9); in this case, the coefficients Ÿ   depend on only x. ½ ¾^§ ¨ ©>ª « For ¡ the second boundary value problem with ¹ (x) ≡ 0, the thus defined Green’s function must not necessarily exist; see Remark 2 in Paragraph 8.2.1-2.

0.8.4. Comparison of the Solution Structures for Boundary Value Problems for Equations of Various Types Table 10 lists brief formulations of boundary value problems for second-order equations of elliptic, ¯ parabolic,£ and hyperbolic types. The coefficients of the differential operators  x and x in the space £ are assumed to be independent of time ‰ ; these operators are the same for the variables 1 , ’,’,’ , problems under consideration. · Below are the respective general formulas defining the solutions of these problems with zero initial conditions ( › = › 0 = › 1 = 0): ”

”

”

0 (x)

=

1 (x, ‰

)= ƒ

2 (x, ‰

)= ƒ

0 (x, y) Š

+

ƒ …

†

(y) ˆ

„ ƒ …

†

(y, ‡ ) ˆ

1 (x, y, ‰

−‡ )Š ²

(y, ‡ ) ˆ

2 (x, y, ‰

−‡ )Š ²

0 „ ƒ

…

†

0

²

‹

‹

‹

Š

‡

+ƒ

Š

‡

+ƒ

Œ

p

(y)¿

„ ƒ Œ

p

(y, ‡ )¿

µ ˆ

1 (x, y, ‰

− ‡ )¶ Š

¦

‹

Š

‡

,

„ ƒ

Œ

p

(y, ‡ )¿

µ ˆ

2 (x, y, ‰

− ‡ )¶ Š

¦

‹

Š

‡

,

0

0

0 (x, y)¶

ƒ

µ ˆ

Š

¦

‹

,

© 2002 by Chapman & Hall/CRC Page 31

TABLE 10 Formulations of boundary value problems for equations of various types Type of equation

Form of equation −

Elliptic Ž

Parabolic

”

x [”

Initial conditions

] = † (x)

Ž ”

¯

not set

−

x [”

] = † (x, ‰ )

−

x [”

] = † (x, ‰ )

= › (x) at ”

„

Hyperbolic

Boundary conditions

= =

” Ž ”

„À„

=0 ‰

0 (x) ›

›

at 1 (x) at ‰ ‰

= 0, =0

¯

¯

x [”

] = p (x) for x

x [”

] = p (x, ‰ ) for x 

¦

x [”

] = p (x, ‰ ) for x 

¦



¦

„

where the ˆ are the Green’s functions, the subscripts 0, 1, and 2 refer to the elliptic, parabolic, and hyperbolic problem, respectively. All solutions involve the same operator ¿ [ ˆ ]; it is explicitly · defined in Subsections 0.8.1–0.8.3 (see also Section 0.7) for different boundary conditions. It is apparent that the solutions of the parabolic and hyperbolic problems with zero initial conditions have the same structure. The structure of the solution to the problem for a parabolic equation differs from that for an elliptic equation by the additional integration with respect to ‰ . ÁOÂ

References for Section 0.8: P. M. Morse and H. Feshbach (1953), V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), E. Butkov (1968), A. G. Butkovskiy (1979, 1982), E. Zauderer (1989), A. N. Tikhonov and A. A. Samarskii (1990), A. D. Polyanin (2000a, 2000c, 2001a).

0.9. Construction of the Green’s Functions. General Formulas and Relations 0.9.1. Green’s Functions of Boundary Value Problems for Equations of Various Types in Bounded Domains 0.9.1-1. Expressions of the Green’s function in terms of infinite series. Table 11 lists the Green’s functions of boundary value problems for second-order equations of various types in a bounded domain ² . It is assumed that  x is a second-order linear self-adjoint £ £ ¯ differential operator (e.g., see Zwillinger, 1998) in the space variables 1 , ’,’,’ , , and x is a zerothor first-order linear boundary operator that can define a boundary condition of· the first, second, or ¯ third kind; the coefficients of the operators  x and x can depend on the space variables but are independent of time ‰ . The coefficients Ã Ä and the functions Å Ä (x) are determined by solving the homogeneous eigenvalue problem ] + à Š= 0, [ Å ] = 0 for x x x [Å

 ¯



¦

.

(1) (2)

It is apparent from Table 11 that, given the Green’s function in the problem for a parabolic (or hyperbolic) equation, one can easily construct the Green’s functions of the corresponding problems for elliptic and hyperbolic (or parabolic) equations. In particular, the Green’s function of the problem for an elliptic equation can be expressed via the Green’s function of the problem for a parabolic equation as follows: ˆ

0 (x, y)

= ƒ

Æ

0

ˆ

1 (x, y, ‰

)Š ‰ .

(3)

Here, the fact that all Ã Ä are positive is taken into account; for the second boundary value problem, it is assumed that à = 0 is not an eigenvalue of problem (1)–(2).

© 2002 by Chapman & Hall/CRC Page 32

TABLE 11 The Green’s functions of boundary value problems for equations of various types in £ bounded £ ¯ domains. In all problems, the operators  x and x are the same; x = { 1 , ’,’,’ , } Equation ¯

Green’s function

] = p (x) for x  ¦ (no initial condition required)

Elliptic equation −  x [” ] = † (x) Parabolic equation ” −  x [” ] = † (x, ‰ ) Ž

·

Initial and boundary conditions x [”

= › (x) ”

¯

at

] = p (x, ‰ ) for x

x [”



Å Æ

Ä

Ä

2Ã

Å

Å Æ

¦

(x)Å Ç

=1

ž

(x, y, ‰ ) = ˆ

Ä Ç

Ä

=0 ‰

ž

(x, y) = ˆ

(x)Å Ä

Ç

Å

=1 Ä

(y) Ä

(y) Ä

Ç

2

Ä

, Ã

Ä

≠0

exp È − Ã Ä

‰ZÉ

„

Ž

= = ”

Hyperbolic equation ” −  x [” ] = † (x, ‰ )

” ¯

at 1 (x) at ›

=0 =0 ‰ ‰

] = p (x, ‰ ) for x

x[

„À„

0 (x) ›

”



ˆ

ž

(x, y, ‰ ) =

Å Æ

Ç

Å

=1 Ä

¦

(x)Å Ä

Ç

Ä

Ä

2Ê

(y) Ã

sin È

‰Ë

Ã

Ä É

Ä

0.9.1-2. Some remarks and generalizations. ½

¾^§

¨

©>ª

Ì

«

Formula (3) can also be used if the domain ² is infinite. In this case, one should make sure that the integral on the right-hand side is convergent. ½ ¾^§ ¨ ©>ª Í « Suppose the equations given in the first column of Table 11 contain − Î x [Ï ] − Ð Ï instead of − Î x [Ï ], with Ð being a free parameter. Then the Ã Ä in the expressions of the Green’s function in the third column of Table 11 must be replaced by Ã Ä − Ð ; just as previously, the Ã Ä and Å Ä (x) were determined by solving the eigenvalue problem (1)–(2). ½ ¾^§ ¨ ©>ª Ñ « The formulas for the Green’s functions presented in Table 11 will also hold for boundary value problems described by equations of the fourth or higher order in the space variables; provided that the eigenvalue problem for equation (1) subject to appropriate boundary conditions is self-adjoint.

0.9.2. Green’s Functions Admitting Incomplete Separation of Variables 0.9.2-1. Boundary value problems for rectangular domains. 1 Ò . Consider the parabolic equation Ó Ó Ï Ô

where each term Î Ø

,Õ

=

Ø

,Õ

[Ï ] ≡ Ù

[Ï ] +

1, Õ Î

+Î

Ö,Ö,Ö

,Õ

[Ï ] + × (x, ),

[Ï ] depends on only one space variable, Ó Ó

£

·

Ô Î

Ô

Ø

(Ú

2 Ó

, ) Ø

Ú

, and time : Ø

Ô Ï

2

+

(Ú Ø

Ø

Ø

(4) Ô

Ô Ó

, )

Ï Ú

¸

+ ¹Ø (Ú Ø

, )Ï , Ø

Û

= 1,

Ü,Ü,Ü

,Ý .

For equation (4) we set the initial condition of general form Ô Ï

= Þ (x) at

= 0.

(5)

Consider the domain ² = { ß Ø ≤ Ú Ø ≤ Ð Ø , Û = 1, Ü,Ü,Ü , Ý } which is an Ý -dimensional parallelepiped. We set the followingÓ boundary conditions at the faces of the parallelepiped: Ô

Ó Ï

(1) o Ø

o

Ó Ú

+à

(2)

Ø

Ó

Ø

Ô

( )Ï = p

(1) Ø

Ô

Ï

Ø

Ú

(1)

Ø

+à Ø

(2)

( )Ï = p

Ú

Ø

=

Ú

Ø

=

(x, ) at

ß

Ø

, (6)

Ô Ø

(2)

(x, ) at

Ð

Ø

.

© 2002 by Chapman & Hall/CRC Page 33

Ô

Ô

By appropriately choosing the coefficients o (1) , o (2) and functions à (1) = à (1) ( ), à (2) = à (2) ( ), we can Ø Ø Ø Ø Ø Ø obtain the boundary conditions of the first, second, or third kind. For infinite domains, the boundary conditions corresponding to ß Ø = − » or Ð Ø = » are omitted. 2 Ò . The Green’s function of the nonstationary Ý -dimensional boundary value problem (4)–(6) can be represented in the product form á á Ô

(x,á y, , â ) = á

Ô ã ·

where the Green’s functions

(Ú Ø

,ä Ø

, , â ), Ø

(7)

=1 Ø

Ô

á

= Ø

Ó Ó

Ô Ø

(Ú Ø

−Î

,Õ Ø

á Ø

[

,ä

, , â ) satisfy the one-dimensional equations Ø

]=0 Ø

(Û

= 1,

,Ý )

Ü,Ü,Ü

á

with the initial conditions Ô

= å (Ú Ø

−ä Ø

=

) at Ø

â

á

and the homogeneous boundary conditions á

Ó

Ø

Ô á

Ó

(1) o

Ó

Ø Ø

Ú

Ø

+à Ø

Ø

= 0 at Ø

= 0 at

( )á

Ú

Ø

=

Ú

Ø

=

ß

, Ø

Ô

Ó

(2) o

Ú

(1)

Ø

+à

Ø

Ø

(2)

()

Ð

. Ø

Ô

Here, ä Ø and â are free parameters ( ß Ø ≤ ä Ø ≤ Ð Ø and ≥ â ≥ 0), and å (Ú ) is the Dirac delta function. Ô It can be seen that the Green’s function (7) admits incomplete separation of variables; it separates in the space variables Ú 1 , Ü,Ü,Ü , Ú æ but not in time . 0.9.2-2. Boundary value problems for a cylindrical domain with arbitrary cross-section. 1 Ò . Consider the parabolic equation Ó

Ô Ó Ï Ô

=

x, Õ Î

[Ï ] + ç

,Õ è

[Ï ] + × (x, é , ),

(8)

Ô where Î x,Õ is an arbitrary second-order linear differential operator in Ú 1 , Ü,Ü,Ü , Ú æ with coefficients dependent on x and , and ç è ,Õ Ô is an arbitrary second-order linear differential operator in é with coefficients dependent on é and . For equation (8) we set the general initial condition (5), where Þ (x) must be replaced by Þ (x, é ). We assume that the space variables belong to a cylindrical domain ê = {x ë ì , é 1 ≤ é ≤ é 2 } with arbitrary cross-section ì . We set the boundary conditions* Ô í

] = p 1 (x, Ô ) 2 [ Ï ] = p 2 (x, ) Ô 1 [Ï í í

at at é é

3 [ Ï ] = p 3 (x, é , ) for x í

=é = Óé ë

2

(x (x

ì

(é

1

ë

), ), ì

ë

ì

1

≤ é

≤

(9) é

2 ),

î

where the linear boundary operators ( à = 1, 2, 3) can define boundary conditions of the first, í î Ô second, or third kind; in the last case, the coefficients of the differential operators can be dependent on . * If ï

1

= −ð

or ï

2

= ð

, the corresponding boundary condition is to be omitted.

© 2002 by Chapman & Hall/CRC Page 34

á

á

ò

á

ó

2 Ò . The Green’s function of problem (8)–(9), (5) can be represented in the product form Ô á

ò

á

ò

á

Ô

ó

á

ó

(x, y, é , ñ , , â ) =

Ô

(x, y, , â )

Ô

( é , ñ , , â ),

(10)

Ô

where = (x, y, , â ) and = ( é , ñ , , â ) are auxiliary Green’s functions; these can be determined from the following two simpler problems with fewer independent variables: á ò á ó á

ò

á

Ó Problem on the cross-section ì : ôõ á õ

Ó

ö

ò

ôõ Ô á

õ õ÷ í

ò

= Î

x, Õ

] for x

[

Ô

= å (x − y) ]=0 3[

ë

, ì

á

ó

õ

Ó

Ô

ö á õ

at = âÓ , for x ë ì ,

ó

Problem on the interval é Ó

õ÷ í

î

ó

= ç

è

,Õ

[

= å (é − ñ ) [ ]=0 Ô

] for at at

é Ô

é

1

≤

1


0. Note that the selection of the function 3 is of a purely algebraic nature and is not connected with the equation in question; there are infinitely many suitable functions 3 that satisfy condition (3). Transformations of the form (2) can often be used at the first stage of solving boundary value problems.

© 2002 by Chapman & Hall/CRC Page 40

TABLE 13    Simple transformations of the form (
, ) =  (
, ) +  (
, ) that lead to homogeneous boundary conditions in problems with one space variables (0 ≤
≤  ) No

Problems

1

First boundary value problem



Boundary conditions 

=  1( ) at  =  2( ) at

2

Second boundary value problem

3

Third boundary value problem

4

Mixed boundary value problem

5

Mixed boundary value problem













=0 =





=  1( ) at =  2( ) at 

2





=  1( ) at





=  2( ) at




=




=  1( ) at  =  2( ) at

=0 =


(



(
, ) = 















=0 =

1( 



)+

−1− 

2







2( 



2








) −  1( )

2( 

2





) −  1( )



) 1( ) + (1 −  − 2  1 −  1  2

2 

=0










(
, ) =  1( ) +

(
, ) =


=


=  1( ) at  =  2( ) at

1



=0




+ +

Function  (
, )



(
, ) =  1( ) +


2( 



1




) 2( )

)





(
, ) = (
−  ) 1( ) +  2( ) 

0.11.2. Transformations That Lead to Homogeneous Initial and Boundary Conditions A linear problem with nonhomogeneous initial and boundary conditions can be reduced to a linear problem with homogeneous initial and boundary conditions. To this end, one should introduce a new dependent variable  by formula (2), where the function  must satisfy nonhomogeneous initial and boundary conditions. Below we specify some simple functions  that can be used in transformation (2) to obtain boundary value problems with homogeneous initial and boundary conditions. To be specific, we consider a parabolic equation with one space variable and the general initial condition 

= (
) at

= 0.

(4)

1. First boundary value problem: the initial condition is (4) and the boundary conditions are given in row 1 of Table 13. Suppose that the initial and boundary conditions are compatible, i.e., (0) =  1 (0) and (  ) =  2 (0). Then, in transformation (2), one can take 





(
, ) = (
) +  1 ( ) −  1 (0) +




 

2(





) −  1 ( ) +  1 (0) −  2 (0) .

2. Second boundary value problem: the initial condition is (4) and the boundary conditions are given in row 2 of Table 13. Suppose that the initial and boundary conditions are compatible, i.e., (0) =  1 (0) and (  ) =  2 (0). Then, in transformation (2), one can set 



(
, ) = (
) +


 

1(



) −  1 (0) +




2

2

 

2(





) −  1 ( ) +  1 (0) −  2 (0) .

3. Third boundary value problem: the initial condition is (4) and the boundary conditions are given in row 3 of Table 13. If the initial and boundary conditions are compatible, then, in transformation (2), one can take 



(
, ) = (
) +

(

2


−1−

2



)[ 1 ( ) −  1 (0)] + (1 −   2 −  1 −  1 2

1




)[ 2 ( ) −  2 (0)]

(

1

< 0, 

2

> 0).

© 2002 by Chapman & Hall/CRC Page 41

4. Mixed boundary value problem: the initial condition is (4) and the boundary conditions are given in row 4 of Table 13. Suppose that the initial and boundary conditions are compatible, i.e., (0) =  1 (0) and (  ) =  2 (0). Then, in transformation (2), one can set 





(
, ) = (
) +  1 ( ) −  1 (0) +


 

2(



) −  2 (0) .

5. Mixed boundary value problem: the initial condition is (4) and the boundary conditions are given in row 5 of Table 13. Suppose that the initial and boundary conditions are compatible, i.e., (0) =  1 (0) and (  ) =  2 (0). Then, in transformation (2), one can take 



(
, ) = (
) + (
−  )

 

1(





) −  1 (0) +  2 ( ) −  2 (0).



References for Section 0.11: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), A. D. Polyanin, A. V. Vyazmin, A. I. Zhurov, and D. A. Kazenin (1998).

© 2002 by Chapman & Hall/CRC Page 42

Chapter 1

Parabolic Equations with One Space Variable 1.1. Constant Coefficient Equations 1.1.1. Heat Equation 

2

= 













2



This equation is often encountered in the theory of heat and mass transfer. It describes onedimensional unsteady thermal processes in quiescent media or solids with constant thermal diffusivity. A similar equation is used in studying corresponding one-dimensional unsteady mass-exchange processes with constant diffusivity. 1.1.1-1. Particular solutions (  , (
) =  (
, ) =

, and 

are arbitrary constants). 

+ ,  (
2 + 2  ) +  ,








(
, ) =  (
, ) = 





(
, ) =  (
, ) =  (
, ) = 







(
, ) =




(
(


3

(
(
(


5



6



2

+

+ 6
) +  ,  4 + 12 
2 + 12  + 20  + 30  7 + 42  



2 2

3

+ 60  + 180  5 + 420 


4




2 2




)+ ,


)+ ,
+ 120  2 3
+ 840 

2 2 2

3 3

2

3 3

(2 )(2 − 1)  =1

2  +1



(
, ) =









)= )= )= )= 

(
, ) =

!





(
, ) =

3% 2









erf #



2




4

2  −2






( )


, 

2  −2 +1

,

$

+ ,

2  $

+ ,

2

) cos(
) +  , ) sin(
) +  ,  ) cos(
− 2   2 ) +  ,  ) sin(
− 2   2 ) +  ,



2 



"



4

(2 − 2  + 2)

!



( )



)+ ,


2


exp # −

exp(−  exp(−  exp(− exp(− 

  



(
, ) =

!

(2 + 1)(2 ) 

exp(   1 "  exp # − 



,  ,  ,  ,



=1 2

(
, ) =

(
(
(
(


+



(2 − 2  + 1)

  

)+ , 
) +  ,






$

+ ,

© 2002 by Chapman & Hall/CRC Page 43



(
, ) =

erfc # 




2

"

+ ,

 $ 

 

(
, ) = 

&('

exp # − )

where  is a positive integer, erf + ≡

2




4

− $

2 "

)

,

erfc # 

exp(−. 2 ) / -

0

2


"

.

2


"

$ 



+ , *

the error function (probability integral),

and erfc + = 1 − erf + the complementary error function (complementary probability integral). 0 Fundamental solution: 2
1  (
, ) = " )  exp # −  $ . 4 2  

References: H. S. Carslaw and J. C. Jaeger (1984), A. D. Polyanin, A. V. Vyazmin, A. I. Zhurov, and D. A. Kazenin (1998).

1.1.1-2. Formulas allowing the construction of particular solutions. 

= (
, ) is a solution of the heat equation. Then the functions

Suppose

1 2 3

= = =

( 

!

1 1

exp( 

+2


+


 "

1,



|3 + 4 |

1

2 1

2

+2

2 ), 1

) (
+ 2 

exp & −

4



+2

2




4 (3 + 4 )

# *



1,

+2

2 ),


! 3



+4

, 5

+ +4 3

1

 

$

1

,

3

−4 5

= 1,

1

where  , 2 1 , 2 2 , 4 , 3 , and are arbitrary constants, are also solutions of this equation. The last 1 formula with 4 = 1, = −1, 3 = = 0 was obtained with the Appell transformation. 

5

References: W. Miller, Jr. (1977), P. J. Olver (1986).

1.1.1-3. Infinite series solutions. A solution involving an arbitrary function of the space variable: 

(
, ) = (
) +



( )  !

 6

=1 

(2 

)

(
),

(7

)

(
) =

/

7

/


7



(
),

where (
) is any infinitely differentiable function. This solution satisfies the initial condition (
, 0) = (
). The sum is finite if (
) is a polynomial. Solutions involving arbitrary functions of time: 



(
, ) =  ( ) +

 6 



(
, ) =


9





=1

( )+




1
(2 )!

2

(  8

1

 6 





=1





(2 + 1)!

)






( ), 2 9

(8  )



( ), 



where  ( ) and 9 ( ) are infinitely differentiable functions. The sums are finite if  ( ) and 9 ( ) are   polynomials. The first solution satisfies the boundary condition of the first kind (0, ) =  ( ) and

  the second solution to the boundary condition of the second kind (0, ) = 9 ( ). 

Reference: H. S. Carslaw and J. C. Jaeger (1984).

© 2002 by Chapman & Hall/CRC Page 44



TABLE 14    Transformations of the form . = (
, ), =  (. , )  (. , ) for which the equation

  − = 0 admits multiplicatively separable particular solutions with  (. , ) = : ( )  (. ) 8

Function  . = (
, )

No 1

.

2 .

3 .

"




−

= = "






=





= exp ? − 41 . 

2



−

1 2.

−

1 2 4.

−

1

1

=0 

@

=0

1



@ 

=0

1.1.1-4. Transformations allowing separation of variables. Table 14 presents transformations that reduce the heat equation to separable equations (the identity transformation with . =
,  = 1 is omitted). A

BDC

E FHG

I

In general, the solution of the equation for  in the first row of Table 14 is expressed 1 in terms of degenerate hypergeometric functions. In the special case = 12  ( = 0, 1, 2,    ), the equation admits solutions of the form  (. ) = (JY>Y ), which is accurate within 0.3%. As r  … , we have m [ +1 − m [  Ž . m 6

Example 1. The initial temperature of the cylinder is uniform,  (  ) = ‘ 0 , and its lateral surface is maintained all the time at a constant temperature, ’ (“ ) = ‘ ” . ™ — — › Solution: — ™ — –—  ‘ ( , “ ) − ‘ ” 2 ˜ ˜ 2 “ = • exp š − œ 2 œ  . 0š ˜ ˜ ‘ 0−‘ ” ) 1( =1

{"|

Reference: H. S. Carslaw and J. C. Jaeger (1984).

1.2.1-4. Domain: 0 ≤

≤ c

. Second boundary value problem. „

The following conditions are prescribed: b = } (c ) at ‚ ƒ

b

= € (d ) | |≠…

at at

b

d c c

=0

(initial condition),

=„ =0

(boundary condition), (boundedness condition).

Solution:  b

(c , d ) =

Here, Ÿ ˆ

(c , ‡ , d ) = „

2 2

‡

f

0

†

+

(‡ ) ˆ (c , ‡ , d ) j }

„

2 2

žŸ ‡ ~

=1

n 02 ( m

‡

+`

Ÿ

)

n 0e

f

€

Ÿ 0 

‰

Ÿ

m „

(‰ ) ˆ (c , „ , d − ‰ ) j

¡

n 0¢ £

Ÿ ¥ ‡

„

.

¡

exp ¢ − ¤

2 £ „ 2 ¡

,

where the are positive zeros of the first-order Bessel function, ¦ 1 ( ) = 0. Below are the numerical £ £ values of the first ten roots: 1 = 3.8317, 2 = 7.0156, 3 = 10.1735, 4 = 13.3237, 5 = 16.4706, £ £ £ £ £ Ÿ Ÿ = 19.6159, = 22.7601, = 25.9037, = 29.0468, 6 7 8 9 10 = 32.1897. £ £ £ £ £  Ž . As §  … , we have +1 −

£ £ Example 2. The initial temperature of the cylinder is uniform,  (  ) = ‘ 0 . The lateral surface is maintained at constant thermal flux, ’ (“ ) = ’¨” . Solution: ™ — › — › — — ™ –—   2 2 1 “ œ ª ˜ ˜ 2 “ ‘ (  , “ ) = ‘ 0 + ’©” 2œ 2 − + œ 2 − • ˜ 2 exp š − œ 2 œ  « . 0š ˜ 4 2 ( ) 0 =1

¬"­

Reference: H. S. Carslaw and J. C. Jaeger (1984).

© 2002 by Chapman & Hall/CRC Page 67

 

1.2.1-5. Domain: 0 ≤

≤

. Third boundary value problem. ®

  The following conditions are prescribed: ¯ ± ² ¯

= ° ( ) at = ´ ( ) at ¯

+³

|¯ | ≠  

Solution: ¯

Here,   ¹

where the

Ÿ

¥  

¥

 

at µ

  ¥

( , )= ¶

·

0

=0 =®

(initial condition), (boundary condition),

=0

(boundedness condition).  

¥

(¸ ) ¹ ( , ¸ , ) º °

+ ¸

¶ ¤

Ÿ   Ÿ

¥

2

( ,¸ , ) =

2 ®

2Ÿ

žŸ ½

2®

(³

=1

£

+

2

(¼ ) ¹ ( , ® , − ¼ ) º

» ´

0

¥

2 ) ¦ 2( 0 £

Ÿ

Ÿ ¥

Ÿ ¸

) £

¦ 0¢ £

. ¼

®

¦ 0¢ ¡

¸ £ ®

exp ¢ − ¤ ¡

2 £ ® 2 ¡

,

are positive roots of the transcendental equation £

−³

¦ 1( Ÿ ) £

£

The numerical values of the first six roots

¦ 0( ®

) = 0. £

can be found in Carslaw and Jaeger (1984).

£

Example 3. The initial temperature of the cylinder is uniform,  ( ¾ ) = ¿ uniform and is equal to ¿ À , which corresponds to Á (  ) = à ¿ À . Æ Solution: Æ Æ

ÅÆ ¿ (¾ , Â ) − ¿ 0 =1− Ä ¿ À −¿ 0

2 Â

exp È − É©Ë Ê

È Ê Ë Ì Í 0

2

=1 Ç

¾

0.

The temperature of the environment is also

Æ

=

, Ì

Ç

(Ã 2

Ë Æ 2Ã Æ . Ë 2 2 + )Í 0( ) Ê Ê

¬"­

References: A. V. Lykov (1967), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980), H. S. Carslaw and J. C. Jaeger (1984).

1.2.1-6. Domain: ®

≤Î ≤

1

2. ®

First boundary value problem.

The following conditions are prescribed: ¯

= ° (Î ¥ ) ¯

= ´ 1 (¥ ) at = ´ 2 ( ) at ¯

Solution: ¥

¯

(Î , ) =

¥

=0

at

= = Î Î

(initial condition),

®

1 ®

2

(boundary condition), (boundary condition).

¥ 2

¶

·

(¸ ) ¹ (Î , ¸ , ) º °

+ ¸

1

¶ ¤

·

Here, ¹

(Î , ¸ , Ð ) =

×

where 0 ( Þ ) and equation

)Ï

Ú 0( ±

,Ð ) =

2Ÿ Õ

ӟ

2 ½ 1 Ÿ =1Õ

2®

(Î ) =

Ï 1 (Î Ú 0 (Þ

2 Ò

± ¸

Ô

)

)¸ 2 ) − Ô 0 (Ö Î

Ô Ô

®

1 Ù Û = ÛÜ

1

0Ø

,Ÿ Ð ) =

·

) are the Bessel functions; the Õ

)Ú 0 (Ö ) − Ô Ô

2 (Î

Î ®

±

1 Ù ¸

,Ð −¼ )º ¼

.

¹

, Ö

=

2

®

Ð 2 ® 1 Ù Ô

2 ®

1

(Î , ¸ , Ð ) Û

Û = ÛÝÜ

,

, ,

2

·

are positive roots of the transcendental

Õ

0(

)Ï

Ñ

(¸ ) exp Ø −

×

Ô ±

Ô Ï 2 (Î

,

» ´ 2 (¼

0

Ÿ

(Î ) Ÿ

)Ú

¶

Ÿ

)×

Ô 0(

−

(Î , ¸ , Ð ) Û ¹

Ÿ

Õ

−Ñ ¼

Ÿ Ÿ

2 Õ 0 (Ö

2 Ÿ 0 (Ô 0Ø

,Ð −¼ )º

1 (Î

Ÿ

Ÿ Õ

Ÿ

Õ

» ´ 1 (¼

0

Ô

) Ú 0 ( ) = 0.

0(Ö Ô

Ô

© 2002 by Chapman & Hall/CRC Page 68

Ÿ

Ÿ

The numerical values of the first five roots = ( Ö ) range in the interval 1.4 ≤ Ö ≤ 4.0 and can be found in Carslaw and Jaeger (1984). See also Abramowitz and Stegun (1964). Ô

Ô

Example 4. The initial temperature of the hollow cylinder is zero, and its interior and exterior surfaces are held all the time at constant temperatures, Á 1 ( Â ) = ¿ 1 and Á 2 ( Â ) = ¿ 2 . Æ Æ Æ Solution: Æ Ë

1 È ¿ ¿ (¾ , Â ) = ln ß ¬"­

ln

1

2

+¿

¾

ln Ë

2

¾

−à Ì

1

ÅÆ

( Í 0 Ê Ä

Æ

)[¿ 2

( Í 0 Ê

=1

Æ

)−¿ Ê ) − Í 02 ( ß

2 Í 0(

1 Í 0 (ß

Ê

)

Ê

)]

exp È −

Æ

2 Â É©Ë Ê 2 Ì 1

( ¾ ). á

Reference: H. S. Carslaw and J. C. Jaeger (1984).

1.2.1-7. Domain: ®

1

≤Î ≤

2. ®

Second boundary value problem.

The following conditions are prescribed: ¯

= ° (Î ) at = ´ 1 (Ð ) at

± ² ¯ ± ² ¯

Î

=0 =®

1

(initial condition), (boundary condition),

Î

=

2

(boundary condition).

Ð

= ´ 2 (Ð ) at

®

Solution: ¯

2

(Î , Ð ) = ¶

−Ñ

(¸ ) ¹ (Î , ¸ , Ð ) º °

·

¸

¶

» ´ 1 (¼

0

1

) ¹ (Î , ®

·

Here,

(Î , ¸ , Ð ) = ¹

Ÿ

2 2 ®

(Î ) =

â×

Õ â

2¸ −®

2

+

2 1

ÓŸ

Ò

2Ÿ ® 12 Õ

Ú 1(

)

½

2Ÿ

=1

Î

0Ø ® Ô

Ô

2 Õ 1 (Ö

2 1 (Ô Ÿ

)−

1(

)Ú

1 Ù

−

) Ô

¶

0

Ÿ Ÿ

)¸ 2 (Ö Ô 1Ÿ

Õ

Ô

+Ñ ¼

Ÿ

Ÿ Õ Õ

−¼ )º

1, Ð

(Î ) Î

Ô Ô

®

×

,

1 Ù

¬"­

=

. ¼

Ö Ÿ

=

®

2 ®

1

2

Ñ ®

Ð 2 Ô 1 Ù

,

,

are positive roots of the Ô

( Ö ) can be found in Abramowitz and Stegun Ô Ô

Ô

−¼ )º

2, Ð Ÿ

(¸ ) exp Ø −

where ( Þ ) and Ú ( Þ ) are the Bessel functions ( ³ = 0, 1), and the Õ Õ transcendental equation Ÿ Ÿ 1 ( ) Ú 1 ( Ö ) − 1 ( Ö ) Ú 1 ( ) = 0. The numerical values of the first fiveÔ rootsÔ (1964).

) ¹ (Î , ®

Ÿ

× Ÿ

0Ø

» ´ 2 (¼

Ô

References: A. V. Lykov (1967), H. S. Carslaw and J. C. Jaeger (1984).

1.2.1-8. Domain: ®

1

≤Î ≤

2. ®

Third boundary value problem.

The following conditions are prescribed: ± ² ¯

−³ +³

± ² ¯

= ° (Î ) ¯

= ´ 1 (Ð ) at = ´ 2 (Ð ) at

1

¯

2

¯

=0

at Ð

= = Î Î

(initial condition),

®

1 ®

2

(boundary condition), (boundary condition).

Solution: ¯

(Î , Ð ) =

2

¶

° ·

(¸ ) ¹ (Î , ¸ , Ð ) º

−Ñ ¸

¶

1

·

Here, ¹

(Î , ¸ , Ð ) =

Ÿ

2Ÿ

Ò

2 ÓŸ

2 ½

ä

=1 ã

0

) ¹ (Î , ®

Ÿ Õ

[³

» ´ 1 (¼

2 0(

Ÿ Õ

ã

®

2)

−

¼

Ÿ

1( ã

−¼ )º

1, Ð

¶

0

Ÿ ®

ã

+Ñ

2 2 )] ¸ å

» ´ 2 (¼

) ¹ (Î , ®

Ÿ

(Î ) å

−¼ )º

2, Ð

¼

.

Ÿ

2

(¸ ) exp(− Ñ

Ð

),

ã

© 2002 by Chapman & Hall/CRC Page 69

where Ÿ

ä

Ÿ Ÿ

and the [³

Õ

Ÿ Õ

+ ³ 22 )[ ³ Ÿ 1 0 (

(Ÿ Î ) ã= [ ³ å

¬"­

2

=(

Ÿ Õ

1 Ú 0(

1)

+

1 )ã ®

Ÿ ®

Ÿ

Ÿ

+

Õ

1(

Ú 1( ã

Ÿ Ÿ 2 1 )] ®

( 1 )] ã 0 ®

Î

Ÿ + ³ 12 )[ ³

2 Õ

−( ) − [ã ³

1 0(

ã roots ofã the transcendental ã ã are positive equation Õ

1ã 0 ( ®

1)

ã

+

1( ® ã

1 )][ ³ 2 Ú 0 ( ®

ã

2)

− [³ ã

−Õ

Ú 1( ®

Ÿ Õ Ÿ ® 2) −

2 0( 1)

®

+ã

ã

ã

< Î

Ÿ

1(

1(

−

1( ®

Ÿ

2 )]

1 )] Ú ã 0 ( ®

ã

®

ã

2 )][ ³ 1 Ú 0 ( ®

ã References: A. V. Lykov (1967), H. S. Carslaw and J.ã C. Jaegerã (1984).

1.2.1-9. Domain: 0 ≤

Ÿ Õ

Î

2

,

),

ã

Õ

2 )]

2 ã 0( ® ã 2)

Ÿ Õ

1)

+

ã

Ú 1( ® ã

1 )]

= 0.

ã

. Cauchy type problem. µ

This problem is encountered in the theory of diffusion wake behind a drop or a solid particle. Given the initial condition ¯ = ° (Î ) at Ð = 0, the equation has the following bounded solution: ¯

where

¬"­

æ

0 (¸

1 2Ñ

(Î , Ð ) =

¶

¸

0

½

exp Ø − Ð

2

2

+¸ 4Ñ Ð

Î

Ù

Î ¸

0Ø æ

2Ñ

°

Ð Ù

(¸ ) º ¸ ,

) is the modified Bessel function.

References: W. G. L. Sutton (1943), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980), Yu. P. Gupalo, A. D. Polyanin, and Yu. S. Ryazantsev (1985).

1.2.1-10. Domain:

≤Î < ®

. Third boundary value problem. µ

The following conditions are prescribed: ¯ ± ² ¯

= ° (Î ) at = ´ (Ð ) at ¯

−³ ¯

| |≠

Î

at µ

=0 =® Ð

Î

ç

(initial condition), (boundary condition), (boundedness condition). µ

Solution: ¯

(Î , Ð ) = ¶

(¸ ) ¹ (Î , ¸ , Ð ) º

−Ñ ¸

¶

(¼ ) ¹ (Î , ® , Ð − ¼ ) º

» ´

0

¼

,

·

where (Î , ¸ , Ð ) = ¸ Õ ¹

ì ð"ñ

° ½

(Î , é ) =

¶

0

½

0 (é Î ï

exp è − Ñ )[é [é

é

2

(Î , é ) ì (¸ , é )é

Ðëê>ì

Ú Õ 1 (é í

)+î

1 (é í

)+î

Ú Õ 0 (é í

)] − Ú 0 (é

0 (é í

)]2

+ [é

º é

,

Î

)[é

Õ

Õ

1 (é í

Ú 1 (é í

)+î

)+î

0 (é í

Ú 0 (é í

)]2

)]

.

Reference: H. S. Carslaw and J. C. Jaeger (1984).

1.2.2. Equation of the Form ò

=

ò

ó ô

õ

ö

2 ò ò

÷

ó

2

+ 1 ÷

ò

+ ó

ò

÷

ø

ù

(÷ , ô )

This equation is encountered in plane problems of heat conduction with heat release (the function ú is proportional to the amount of heat released per unit time in the volume under consideration). The equation describes one-dimensional unsteady thermal processes having axial symmetry.

© 2002 by Chapman & Hall/CRC Page 70

1.2.2-1. Domain: 0 ≤

≤ Î

. First boundary value problem. í

The following conditions are prescribed: û = ü (Î ) at ý = 0 û = þ (ý ) at Î = í |û | ≠ ÿ at Î = 0 Solution: û

(Î , ý ) =

where 

( ) (Î , , ý ) ü




0









− 



(boundedness condition).

( ) (Î , ý − ) þ



0

(initial condition), (boundary condition),









+ 



( , ) (Î , , ý − ) ú




0

0

















, 

 

2



(Î , , ý ) = 



2 2( 1



Here, the ten roots

0 

)



=1 





Î





 

exp −



0 



 







1.2.2-2. Domain: 0 ≤

2 

ý

2

, 

( ,ý ) = 



( , ,ý ) 







≤ 

ý



û



.

= 

) = 0. The numerical values of the first




. Second boundary value problem.

The following conditions are prescribed: û = ü ( ) at 

0( 

 





are positive zeros of the Bessel function, are given in Paragraph 1.2.1-3.







= þ (ý ) û | |≠ÿ

at at

=0

(initial condition),

= =0

(boundary condition), (boundedness condition).





Solution: û

( ,ý ) =

( ) ( , ,ý ) ü








0

Here,











2

( , ,ý ) = 





2

+



0











( , ) ( , ,ý − ) ú








0



0 







 

2 0( 

=1



0 

) 





0 





exp −







where the are positive zeros of the first-order Bessel function, of the first ten roots can be found in Paragraph 1.2.1-4.



ý



2



1( 











.











2



2

+





2

+ 

( ) ( , ,ý − ) þ



, 

) = 0. The numerical values 





References: A. V. Lykov (1967), H. S. Carslaw and J. C. Jaeger (1984).

1.2.2-3. Domain: 0 ≤

≤ 

. Third boundary value problem.

The following conditions are prescribed: û = ü ( ) at û + û = þ (ý ) at |û | ≠ ÿ at Solution:

ý





û

( ,ý ) = 




0

Here,









+ 



0



=0

(boundedness condition).

( ) ( , ,ý − )

( , ,ý ) = 





2



2

2

 



=1

( 

2

2

+

+

þ 

(initial condition), (boundary condition),











ú


0

0

( , ) ( , ,ý − ) 











  

)





0



 





0 













.



 

2 ) 2( 0 



 



where the







( ) ( , ,ý ) ü

=0 =





exp − 





2 2

ý 

,

are positive roots of the transcendental equation 1( ) − 0 ( ) = 0. The numerical values of the first six roots can be found in Carslaw and Jaeger (1984). 



















References: A. V. Lykov (1967), H. S. Carslaw and J. C. Jaeger (1984).

© 2002 by Chapman & Hall/CRC Page 71

1.2.2-4. Domain:

1

≤

≤ 

2.

First boundary value problem.

The following conditions are prescribed: û

=ü ( )

=0

at



û

= þ 1 (ý ) at û

= þ 2 (ý ) at

ý

= 

= 

(initial condition), 1

(boundary condition),

2

(boundary condition).



Solution: û

2

( ,ý ) = 

( ) ( , ,ý ) ü














2

+ 



0

1


+ 

þ 1( 

0

) 

1( 

,ý − ) 



Here,




− 





















1



þ 2( 

0

) 

2( 

,ý − ) 





. 

 



 



( , ,ý ) = 



( , ) ( , ,ý − ) ú








( ) 









= 

2

2 1

  "

2 0(

where 0 ( ) and equation

2 0(

) ) − 02 (

 



%

0(

1( 

)

( )= 



1

≤

≤ 

  



= 



0( $

) 



0 





1 

0(

2.

2( 

,ý ) = 

  

1 

,







= 





) 0( 

$

)− 

!

0(

2

= 

0( 




) 



0 $



=

,





1

2

1

,

are positive roots of the transcendental 

) 0 ( ) = 0.





−



#

The numerical values of the first five roots

,







) are the Bessel functions, the %

,ý ) = 

,



1.2.2-5. Domain:

, 




!

$

ý

2 1



!





2 













=1 2 2

( ) exp − 

!

$



( ) can be found in Carslaw and Jaeger (1984). 

Second boundary value problem.

The following conditions are prescribed: û

=ü ( )







ý

= þ 1 (ý ) at = þ 2 (ý ) at û



=0

at



û

= = 



(initial condition), (boundary condition), (boundary condition).

1

2

Solution: û

2

( ,ý ) = 

( ) ( , ,ý ) ü














2

+ 



0

1


− 

1( 

0

) ( , 



1,



− ) '



( , ) ( , , − ) 




1






+ 



2( 

0

(

Here,







) ( , 







2 2− 2 

'



2 1

+

2

2 



2 

 

2 1

2 1(



=1





2,











− ) '





. 

(

 

( , , )=



 



'

&



2 1(

) ) − 12 ( !

( )



)









!









( ) exp − 









2 2 1

'



,

( )= 



$

1(

) 



0 



−





1



1(

) 

$



0 





1

where ( ) and ( ) are the Bessel functions of order = 0, 1 and the transcendental equation 1 ( ) 1 ( ) − 1 ( ) 1 ( ) = 0. 

=

,

)



)

%

$

%









$

!



=

1

,

are positive roots of the





!

The numerical values of the first five roots (1964).



2



$



( ) can be found in Abramowitz and Stegun

© 2002 by Chapman & Hall/CRC Page 72

1.2.2-6. Domain:

≤

1

≤ 

2.

Third boundary value problem.

The following conditions are prescribed: *

*

= ( ) at = 1 ( ) at *

+

− *





1 

+



*

'

=

2( (



) at '

=0 =

1

(initial condition), (boundary condition),

=

2

(boundary condition).

'

2 





 (

The solution is given by the formula from Paragraph 1.2.1-8 with the additional term 2 

( , ) ( , , − )






0

1









'









, 

&

which takes into account the nonhomogeneity of the equation.


1.2.2-7. Domain: 0 ≤

< 

. Cauchy type problem. ,

The bounded solution of this equation subject to the initial condition *

= ( ) at +



=0 '

is given by the relations *

( , )= 

'

where -

0( %





'

+



( , , )= 



( , , ) ( ) 



0 





2



0 2



'





'















,

&

+ 4





( , , − ) ( , ) 



0

exp −



'

+ 



2 





0 -

2 

'

,







'

) is the modified Bessel function.

References: W. G. L. Sutton (1943), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

1.2.2-8. Domain:

≤

< 

,

. Third boundary value problem.

The following conditions are prescribed: *

* *

= ( ) at = ( ) at +

− 





*

'



at

( ,

(initial condition), (boundary condition),

'

| |≠

=0 =





.

(boundedness condition). ,

The solution is given by the formula from Paragraph 1.2.1-10 with the additional term 

( , ) ( , , − ) 



0









'











,

&

which takes into account the nonhomogeneity of the equation.


1.2.3. Equation of the Form /

2

= 0

2

3

/

0

2

+ 2 /

0

This is a sourceless heat equation that describes unsteady heat processes with central symmetry. It is often represented in the equivalent form /

1

/

4

*

/

4

5

*

= 

'



4

2





2 





 





.





A similar equation is used for the analysis of the corresponding one-dimensional unsteady diffusion processes.

© 2002 by Chapman & Hall/CRC Page 73

1.2.3-1. Particular solutions ( ,

, and 6

are arbitrary constants). 

*

1 

( )=

+



*

,

6



( , )= *



+ 

'

( , )= 

+ 

'

2

( 6

( 6

+ 6 ),





4

'

2

+ 20





+ 60

'7

2 2 

),

'

*

( , )= 

+

'

2 6

;

+ :

89


=

=1 )

*

*

(2 + 1)(2 ) :






) 8

+

−+

2 ) 8

+

*

, 9

*

8

2 )

, 9

*

2 9

,

where ? is an arbitrary positive integer, : (& , @ ) is the incomplete gamma function, ; are the Bessel functions, and + ( @ ) and + ( @ ) are the modified Bessel functions.



=

+

( @ ) and < +

(@ )

>

References: W. G. L. Sutton (1943), A. D. Polyanin (2001a).

1.2.5-2. Infinite series solutions. A solution containing an arbitrary function of the space variable: (* ,  ) =  (* ) + 

 .

=1 -

1  ? ! -

A

-

[  (* )], A

≡

  *

2 2

+

1 − 2& 

*

 *

,

where  (* ) is any infinitely differentiable function. This solution satisfies the initial condition  (* , 0) =  (* ). The sum is finite if  (* ) is a polynomial that contains only even powers. A solution containing an arbitrary function of time: 

(* ,  ) =  ( ) +

 .

-

=1

4 -

?

1 * ! (1 − & )(2 − & ) 67676 (? − & )

2-



(- )

( ),

where  ( ) any infinitely differentiable function. This solution is bounded at * = 0 and possesses the properties  B   (0,  ) =  ( ), (0,  ) = 0.

© 2002 by Chapman & Hall/CRC Page 82

1.2.5-3. Formulas and transformations for constructing particular solutions. Suppose  =  (* ,  ) is a solution of the original equation. Then the functions 



1

= '

2

= '



(C D

, *

2 D

−1

|  + F4 |+

+ E ), 

exp  −

2

F4*



4(  + F4 ) 

8

C

*

,

+ F4 

G

+  + F4 

, 9



−F 

G

= 1,

where ' , E ,  , F , and are arbitrary constants, are also solutions of this equation. The second G formula usually may be encountered with  =  = 0, F = 1, and = −1. G The substitution  = * 2+ H (* ,  ) brings the equation with parameter & to an equation of the same type with parameter −& :    2H H H 1 + 2&   =  2 + . 

1.2.5-4. Domain: 0 ≤

< *

*

*

*

. First boundary value problem. 

The following conditions are prescribed:

Solution for 0
0.

This is a special case of equation 1.8.9 with  (Ï ) = 1,  (Ï ) = , (Ï ) = − 4Ï 2 , and Õ (Ï , Ð ) = 0. The Ö transformation (Ï , Ð ) = Î

1

cos ó 2  Ô

Ù

=

Ð

exp ù

cos ó 2  Ð

Ô

4.




=

2


The substitution  > 0).

 2 + 

 2 + ( Ù

= Ï

+ Ö



2 Ï

tan ó 2  Ô

Ð

ô

− Ö

Ð ú ÷

( Ù , ø ),

Ô

Ï

leads to the constant coefficient equation

 Ô

2

ô

û

ô

, ø

=

Ô

2 

tan ó 2  Ô

Ð

ô

,

Ô

ü ÷

= û

ý4ý7÷

, which is considered in Subsection 1.1.1.

+  )
.

(2  ) leads to an equation of the form 1.3.1.2 (for  < 0) or 1.3.1.3 (for

© 2002 by Chapman & Hall/CRC Page 86




5.

2


= 

 –2 ô
.

 2 + ó  +

1 ß . Particular solutions (Ï , Ð ) = Î

à

2

( ä − ã

)

¸Ñ   (Ó Ï

Ô —

) +    (Ó Ï

1  2= −

) ,

Ö , 4 are arbitrary constants.

Ï

where   ( Ù ) and   ( Ù ) are the Bessel functions; Ñ ,  , and Ó 2 ß . Domain: 0 ≤ Ï <  . First boundary value problem. The following conditions are prescribed: =  (Ï ) at =0 at Î

 õ

Solution: 

where

− 34

= 0 (initial condition),  = 0 (boundary condition). Ð

( ,  ) = < < #

1 4

ä à



Ú

2 — 



0

 2+ 2

exp  − é

4 



 

! "

2  !

 ( )ö

1  2= − # , 4

,

.

3 $ . The transformation 

( ,  ) = % &  ' ( ( , ) ),

) =  ,

2

where * is a root of the quadratic equation * — − * + = 0 , leads to an equation of the form 1.2.5: ( û

= )

û

2 û

#

2*

(

( û

+  2 

û

.

 û

See also Miller, Jr. (1977).


6.



=

2


 2 + + –

 2 +  +   –2 ,
,

 > 0. 2

This is a special case of equation 1.8.1.2 with  ( ) = − - number) 

( ,  ) = ( ( / , ) ) exp 0 1 2  2

2

+(

1 3

+ )54 ,

1 3 2

8 2(

= )

*

9 ;

2

: 9 =

2

+ (>

2

=

–? +@

–2

) ,

(6 , ) = C 7

2

2

1 D D

cos + 2

/ =

cos + 2

2

1 3 2 7

leads to an equation of the form 1.3.1.5:8

8 ( )

1 3

2

8

=

8 2(

+ / 2

*

−2

#

) = 1 3

, ,

− +* 6

6 2 tan + 2

D exp 0 1 2

, D

1 3 2 7 6

1 4

exp + 4

1 3 2

1 3 2 7

, +. ,

3

This is a special case of equation 1.8.1.2 with A (6 ) = 6 B

) =

> > 0.

:

=

, ,

/ −2 ( .

+ / 2

See also Miller, Jr. (1977). =< 9

1 3 2 7

. The transformation ( . is any

8

8 (

9 :

−2

#

/ = 6 exp + 2

#

leads to an equation of the form 1.3.1.5:8

7.

− +* 

1 3 2 7

tan + 2

1 2

. The transformation , −

1 3 2 7

#7

4 ( ( / , ) ),

, ,

/ −2 ( . 3

© 2002 by Chapman & Hall/CRC Page 87

1.3.1-2. The function A depends on time 8.

9 :

=< 9

9.

9 :

=< 9

2

:

+ +>

2

+? ,

alone.

.

: 9 = case of equation 1.8.1.1 with A ( ) = This is9 ; a special + . 7 2-7 # 1 $ . Particular solutions ( . , E , and F are arbitrary constants): B (6 , ) = ( . 6 + E ) exp + 12 2 + , , 7 2-7 #7 B (6 , ) = . (6 2 + 2 ) exp + 12 2 + , , 7 2-7 #7 3 7 B (6 , ) = . exp GHF 6 + 12 2 + ( + F 2 ) , 7 2-7 # 7JI 3 B (6 , ) = . exp G 12 2 + ( − F 2 ) cos(F 6 ), 7 2-7 # 7 I 3 B 8 K (6 , ) = . exp G 12 2 + ( − F 2 ) sin(F 6 ). 7 2 7 # J 7 I 3 8 L L 2 $ . The substitution B (6 , ) = ( (6 , ) exp + 12 2 + , leads to a constant coefficient equation, ( = 7 7 2-7 #7 ( , which is considered in Subsection 1.1.1. 3

9 ;

2

: 9 =

+>

2

;

7

.

;M :

This is a special case of equation 1.8.1.1 with A ( ) = ' . 7 2-7 1 $ . Particular solutions ( . , E , and F are arbitrary constants): B

(6 , ) = ( . 6 + E ) exp N 7

B

2

(6 , ) = . (6 7

B

' +1 O , 2 * +17

+ 2 ) exp N 3 7

(6 , ) = . exp NPF 6 + F 7

B

' +1 O , 2 * +17

2

7

3

+

' +1 O , 2 * +17

' +1 − F 2 O cos(F 6 ), 2 7 * +17 3 B +1 2 O ' (6 , ) = . exp N 2 − F sin(F 6 ). 8 K 7 7 * +17 3 8 L L , 2 $ . The substitution B (6 , ) = ( (6 , ) exp + +1 ( = & ' +1 leads to a constant coefficient equation, 7 7 7 ' ( , which is considered in Subsection 1.1.1. 3

(6 , ) = . exp N 7

1.3.1-3. The function A depends on both 6 and . 10.

9 :

=< 9

7

2

:

+ (>

2

=

+? ;

+Q ) . :

= This is9 a; special9 case of equation 1.8.1.6 with A ( ) = and R ( ) = 7

2

1 $ . Particular solutions ( . and F are arbitrary constants): B (6 , ) = . exp + 6 + 13 2 3 + 12 2 + S , , 7

B

7

(6 , ) = . 7

B

(6 , ) = Z 7

#7

+S .

#7 7 6 + 13 2 3 + 12 2 + S , , # 7 -2 7 7 3 2-7 3 2 7 2 3W 2 Y 1 2 U exp( 2 − F ) T U V 1W 3 X , U = 6 +S +F , # 7 2 7 3 2 1 3 2 U 3W 2 Y , U = 6 + S + F , exp( 12 [ 2 − F ) T U \ 1 W 3 X 2 7 7 3 2 1 3

(6 , ) = . (6 + B

7

2-7

2

3 2 7

) exp +

where V 1 W 3 (U ) and \ 1 W 3 (U ) are the Bessel functions of the first and second kind. 2 ] . The transformation B (6 , ) = ^ ( _ , ) exp `Ja 8 6 K + 13 b a8 2e-e 3 + 12 [ 2 + S , _ = 6 + b a 2 7 7 7 7 7 7 7dc ^ , which is considered in Subsection 1.1.1. leads to a constant coefficient equation, ^ = b

© 2002 by Chapman & Hall/CRC Page 88

11.

2

=< 9

9 :

:

2

+ (> =

+? ) . :

;

= This is9 a; special9 case of equation 1.8.1.3 with A ( ) = a + [ . 7 7 1 ] . Particular solutions ( Z and F are arbitrary constants): B

(6 , ) = Z exp G 6 ` 7

B

(6 , ) = Z G 6 + b ` 7 (6 , ) = Z exp G 6 ` B

7

4 1 2 1 2 5 1 1 2 3 2 a 7 + [ 7dc + b ` 20 a 7 + 4 a [ 7 + 3 [ 7 c I 2 1 3 1 2 3 a 7 + [ 7 c I exp G 6 ` 2 a 7 + [ 7dc + b f ( 7 )I , 2 1 2 1 3 2 a 7 + [ 7 + F c + b F ` 3 a 7 + [ 7 + F 7dc + b

, f ( 7 )I ,

+ 14 a [ 4 + 13 [ 2 3 . 7 7 2 ] . The transformation where f ( ) = 7

B

1 2 5 20 a 7

1 2

(6 , ) = ^ ( _ , ) exp Gg6 ` 7

7

a 2+[ + ` 1 a 2 5 + 1 a 4 + 13 [ 2 3 , 7 c I 7 c 8 b K 20 78 e-e 4 [ 7 7 ^ =b

leads to a constant coefficient equation, 12.

9 :

=< 9

2

:

2

+ (>

+?

= ;

+Q =

_ = 6 + b ` 13 a 3 + [ 2 7 c 7

^ , which is considered in Subsection 1.1.1.

+h ) . :

;

= This is9 a; special9 case of equation 1.8.1.6 with A ( ) = a + [ and R ( ) = S + i . 7 7 7 7 1 ] . Particular solution: B

(6 , ) = exp GH6 ` 7

1 2

1 2 5 a 2+[ + b ` 20 a + 14 a [ 4 + 13 [ 2 3 + 12 S 2 + i . 7 7 c 7 c 7 7 7 J7 I

2 ] . The transformation B

4 2 1 2 5 1 1 2 3 1 2 + b ` 20 8 a K 7 + 48 e-a e [ 7 + 3 [ 7 c + 2 S 7 + i 7JI , _ = 6 + b ( a 7 + 2 [ 7 ) ^ , which is considered in Subsection 1.1.1. leads to a constant coefficient equation, ^ = b 1 2a

(6 , ) = ^ ( _ , ) exp Gg6 ` 7

13.

7

9 :

=< 9

9 ;

2

: 9 =

2

+ (– >

2

+[

7

2

=

7 c

+?

+Q ) . :

;

This is a special case of equation 1.8.1.7 with A ( ) = [ + S . 7 7 1 ] . Particular solutions ( Z is an arbitrary constant): B

1 2k

(6 , ) = Z exp j 7

B

(6 , ) = Z 6 exp j 7

1 4 , 6 2 + [ 2 + `ml b a + S c 7 2 7 a

b

1 2k

a b

1 6 2 + [ 2 + ` 3l b a + S 4 . c 7 n 2 7

2 ] . The transformation ( Z is any number) B

(6 , ) = ^ ( _ , ) ) exp j 7

_ = 6 exp ` 2 l b a , 7dc 8 o

leads to the constant coefficient equation 14.

9 :

=< 9

2

:

2

+ (– > =

=

+? ;

1 2k

a b

1 6 2 + [ 2 + `ml b a + S 4 , c 7 2 7 1

) = 8 e-e 4 k ^ =

b

a

exp ` 4 l b a +Z 7dc

^ , which is considered in Subsection 1.1.1.

+Q ) . :

= This is9 a; special9 case of equation 1.8.1.8 with A ( ) = [ + S . 7 7

© 2002 by Chapman & Hall/CRC Page 89

15.

2

=< 9

9 : 9 ;

:

2

9 =

+>

.

= ; M :

This is a special case of equation 1.8.1.3 with A ( ) = a 7

.

7qp

1 ] . Particular solutions ( Z and F are arbitrary constants): B

(6 , ) = Z exp j 7

B

(6 , ) = Z j 6 + 7

B

(6 , ) = Z exp j 7

a 6

* +1 7 p

+1

+

2 b a

2b a ( * + 1)( * + 2) 7 p a 6

* +1 7 p

+1

2 +3

( * + 1)2 (2 * + 3) 7 p +2

a

4 exp j

+F 6 +

(* +

+1

6

* +1 7 p 2 b a

1)2 (2 *

4 , 2 b a

+

2 +3

( * + 1)2 (2 * + 3) 7 p

2 +3

+ 3) 7 p

+

2 b a-F ( * + 1)( * + 2) 7 p

4 ,

+2

+b F

2

7

4 .

2 ] . The transformation B

(6 , ) = ^ ( _ , ) exp j 7

7

a 6

* +1 7 p

+1

+

2 2 +3 b a 4 , e-e * + 3) 7 p 8 (K * + 1)28 (2

^ =b

leads to a constant coefficient equation, 16.

2

=< 9

9 :

:

2

2

+ (>

+?

;-r

=

= ;-s

+Q

_ =6 +

2b a ( * + 1)( * + 2) 7 p

+2

^ , which is considered in Subsection 1.1.1.

) . :

;M

= This is9 a; special9 case of equation 1.8.7.5 with t (u ) = b , A (u ) = R (u ) = 0, v (u ) = a-uqw , x (u ) = [ uzy , and { (u ) = S u . p

1.3.2. Equations of the Form 1.

‚ ƒ

=… ‚

2

ƒ

2

+ (‡

+ˆ )‚ ƒ „

=

| } | ~



2 |

| €

+  (€ , ~ ) |

} 2

} | €

.

‚ † ‚ † This is‚ „ a special case of equation 1.8.2.1 with ‰ (u ) = a-u + [ . 1 ] . Particular solutions ( Z , Š , and ‹ are arbitrary constants): Œ

( , u ) = 2 Z  + Z ( a-u 2 + 2 [ u ) + Š , 2 Œ ( , u ) = Z `Ž + 12 a-u 2 + [ u + 2 b Z u + Š , Œ

c

( , u ) = Z exp g‹  + 12 ‹ a-u 2 + ( b ‹

2

+ ‹ [ )u5 + Š .

2 ] . The substitution _ =  + 12 a-u 2 + [ u leads to a constant coefficient equation, ‘ ’ Œ = b ‘ is considered in Subsection 1.1.1. 2.

‚ ƒ

=… ‚

2

ƒ

2

+‡ †

e-e Œ

, which

.

‚ ƒ

‚ „ ‚ † † This equation is a special ‚ case of equation 1.8.2.2 with ‰ ( ) = a- and a special case of equation 1.8.2.3 with ‰ (u ) = a .

1 ] . Particular solutions ( Z , Š , and ‹ are arbitrary constants): Œ

( ) = Z “ exp ” − ’

Œ Œ

( , u ) = Z  i — + Š , ’

a

2b

 2• –  + Š , ’

( , u ) = Z a- 2 i 2 — + Z b i 2 — + Š , ’ ’ Œ ( , u ) = Z exp ` 2 a-‹  i — + 2 b a-‹ 2 i 2 — c

+Š .

© 2002 by Chapman & Hall/CRC Page 90

2 ] . On passing from u ,  to the new variables ( Z and Š ˜

= ™

2

2a

’ i 2— + Š ,

are any numbers) š =

’  i — , ™

Œ

for the function Œ (˜ , š ) we obtain a constant coefficient equation, ‘ in Subsection 1.1.1. 3 › . Domain: − œ <  < œ . Cauchy problem. An initial condition is prescribed: Œ = ‰ ( ) at

e-e o

=b ‘

Œ

, which is considered

u = 0.

Solution: Œ ª¬«

3.

( , u ) = j

2 ž Ÿ

’

2

 ¢¡ —

− 1 £¤

−1 ¥ 2

¦

Ÿ

§

exp ¨ −

−§

’

2

−© £ ¤ ‰ (© ) – © . ’ 2 2ž   ¡ — − 1 £   ¡ —

References: W. Miller, Jr. (1977), A. D. Polyanin, A. V. Vyazmin, A. I. Zhurov, and D. A. Kazenin (1998).

‚ ƒ

2

=… ‚

‚ „

ƒ

2

‚ †

+ (‡

2

„

+ˆ )‚ ƒ

.

‚ †

Ÿ

This is a special case of equation 1.8.2.1 with ‰ (­ ) = ­ 2 + ® . 1 › . Particular solutions ( , Š , and ‹ are arbitrary constants): ™

Œ

( , ­ ) = Œ

( , ­ ) = Œ

( , ­ ) =

Ÿ    + 13 ­ 3 + ®­d£ + Š , Ÿ 2    + 13 ­ 3 + ® ­d£ + 2 ž ­ +Š , ™ 1 Ÿ 3 exp g‹  + 3 ‹ ­ + ‹ ( ž ‹ + ® )­5 + Š . ™ ™ ™

Ÿ

2 › . On passing from ­ ,  to the new variables ­ , š =  + 13 ­ 3 + ®­ , we obtain a constant coefficient equation, ‘ ’ Œ = ž ‘ ¯-¯ Œ , which is considered in Subsection 1.1.1. 4.

‚ ƒ

2

=… ‚

‚ „

ƒ ‚ †

2

+ (‡ †

„

+ˆ )‚ ƒ

. Ÿ

‚ †

This is a special case of equation 1.8.2.3 with ‰ (­ ) = ­ + ® . 1 › . Particular solutions ( , Š , and ‹ are arbitrary constants): ™

Ÿ  exp   12 ­ 2 + ®­d£ + Š , ™ Ÿ Ÿ Œ ( , ­ ) =  2 exp   ­ 2 + 2 ®­°£ + 2 ž “ exp   ­ 2 + 2 ®­°£ – ­ + Š , ™ ™ Ÿ Ÿ Œ ( , ­ ) = exp ±²‹  exp   12 ­ 2 + ®­ £ + ž ‹ 2 “ exp   ­ 2 + 2 ®­ £ – J­ ³ + Š . ™ Œ

( , ­ ) =

2 › . On passing from ­ ,  to the new variables ( is any number) ˜

=“

™ Ÿ 2 – exp   ­ + 2 ®­°£ ­ +

™

,

Ÿ š =  exp   12 ­ 2 + ®­d£ ,

for the function Œ (˜ , š ) we obtain a constant coefficient equation, ‘ ´ Œ = ž ‘ ¯-¯ Œ , which is considered in Subsection 1.1.1. 5.

‚ ƒ ‚ „

= ‚

2

ƒ ‚ †

2

+ (… †

+‡ „

+ˆ )‚ ƒ ‚ †

. Ÿ

This is a special case of equation 1.8.2.7 with ‰ (­ ) = ž and µ (­ ) = ­ + ® . See also equation 1.8.2.4.

© 2002 by Chapman & Hall/CRC Page 91

6.

‚ ƒ

2

=… ‚

ƒ

2

+‡ †

.

‚ ƒ

„ ‚ † „ ‚ † Ilkoviˇ‚ c’s equation. It describes heat transfer to the surface of a growing drop that flows out of a thin capillary into a fluid solution (the mass rate of flow of the fluid moving in the capillary is assumed Ÿn¶ constant). This equation is a special case of equation 1.8.2.3 with ‰ (­ ) = ­ .

1 › . Particular solutions ( , Š , and ‹ are arbitrary constants): ™

Œ

( , ­ ) =

 ­°— + Š , Ÿ ( , ­ ) = (2 + 1) 2 ­ 2 — + 2 ž ­ 2 — +1 + Š , ™ ™ ž ‹ 2 2 +1 ¸ Œ ( , ­ ) = exp · ‹  ­d— + Ÿ ­ — +¹ . ™ 2 +1 ™

Œ

2 › . On passing from ­ ,  to the new variables ˜

=

1 2 +1 Ÿ ­ — , 2 +1

š = ­ — ,

for the function º (˜ , š ) we obtain a constant coefficient equation, » ´ º = ž » ¯-¯nº , which is considered in Subsection 1.1.1. 3 › . The solution of the original equation in the important special case where the drop surface has a time-invariant temperature º s and the heat exchange occurs with an infinite medium having an initial temperature º 0 , namely, º

=º º

=º º

¼

0 s

º 0

at

­ =0

(initial condition),

at at

 =0

(boundary condition), (boundary condition),



¼

œ

is expressed in terms of the error function as follows: −º º 0−º º

ª¬«

Ÿ

s s

2 +1  ¸ , 2 ž ­

= erf · ½

½

¦

2

erf © =

½

½

¾



0

exp   − ¿ 2 £ À ¿ .

Reference: Yu. P. Gupalo, A. D. Polyanin, and Yu. S. Ryazantsev (1985).

7.

Á Â

2

=Ä Á

Á Ã

 Á Å

2

+ (Æ

+É

ÃÈÇ Å

)Á Â

Ã-Ê

.

Á Å

This is a special case of equation 1.8.2.7 with Ë (­ ) = Ì-­qÍ and µ (­ ) = ®­ y . 8.

Á Â

2

=Ä Á

Á Ã

 Á Å

2

+ · É Å

Æ ¸

+

.

Á Â

Å

Á Å

On passing from ­ ,  to the new variables Î , Ï by the formulas Î =  Ð Ñ¢Ò ,

Ï = Ó

2®

Ð 2 Ñ¢Ò + const,

we obtain a simpler equation of the form 1.2.5: » º » Ï

For 9.

Ô

= 1 and Á Â Á Ã

=Ä Á

=

» 2º

+ Ô » Î 2 Î

» º » Î

, Ô

Ì

=

.

Ó

= 2, see also equations from Subsections 1.2.1 and 1.2.3. Ô

2

 Á Å

2

+ · É

Ã-Õ Å

Æ ¸

+ Å

Á Â

.

Á Å

This is a special case of equation 1.8.2.6 with Ë (­ ) = ®­×Ö .

© 2002 by Chapman & Hall/CRC Page 92

1.3.3. Equations of the Form 1.

Á Â

=Ä Á

Á Ã

2

 Á Å

2

+Æ Á Â

+ (É

Á Å

=Û

Ø Ù Ø Ú

2 Ø

Ø Ü

Ù 2

+ Ý (Ü , Ú ) Ø

+ Þ ( Ü , Ú )Ù + ß ( Ü , Ú )

Ù Ø Ü

+à ) . Â

Å

This is a special case of equation 1.8.7.4 with á (­ ) = , Ë (­ ) = 0, µ (­ ) = Ì , â (­ ) = ® , and ã (­ ) = À . Ó

1 ä . Particular solutions ( å and º ( , ­ ) = å

are arbitrary constants): Ô

Ì

exp ¨æ®­m −

 +

2

Ì 2 ¸ 1 23 ­Jç , ® ­ + · À − 3Ó 4

Ó Ó 1 23 Ì  + è 2) exp ê mè  − è + ë Ó é é 2 3Ó é Ó Ì ¸ 1 23 è exp êì ë è + − + è + Ô Ó éÔ é 2 3Ó é Ó Ì 2 î 3ð  ¸ í î ï 1ð 3 ë exp ë − è − Ô 2 3 Ó é ½ Ó Ì 2 î 3ð ¸ í î ñ  exp ë − è − 1ð 3 ë Ô 2 3 Ó é ½ Ó î( ) are the Bessel functions of the

º ( , è ) = å ( + º ( , è ) = å º ( , è ) = å º ( , è ) = å

where ï 1 ð 3 (î ) and ñ 1 ð respectively.

À − 2

3

+ë

2¸

,

2¸

,

Ì 2 ¸ èJç ,

4

Ó 2

Ó Ô

+À −

Ì 2 ¸ èJç ,

4

Ó 2 î =  + +À − Ì , é Ô 4 Ó 2 î =  + +À − Ì , é Ô 4 Ó

first and second kind of order 1 ò 3,

2 ä . The transformation Ì Ì 2 ¸ 1 2 3  + è ç , º ( , è ) = ó ( Î , è ) exp ê mè  − è + ë À − 2 3Ó é 4 é Ó Ó

Î = +

Ó é

è 2

leads to a constant coefficient equation, » ó = » ô-ônó , which is considered in Subsection 1.1.1. Ò

2.

Á Â Á Ã

=Ä Á

2

 Á Å

2

+Æ Å

+ (É

Á Â

+à ) . Â

Å

Á Å

Ó

This is a special case of equation 1.8.7.4 with á (è ) = , Ë (è ) = Ì , õ (è ) = 0, â (è ) = , and ã (è ) = À . 1 ä . Particular solutions ( å and

Ó

Ô

2

º (ö , è ) = å

exp ê − é ö + ë À + Ó é 2 ¸ èJç ,

º (ö , è ) = å

ë ö −

º (ö , è ) = å

é

are arbitrary constants):

Ì

2 Ì

Ì

Ó 2é ¸

2

exp ê − é ö + ë Ì + À + Ó é 2 ¸ è ç , Ì

Ì

2

2 2 exp ê Ó Ô Ð 2 ÷ Ò + Ð ÷ Ò ë ö − Ó 2é ¸ − é ö + ë À + Ó é 2 ¸ èJç . Ô 2Ì Ì Ì Ì

See 1.3.4.7 for more complicated solutions. 2 ä . The transformation 2

º (ö , è ) = ó ( Î , Ï ) exp ê − é ö + ë À + Ó é 2 ¸ èJç , Ì Ì

Ï = Ó

2Ì

Ð 2÷ Ò ,

2 Î = Ð ÷ Ò ë ö − Ó 2é ¸ , Ì

leads to a constant coefficient equation, » ø ó = » ô-ônó , which is considered in Subsection 1.1.1.

© 2002 by Chapman & Hall/CRC Page 93

3.

2

=Ä Á

Á Â Á Ã

Â

2

Á Å

+ (Æ

+É )Á Â Å

+ (à

+ù ) . Â

Å

Á Å

For Ì = 0, see equation 1.3.3.1. For Ì ≠ 0, the substitution Î = ö + ò Ì leads to an equation of the é form 1.3.3.2: » º

» è

4.

= Á

Á Â

2

Â

2

+ 2( Ä

» 2º

=

Ó » Î

+Æ )Á Â Å

» º

+ ÌÎ

2

+ ( À Î + ú )º ,

» Î 2

+ (Ä

2

+ 2Ä Æ

Å

ú =Ð − é

À

Ì

.

+É ) . Â

Å

Á Ã Á Å Å This equation is a special caseÁ of equation 1.8.6.5 and a special case of equation 1.8.7.5. The substitution º (ö , è ) = ó (ö , è ) exp û − 21 ö 2 − Ì-ö ü leads to a constant coefficient equation of the form Ó 1.1.3 with ý ≡ 0, namely, » ó = » þ þ ó + ( − − Ì 2 )ó . Ò

5.

= Á

Á Â

2

Â

2

+ (Ä

é

+Æ )Á Â Å

Ó

2

+ (É

+à

Å

+ù ) . Â

Å

Á Ã Á Å This equation is a special case Á ofÅ equation 1.8.6.5 and a special case of equation 1.8.7.5.

1 ä . The substitution

º (ö , è ) = ó (ö , è ) exp û 12 å ö 2 ü , 2

where å is a root of the quadratic equation å » ó

» 2ó

=

» è

+ ÿ (2 å » ö 2

+ å + = 0, yields an equation of the form 1.8.7.4, Ó

é

» ó

+ )ö + Ì
Ó

» ö

+ ÿ ( å Ì + À )ö + å + Ðó ,

which is reduced to a constant coefficient equation. 2 ä . The substitution

º (ö , è ) = ó (ö , è ) exp û 12 å ö 2 +  ö +  è°ü

leads to an equation of the analogous form:  



 2 ó

ó è

= 

+ ÿ (2 å ö 2

+ )ö + 2  + Ì Ó

+ ÿ (å

2



+å

ó ö

2

+ )ö é

Ó

+ (2 å  + å Ì +  Ó

+ À )ö + 

2

+ Ì +å − +Ð ó .

By appropriately choosing the coefficients å ,  , and  , one can simplify the original equation in various ways. 6.

= Á

Á Â

2

Â

2

+ (Ä

+Æ Å

Ã

+É )Á Â

+ (s

Á Å Á Å This isÁ Ã a special case of equation 1.8.7.5. 

7.

=





2

 2

+ (





+ 





)

2

Å

+s





+



ÃÅ

2

+ Ã

+

+ Å

Ã

+

) .

.

This is a special case of equation 1.8.3.6 with  (è ) = -è , õ (è ) = è , and â (è ) = ãè . 8.



=

2

 2





+ 

1 . The transformation



 



+ ë  +

é

 

2

.

!



(ö , è ) = " #%¢ö$ & ó (ö , ' ),

where ) is a root of the quadratic equation ( ) 1.2.5:   

ó

'

= 

2

ó

ö 2

2

=( è, '

+ (  − ( ) ) + ú = 0, leads to an equation of the form

+ ë 2) + (







ö

1 ó 

ö

.

© 2002 by Chapman & Hall/CRC Page 94

!

2 . If (ö , è ) is a solution of the original equation, then the functions !

=* " #

1

!

where * and  

!

(, ( ö , ( 2 ' ),

exp ë −

2

ö

=è + , '

+ ' + 2é

4( '

é

(

 !

ö

ë , '

,−

( '



1

,

( 2'

= )

 1 , − 2 2(

are arbitrary constants, are also solutions of this equation.

=



−1

=* ' &

2

9.

(1− + 2 ) ø



2

 2



+





The transformation





 

+ ë  +

+ - ( , ). 

2



 !

(ö , è ) = " #%$ ö & ó (ö , ' ),

=( è, '

2

where ) is a root of the quadratic equation ( ) + (  − ( ) ) + ú = 0, leads to an equation of the form 1.2.6:     2 ó  1 ó 1 − − ó   =  2 + ë 2) + + . (ö , ' ), . (ö , ' ) = " #%¢ö$ & ý ( ö , è ). ö

'

ö

(

ö

(

1.3.4. Equations of the Form 1.



=

2





= (2

/ 0

2

+4 )/ 3

/ 1  2

+ 5 (3 , 1 ) /

0 3 2

/

0 /

3

+ 6 ( 3 , 1 )0

.

 This is a special case of equation 1.8.6.1 with  (7 ) = ( 7 . See also equation 1.3.6.6 with 8 = 0.

1 . Particular! solutions ( * , 9 , : , and ; are arbitrary constants): (7 ) = * 7 + 9 , ! 2

(7 , è ) = 2 * ( è 7 −

, ,

+: ,

+9 E

7

+: , (( è + 9 ( è +9 E (7 , è ) = * ( è + * (7 ln 7 − 7 ) + 9 , ! (7 , è ) = * ( 2 è 2 + 2 * ( è (7 ln 7 − 7 ) + * (7 2 ln 7 − 52 7 2 ) + 9 , !

(7 , è ) =

+9 ,

[8 (8 − 1) @A@A@ (8 − B )]2 (( è ) 7 C 8 (8 − B ) B !

=1

!

3

!

U

)2

exp D −

(7 , è ) = " F $HG 7 IKJ L

1M

(R , V ) = W F X G R IKJ Y

1M

2

G

G N

2

−; 7 O + P Q

G T R O

G N

+P Z

2

1M 1M

G

G N

2 G N

where L 1 ( [ ) and Q 1 ( [ ) are the Bessel functions, Y 1 ( [ ) and Z

−; R O S

for T < 0, for T > 0,

G T R O S 1([

) are the modified Bessel functions.

2 \ . A solution containing an arbitrary function of the space variable: U

=

(R , V ) = ] (R ) + ^ C

=1

(N V )C C [ ] (R )], _ ! ` `

2

≡R a

a 2, R

where ] (R ) is any infinitely differentiable function. This solution satisfies the initial condition U (R , 0) = ] (R ). The sum is finite if ] (R ) is a polynomial.

© 2002 by Chapman & Hall/CRC Page 95

3 \ . A solution containing an arbitrary function of time: =

U

(R , V ) = J + R b (V ) + ^ =2

C

1 [( − 1)!]2 N C _

_

−1

( −1)

R C b C X

(V ),

where b (V ) is any infinitely differentiable function and J is an arbitrary number. This solution possesses the properties U U (0, V ) = J , c d (0, V ) = b (V ). 4 \ . Suppose U = U (R , V ) is a solution of the original equation. Then the functions U U

1

= J U ( e R , e V + f ),

2

= J exp I − N

R g

(h + V )

S

U

R M

(h +

g

g

V )2

, i

+e V O , h + V

e h

−

g

g i

= 1,

where J , f , , h , and e are arbitrary constants, are also solutions of the equation. g

2.

2

=m n j

j k j l

j

k n 2

+ (– o n + p ) . k

The transformation U

(R , V ) = q ( [ , r ) exp MAs N

+ uvV 0), (œ > 0).

© 2002 by Chapman & Hall/CRC Page 105

The solution of the original equation subject to these boundary conditions is given by ­

(œ ,  ) = 1 − ›

®

( ) = |

¨ ‡

=1 |

exp ¡ − Ÿ |

1  exp ¡ − ‰ 2 |

2 

2 ‰

3 ¡ 4

¢ £

œ ¢¤®

|

1 4‰ |

−

( ), |

,

(1)

3 2; ‰ |

2 

¢

,

where the function ® and the coefficients ‡ and ‰ are independent of the parameter Ÿ . | | | The eigenvalues ‰ are solutions of the transcendental equation |

‰

3 4 ¡

£ |

The series coefficients ‡

0 ¯ 1

= |

0 ¯

3 2; ‰ |

,

(1 −  2 ) ®

( ) ° |

2

(1 −  ) ±²® |

Eigenvalues ‰

1 2 3 4 5

‰

(  )³

2

exp ¡ −Œ

,

1 2; ‰ |

= 0. ¢

œ

¢ ° Œ

Š

where

and coefficients ‡ |

= 1, 2,

ª¤ª¤ª

° 

‡ |

|

.

in solution (1) ‰

|

6 7 8 9 10

‡ |

22.3181 26.3197 30.3209 34.3219 38.3227

|

−0.1873 0.1631 −0.1449 0.1306 −0.1191

0 is given by ´

›

¯

,

1.3382 −0.5455 0.3589 −0.2721 0.2211

The solution asymptotics as Ÿ

¶ ¨

1 4‰ |

−

Š ‡

2.2631 6.2977 10.3077 14.3128 18.3159

where erfc § =

3 4 ¡

TABLE 16 and coefficients |

|



2

Table 16 shows the first ten eigenvalues ‰

Š

−£ ¢

are calculated from |

1 ‡

1 4‰ |

−



= erfc ~

2µ

Ÿ œ 

,

is the complementary error function.

1.3.8-3. Dissolution of a plate by a laminar fluid film. The dissolution of a plate by a laminar fluid film, provided that the concentration at the solid surface is constant and there is no mass flux from the film into the gas, satisfies the boundary conditions ›

« ¬

› ›

= 0 at = 0 at 

=0 =0

(0 <  < 1), (œ > 0),



=1

(œ > 0).

œ

= 1 at

The solution of the original equation subject to these boundary conditions is given by (œ ,  ) = 1 − ›

· |

( ) =

­ ¨

‡

=0 | exp ¡ − 12 ‰ |

exp ¡ − Ÿ | 

2

¢ £

¡

1 4

‰

2 |

−

œ ¢¤·

1 4‰ |

|

,

( ), 1 2; ‰ | 

(2) 2 ¢

,

© 2002 by Chapman & Hall/CRC Page 106

where the functions · The eigenvalues ‰

and the constants ‡ and ‰ are independent of the parameter Ÿ . | | are solutions of the transcendental equation |

|

1 4 ¡

£

1 4‰ |

−

1 2; ‰ |

,

= 0. ¢

The following approximate relation is convenient to calculate ‰ ‰

=4 |

Š

+ 1.68

(

Š

: |

= 0, 1, 2,

ª¤ª¤ª

).

(3)

The maximum error of this formula is less than 0.2%. The coefficients ‡ are approximated by |

‡

0

= 1.2, ‡

−7 ¸ 6

= (−1) | 2.27 ‰ |

Š

for

|

= 1, 2, 3,

ª¤ª¤ª

,

where the eigenvalues ‰ are defined by (3). The maximum error of the expressions for ‡ | than 0.1%. The solution asymptotics as Ÿ œ ´ 0 is given by 1 ¹ 1 2 (1 −  )3 ¡ ¢ , › = ¹ 1 , = , 3 9º º where ( ¥ , § ) = function, and

¹

1¯ 3¢

¡

¾²¿

¶ ¨

−¼ »

Œ ½

−1

is less

Ÿ œ

3¢

¡

¹

|

¹

¹

is the incomplete gamma function, ( ¥ ) = ( ¥ , 0) is the gamma

° Œ

≈ 2.679.

References for Subsection 1.3.8: Z. Rotem and J. E. Neilson (1966), E. J. Davis (1973), A. D. Polyanin, A. M. Kutepov, A. V. Vyazmin, and D. A. Kazenin (2001).

1.3.9. Equations of the Form p ( q , ™ ) À À

1.

Å Æ

Ç

È

+

É Æ

È

Ê Æ

Ë Ì

È

2

=

É Ì

È

È

É Ì 2 È

+ à ( , ™ ) À

Á Â

Á À

= ™

2 À À

Á

™

2

+ Ä ( , ™ )

.

This is a special case of equation 1.9.1.1 with Í (œ ) = Ÿ œ Î and Ï (œ ) = Ðkœ Ñ . The transformation Ò Ð 1 2Ð − +1 Ù ° − +1 Ù Ñ Î Ñ Î = exp Ô − Õ Ú = Û exp Ô − Õ Ø , Ÿ Ó ( Ö − × + 1) Ø ( Ö − × + 1) Ø Î Ø leads to a constant coefficient equation, Ü Ý]Þ = Ü ßkß¤Þ , which is considered in Subsection 1.1.1. 2.

Å à

á

â

ã â

+ à

ä à

å æ

â

ã â

2

= æ

â

–ç

ã æ 2 â

à è ã

. Õ

This is a special case of equation 1.9.1.2 with é ( ) = 3.

Å à

è

æ á

–1 â

ã â

+ à

ä à

å æ á

â

ã â

= æ

Ø

2 â

ã æ 2 â

Ø ê

ñ

=

Ü

4. Å

ó

ô

á æ à

õ

â â

ã à

+ ô

ä à

ó

ô

å æ à

õ

â â

ã æ

Õ

ì

, and î ( ) = ï

Ø

Ø í

=

2 â â

ã æ 2

Ø

= Û exp Ô − Õ Ú

2 Þ

Ü Ú 2 ê

and ë ( ) =

Ø ð

Ü

1− Ú

Ø ð

.

ì

.

Ø í ì

(Ö − ò

+ 1) Ø

− í

+1 Ù ð

.

.

This is a special case of equation 1.9.1.4 with é ( Ú ) = ö

Ø

.

This is a special case of equation 1.9.1.3 with é ( ) = Ø The transformation ñ 1 ì ( × + 1) − +1 Ù ° = Õ exp Ô − Õ Ø , ( Ö − ò + 1) Ø í ð Ó Ø ð leads to a simpler equation of the form 1.3.6.6: Ü Þ

,ë ( )=

Õ Ú

ê

See Subsection 1.9.1 for other equations of this form.

and ë ( Ú ) =

ì0Ú í

.

© 2002 by Chapman & Hall/CRC Page 107

1.4. Equations Containing Exponential Functions and Arbitrary Parameters 1.4.1. Equations of the Form ÷

= ø

÷

1.

2

=

â

ã â

Å

â

+

ã

ý

â

à 2




þ]ä ÿ



+ç

2 ÷

ù

÷

. ã

ú

ø

+ ü ( û , ù )ø

2

û

ñ Ý

This is a special case of equation 1.8.1.1 with é ( ) = ì  

+ï .

1  . Particular solutions (  ,  , and are arbitrary constants): ñ

ñ

( , ) = ( Þ

Ø

Ø

Þ

Ø

Ü Ý

=

2. â

Ü

â

Ý

+ 2 Õ ) exp ó   ì

2

+Õ Ø

,

+ï

ñ

ñ

õ ñ

Ý

+  

,

+ï

Ý

+

Ø

. õ

ñ

ì

Ø

õ ñ

leads to a constant coefficient equation, ï

   , which is considered in Subsection 1.1.1. 2

= ã

ì

2 Ø

( , ) =  ( , ) exp   Þ

+ï

ñ

( , ) =  exp ó

ñ

Õ

ñ

Ý

+  ) exp ó   Ø

ñ

( , )= ( Þ

2  . The substitution

ì

Å

â â

à 2

ý




+ (ä

ã

ÿ

+

) .

ç à

ñ

ã

ñ Ý

This is a special case of equation 1.8.1.6 with é ( ) = ï and ë ( ) = ì   . 1  . Particular solutions (  and are arbitrary constants): ñ

ñ

( , ) =  exp ó Þ

Ø

Ø

ñ

þ

+ Ø

Õ ï



2

exp ó ï

ñ ñ

( , ) =  ( Ú , ) exp ó Þ

ï

Ø

ì

+   Ø

Õ

leads to a constant coefficient equation, Ü Ý = 3. â

=

â

ã ý

2 Å

â â

ã à 2

+ (ä

 –ç ) ÿ

ã

Ý

1 + Õ ñ3

Ý

1 + Õ 3 Ý

Ø

2  . The transformation ñ

, õ

+   Ø

ñ

( , ) =  exp Ô ( ï + ) +   Ø

ï

2 3

ì

ì

Þ

ñ

1 + Õ ñ 3 Ý

+  

Øñ

ñ

( , )= Þ

ï

ì

2 3 ï

ñ ï

2 3

,

õ ñ

ñ

2

Õ

+ ï

Õ

+

2 Ù

ñ ñ

1 + Õ 3 ï

.

2 3 õ

,

= Ú

+Õ Ø

2 ï

, which is considered in Subsection 1.1.1.

Ü ßkß

. 

This is a special case of equation 1.8.1.2 with é ( ) = ì  − ï . Ø Particular solutions (  and are arbitrary constants): ñ

ñ

( , ) =  exp(− )   Þ



Ø

ñ

ó



 

  2 ,

 =

  

  2 ,

 =

ì Õ



ñ

( , ) =  exp(− )   Ø

2

ó

2 ì



õ

õ

2 2

− ï

 

ï



− 

, ,

where   ( ) and   ( ) are the Bessel functions of the first and second kind, respectively.

© 2002 by Chapman & Hall/CRC Page 108

4.

2

=

  





Å

 

ý

+(

2


ÿ



+! ) . 

This is a special case of equation 1.8.1.6 with " (# ) = $  % and & (# ) = ' . 1  . Particular solutions (  and are arbitrary constants): 

( , # ) =  exp ( )

$

Ø



( ,# ) = . ( Ø



*+ Ø



Ø

2 $ + ) 2

( , # ) = . exp / Ø

$ 2 ) 3 2 % + '#-, , 2 * +

%,

exp ( )

* + $

( ) Ø



% +

% +0 ,

*+



$ 2 ) 3 2 % + '#-, , 2 * + Ø * +  2 2  $10 $ % + (  0 2 + ' )#32 . + ) 3 2 % + ) 2 2 * + *+ $

% +

2 4 . The transformation 

( , # ) = 5 ( 6 , # ) exp ( )

$

Ø

*+ Ø

% +



$ 2 ) 3 2 % + '#-, , 2 * +

6 = Ø

2 $ + ) 2

*+

%

leads to a constant coefficient equation, 7 5 =  7 8985 , which is considered in Subsection 1.1.1. %

5.

 

2

=





Å

 :

 

2

+ ( ; < = +! ) . 



This is a special case of equation 1.8.1.3 with " (# ) = $

* +

% +' .

1 4 . Particular solutions ( . and 0 are arbitrary constants): >

( , # ) = . exp / Ø

>

( ,# ) = . / Ø

>

Ø

Ø

where @ (# ) =

1 2 ) −3 2 2$ *

+

Ø

+? ( )

( , # ) = . exp / % + 2 $'

$

( )

Ø

% + '#-,

*+

2$

( )

2

% + '# 2 , 2 exp /

* +

$

% + '# + 0 ,

*+

) −3 ) ( # − 1)

2 4 . The transformation >

( , # ) = 5 ( 6 , # ) exp / Ø

Ø

( )

+ ? @ ( # )2 ,

$ * +

* +

Ø

( )

+? 0 ( )

$

% + '#-,

* +

2$ 2

+ ? @ ( # )2 ,

% + '# 2 + 0 #-,

*+

+ ? @ ( # )2 ,

% + 13 ' 2 # 3 .

% + '# ,

+ ? @ ( # )2 ,

6 = Ø

+? ( )

2$ 2

* +

% + '# 2 ,

leads to a constant coefficient equation, 7 5 = ? 7 8985 , which is considered in Subsection 1.1.1. %

6.

 

2

=





Å

 :

 

2

+

2



( ; < = +! ) . 

This is a special case of equation 1.8.7.5 with A (# ) = ? , " (# ) = & (# ) = B (# ) = C (# ) = 0, and D (# ) = $ 7.

   :

=

2 Å



  

2

* +

% +' .

+( ; < = +! ; E = ) . 

This is a special case of equation 1.8.1.1 with " (# ) = $

* +

% +'

* F

%.

© 2002 by Chapman & Hall/CRC Page 109

1 4 . Particular solutions ( . , G , and H are arbitrary constants): >

( , # ) = (. Ø

>

( ,# ) = . ( Ø

>

2 Ø

Ø

$ 2 4 . The substitution > ( , # ) = 5 ( , # ) exp ( ) 7 %

8.

% +

* +

'

% +

* +

Ø

$

% , ,

' 0 * F

% +

*+

% , ,

'

% , .

0 * F

leads to a constant coefficient equation,

% ,

0 * F

0 * F

5 = ? 7   5 , which is considered in Subsection 1.1.1.  

2

=





Å

 :

+( ; <  +! ; E = +I ) .

 

2

>

The substitution

( , # ) = 5 ( , # ) exp J Ø



'

O P

2

=

O

P

Å

7 25

=?

7 L

leads to an equation of the form 1.4.1.3:

%-K

0 * F

Ø

7 5

9.

$

+ ? H 2# + ) Ø

'

% +

*+

+ 2 ? # ) exp ( )

( , # ) = . exp ( H

Ø

$

+ G ) exp ( ) Ø



+ (M

2

7

+ ° )5 .

*N+

Ø

+ (R ; <  +E = + S ) .

2

P

) Q For O =: ) 0, see Oequation 1.4.1.1; for 0 = 0, see equation 1.4.1.3. For ≠ 0, the transformation >

( , L ) = 5 (6 , L ) Ø

 * T

6 =

,

0

+ ) L, Ø

where U

=

0

2?

) ,

leads to an equation of the form 1.4.1.3: 7 5

=?

7 L

7 25

1.4.2. Equations of the Form 1.

O P

2

=

O :

Å

O

P O Q

2

+ (R ; < = + S ) O P

8

+ (M 7 6 2 *N+

=

W X W Y

Z

+ V + ? U 2 )5 . 2 W

W [

+ \ ([ , Y ) W

X 2

X W [

.

O Q

This is a special case of equation 1.8.2.1 with ] (L ) = M

+V .

* + ^

1 4 . Particular solutions ( . , G , and 0 are arbitrary constants): >

( ,L ) = . Ø

>

( ,L ) = . ( Ø

>

equation, 7

>

= ? 7 898

>

Ø

Ø

+ )

( , L ) = . exp /_0 Ø

2 4 . On passing from L ,

+. ( ) Ø

M

M *+ ^ Ø

+ VL-, + G ,

* + ^

2

+ VL-,

+0 )

M *+ ^

to the new variables L , 6 = Ø

+ 2? . L + G ,

+ (? 0 + )

2

M *+ ^

+ V0 )L32 + G . + VL , we obtain a constant coefficient

, which is considered in Subsection 1.1.1.

^

© 2002 by Chapman & Hall/CRC Page 110

2.

O P

2

=

O

P

Å

O :

+ (R ; <  + S ) O P

2

O Q

.

O Q



This is a special case of equation 1.8.2.2 with ] ( ) = M

+V .

* +

Ø

1 4 . Particular solutions ( . and 0 are arbitrary constants): >

)

V

( , L ) = . exp(− 0 L + ` >

( , L ) = . exp(− 0 L + ` Ø

M

) a ( ` , 2` + 1 + ) ; − ) ?

Ø

Ø

)

?

V

Ø

M

) b ( ` , 2` + 1 + ) ; − ) ?

)



?

, ,

*+  * +

(1) , ,

)

where ` = ` ( 0 ) is a root of the quadratic equation ? 2 ` 2 + V ` + 0 = 0; a ( c , H ; 6 ) and b ( c , H ; 6 ) are degenerate hypergeometric functions. [Regarding the degenerate hypergeometric functions, see Supplement A.9 and the books by Abramowitz and Stegun (1964) and Bateman and Erd e´ lyi (1953, Vol. 1)]. d e9f

g hji k

In solutions (1), the parameter ` can be considered arbitrary,and then 0 = − ?

2 4 . Other particular solutions ( . and G >

( )=. +G l

are arbitrary constants):

m (n ) o n ,

Ø

>

(n , w ) = x q w + x l >

(n , w ) = x q wy (n ) + x l

) 2 2 ) ` −V ` .

m (n ) p l

M V m (n ) = exp p − q r st u − q n v , o n m (n )

m (n ) p l

v o n , y (n ) o n

v o n ,

m (n )

y (n ) = l

m (n ) o n .

3 z . The substitution { = s t u leads to an equation of the form 1.3.5.4: | > |

3.

 €

=

2

 Å

 :

€  

2

w

| 2> | > q r 2 2 | r q r | = { + { ( }{ + ~ + ) . { 2 {

+ (‚ ; < = + ƒ )  € 

.

 

This is a special case of equation 1.8.2.3 with „ (w ) = } s t … + ~ . 1 z . Particular solutions ( x , † , and ‡ are arbitrary constants): } m (w ) = exp p r  s t … + ~w-v ,

>

(n , w ) = x n m (w ) + † , >

(n , w ) = x n 2 m >

(n , w ) = x exp ˆ_‡ n m (w ) + q ‡

2

(w ) + 2 x q l

m 2 (w ) o w + † , 2

l

m 2 (w ) o w3‰ + † .

2 z . On passing from w , n to the new variables ( x is any number) Š

=l

m 2 (w ) o w + x ,

{ = n m (w ),

where

} m (w ) = exp p r s t … + ~w-v ,

for the function > (Š , { ) we obtain a constant coefficient equation, in Subsection 1.1.1.

| ‹ >

=q

| Œ9Œ >

, which is considered

© 2002 by Chapman & Hall/CRC Page 111

4.

 €

2

=



€

Å

 :

 

2

+ (‚ ; < = + ƒ ;  = )  €

.

 

This is a special case of equation 1.8.2.1 with „ (w ) = } s t … + ~ s Ž … . 1 z . Particular solutions ( x , † , and  are arbitrary constants): }

~

>

(n , w ) = x n + x p r s t … + s Ž … v + † , ‡ >

(n , w ) = x p n + r s t … + s Ž … v ‡ >

q 2 (n , w ) = x exp p  n +  r st … +  s Ž … +  -w v + † . ‡

}

2

~

}

+ 2q x w + † , ~

}

~

2 z . On passing from w , n to the new variables w , { = n + r st … + s Ž … , we obtain a constant coefficient ‡ | Œ9Œ > | equation, > = q , which is considered in Subsection 1.1.1. …

5.

 €

=

2

 Å

€

2

+ (‚ ; < = +  + ƒ )  €

.

 For r  =: 0, see equation 1.4.2.2; for ‡  = 0, see equation 1.4.2.1. For ‡ ≠ 0, the substitution { = n + (r ‘ ‡ )w leads to an equation of the form 1.4.2.2: | > | 2> | > Œ r q | = | 2 + p } s Ž +~ − v | . ‡ w { {

6.

 €

2

=’ 

 :

€

2

 

+ (‚ ; < = + ƒ 

) €

.

 

This is a special case of equation 1.8.1.6 with „ (w ) = ~ and “ (w ) = } s t … . 7.

 €

2

=’ 

 :

€  

2

+ (‚

; < = +ƒ ) € 

.

 

This is a special case of equation 1.8.1.6 with „ (w ) = } s t … and “ (w ) = ~ . 8.

 €

2

=’ 

 :

€  

2

+ (‚ 

; < = +ƒ ;  = ) €

.

 

This is a special case of equation 1.8.1.6 with „ (w ) = } s t … and “ (w ) = ~ s Ž … . 9.

 €

2

=’ 

 :

€  

2

+ (‚ ; < = + ƒ ;  = )  € 

.

 

This is a special case of equation 1.8.2.3 with „ (w ) = } s t … + ~ s Ž … . 1 z . Particular solutions ( x , † , and  are arbitrary constants): } ~ m (w ) = exp p r  s t … + s Ž … v , ‡

>

(n , w ) = x n m (w ) + † , >

(n , w ) = x n 2 m >

(n , w ) = x exp ˆ” n m (w ) + q 

2

(w ) + 2 x q l

m 2 (w ) o w + † , 2

l

m 2 (w ) o w3‰ + † .

2 z . On passing from w , n to the new variables ( x is any number) Š

=l

m 2 (w ) o w + x ,

{ = n m (w ),

where

} ~ m (w ) = exp p r  s t … + s Ž … v , ‡ | ‹ > q | 9Œ Œ >

for the function > (Š , { ) we obtain a constant coefficient equation, in Subsection 1.1.1.

=

, which is considered

© 2002 by Chapman & Hall/CRC Page 112

10.

 €

2

=’ 

 :

€  

2

+ p ƒ 

‚

; < = +

v

.

 €



 

This is a special case of equation 1.8.2.6 with „ (w ) = ~ s t … .

1.4.3. Equations of the Form 1.

 €

2

=’ 

€

 :   The substitution

˜

2 •

• ™

– 2

+ š (™ , — ) •

– • ™

+ › ( ™ , — )–

+ ( ƒ ; < = + s) .

+‚  €

2

=

• – • — €

 

}

>

} 2

~

(n , w ) = œ (n , w ) exp ˆ − q n + r st … + p  − q v w‰ 2 4 | q | leads to a constant coefficient equation, œ = œ , which is considered in Subsection 1.1.1. 2.

 €

=’ 

 :

€

+ ( ƒ ; <  + s) .

+‚  €

2

  >

€

 

(n , w ) = œ (n , w ) exp ž − 2Ÿ   n ¡ leads to an equation of the form 1.4.1.3:

The substitution

|

œ |

3.

 €

2

=’ 

u u

…

2

€

+ (ƒ £ ¤ ¥ + s£ ¦ ¥ ) .

+‚  €

2

 ¢   The substitution

|

2 œ } 2 = q | 2 + p ~ s t u +  − q v œ . 4 w n

€

§  

}

~

} 2



(n , w ) = œ (n , w ) exp p − q n + r st … + s ¨ … − q w-v 2  4 | q | leads to a constant coefficient equation, œ = œ , which is considered in Subsection 1.1.1. 4.

 €

=’ 

 ¢

€  

+ (ƒ £ ¤  + s£ ¦ ¥ ) .

+‚  €

§2

€

 

(n , w ) = œ (n , w ) exp ž © s ¨ … ¡ leads to an equation of the form 1.4.3.2:

The substitution

|

|

5.

 €

2

=’ 

 ¢

u u

…

2

€  

+ª

2



 €

¨

œ

=q | w

+ ƒ £ 2¤ ¥ €

 

|

| 2 œ n

2

+} |

§

n

+~ s t u œ .

.

On passing from w , n to the new variables ( x and † Š

œ

are any numbers)

x 2

= r s 2t … + † , 2

{ = x n sNt … ,

for the function (Š , { ) we obtain a constant coefficient equation of the form 1.1.3: § § |

| Š

6.

 €  ¢

=’

2 2 

€  

2

+ ž«’

1£

¤  + ‚ 1¡  €

§ | 2 q | −2 = + ~1x . { 2

+ ž«’

0£

2

¤  + ‚ 0£ ¤  + ƒ 0¡

 

€

.

The substitution { § = s t u leads to§ an equation of the form 1.3.5.4: § |

|

| | 2 r 2 2 | r q q r = 2 { + { ž 1{ + } 1 + 2 ¡ | { 2 {

w

q

+ž

q

§ 0{

2

+ } 0{ + ~ 0¡

.

© 2002 by Chapman & Hall/CRC Page 113

1.4.4. Equations of the Form 1.

 €

2

=’



€



+ ž-‚ £ ¤ ¥ + ƒ

2



¡

=

• – • —

˜

2 •

™ ¬

• ™

– 2

+ š (™ , — ) •

– • ™

+ › ( ™ , — )–

. €

 ¢   This is a special case of equation 1.8.8.1 with „ (w ) = q , “ (w ) = 0, ­ (w ) = ~ , and  (w ) = } s t … .

2.

 €

2

=’



€



+ ž‚

2

£ ¤ ¥ +ƒ ¡ 

. €

 ¢   This is a special case of equation 1.8.8.1 with „ (w ) = q , “ (w ) = 0, ­ (w ) = } s t … , and  (w ) = ~ .

3.

 €

2

=’

 

 ¢

€

+‚

2

 

 € 

+ ž«ƒ £ ¤ ¥ + ® ¡

 

. €

This is a special case of equation 1.8.8.1 with „ (w ) = q , “ (w ) = } , ­ (w ) = 0, and  (w ) = ~ s t … + o . 4.

 €

2

2

=’

€





2

+ ž‚ £ ¤ ¥ + ƒ ¡

. €

 ¢   This is a special case of equation 1.8.8.2 with „ (w ) = q , “ (w ) = 0, and ­ (w ) = } s t … + ~ .

5.

 €

=’

4



2



€

2

+ ž‚ £ ¤ ¥ +ƒ ¡ €

.

 ¢   This is a special case of equation 1.8.8.4 with „ (w ) = q and “ (w ) = } s t … + ~ .

6.

 €

=’

 ¢

2

€

 ¯



+

2

 

ž‚ £ ¤ ¥ + ƒ ¡  € 

.

 

This is a special case of equation 1.8.4.5 with „ (w ) = } s t … + ~ . 7.

 €

=’

2

 ¯



€

+‚

2



£ ¤ ¥  €

+ƒ £ ¦ ¥ €

.

 ¢     This is a special case of equation 1.8.8.7 with „ (w ) = q , “ (w ) = } s t … , and ­ (w ) = ~ s ¨ … .

1.4.5. Equations of the Form 1.

 €  ¢

=’ £ ¤  

2

€  

2

• – • —

=

˜ ° ± ²

•

2 • ™

– 2

+ š (™ , — ) •

– • ™

+ › ( ™ , — )–

.

This is a special case of equation 1.8.6.1 with „ (n ) = q s t u . Particular solutions ( x , § † , and  are arbitrary constants): (n ) = x n + † , q r 2 ž w + s −t u ¡ + † , q r 3 r w-n + n s −t u + 2 s −t u ¡ + † , § (n , w ) = x ž (n , w ) = x ž 2 q 2 r 4 w 2 + 4 q r 2 w s −t u + s −2t u ¡ + † , § 2 ´ (n , w ) = x exp(− w ) ³ 0 p r exp ž − 21 µ ¶ ·¸ , §

§ (n , w ) = x

¹

(¶ , w ) = º exp(− w ) »

0

¼ µ

2 ½

´

exp ž − 21 µ ¶ · ¸ ,

where ¾ 0 (¿ ) and » 0 (¿ ) are the Bessel functions.

© 2002 by Chapman & Hall/CRC Page 114

2.

À Á

2

=Ã Ä Å  À

À Â

Á

+Ç

À Æ

2

Á

.

1 È . Particular solutions ( º , É , and  are arbitrary constants): ¹

(¶ , w ) = Ê _Ÿ Ë ( º ¶ ´ + É ), (¶ , w ) = º Ê Ÿ_Ë ž ´ µ 2 w + Ê −Ì Í · + É Ê Ÿ_Ë , ¹ (¶ , w ) = º Ê Ÿ_Ë ž ´µ 3 w ¶ + µ ¶ Ê ´ −Ì Í + 2 Ê −Ì Í · + É Ê Ÿ_Ë , ¹ (¶ , w ) = º Ê Ÿ_Ë ž 2 2 µ 4 w 2 + 4 µ 2 wÊ −Ì Í + Ê −2Ì Í · , ¹

¹

(¶ , w ) = º exp(− w ) ¾ ¹

(¶ , w ) = º exp(− w ) »

0

0

2 ¼

exp ž − 12 µ ¶ ·¸ ,

½ µ

¼

 ´+ Î

µ

2

 ´+ Î

exp ž − 12 µ ¶ · ¸ ,

½

where ¾ 0 (¿ ) and » 0 (¿ ) are the Bessel functions. ´

2 È . The substitution ¹ (¶ , w ) = Ê Ÿ_ËÏ (¶ , w ) leads to an equation of the form 1.4.5.1: Ð Ï = Ê Ì Í Ð Í Í Ï . Ë 3.

À Á À Â

2

=Ã Ä Å  À

Á À Æ

+ (Ç Ä Ñ Ò + Ó ) .

2

Á

1 È . Particular solutions ( º , É ,  are arbitrary constants): Î

¹

Ê Ô Ë + Õw ¸ , (¶ ) = ( º ¶ + É ) exp ¼  ´

¹

(¶ , w ) = º ž µ ¹

(¶ , w ) = º ž µ ¹

(¶ , w ) = º ¹

(¶ , w ) = º exp Ö

´

¹

2

Î Ê Ô Ë + Õ w ¸ , w + Ê −Ì Í · exp ¼ 

Î Ê Ô Ë + Õw ¸ , w ¶ + µ ¶ Ê −Ì Í + 2 Ê −Ì Í · exp ¼  ´ ´ Î Ê Ô Ë + Õw ¸ , ž 2 2 µ 4 w 2 + 4 µ 2 wÊ −Ì Í + Ê −2Ì Í · exp ¼  3

(¶ , w ) = º exp Ö

Î

2

Ê Ô Ë + ( Õ −  )w3× ¾ 0 ¼

 Î 

µ

Ê Ô Ë + ( Õ −  )w × » 0 ¼

2 µ

´

exp ž − 21 µ ¶ ·¸ ,

´

exp ž − 21 µ ¶ ·¸ .

½ ½

¹ 2 È . The (¶ , w ) = exp ž Ÿ Ê Ô Ë + Õ w · Ï (¶ , w ) leads to an equation of the form 1.4.5.1: ´ substitution Ô Ì Í Ð Ï = Ê Ð Í Í Ï . Ë

4.

À Á À Â

=à ¼ Ä Å  À

2

Á À Æ

2

+Ø Ä Å  À Á

¸ .

À Æ

This equation describes heat transfer in a quiescent medium (solid body) in the case where thermal diffusivity is an exponential function of the coordinate. The equation can be rewritten in the divergence form Ð

¹

Ð w

´

=

Ð Ð ¶

Ì Í Ð ¼ Ê

¹

Ð ¶ ¸

that is more customary for applications.

© 2002 by Chapman & Hall/CRC Page 115

1 È . Particular solutions ( º , É , Ù , and  are arbitrary constants): ¹

(¶ , w ) = º exp ¼ − ¹

´

¹

(¶ , w ) = º

(¶ , w ) = 2 º ´

¹

´µ

Ê −Ì Í ´

+É , +Ù ¸

2w

µ

w − º (µ ¶ + 1) Ê −Ì Í + É , 2 −Ì Í +´ º Ê −2Ì Í + É , µ wÊ 2

(¶ , w ) = º ´ 2 µ 4 w 2 − 2 º µ 2´ w (µ ¶ + 1) Ê −Ì Í − º (µ ¶ + 52 ) Ê −2Ì Í + É , ¹ (¶ , w ) = º 2´ µ 4 w 2 Ê −Ì Í + º µ ´ 2 wÊ −2Ì Í + 16 º Ê −3´ Ì Í + É , ¹ 1 (¶ , w ) = 2 º 3 µ 6 w 3 Ê −Ì Í + 3 º 2 µ 4 w 2 Ê −2Ì Í + º µ 2 wÊ Ü −3Ì Ü Í + 12 º Ê −4Ì Í + É , ÝÛ Ú Ü −1

(¶ , w ) = Ê − Ú Ì Í + ¹

(¶ , w ) = Ê Ô Ë − Ì Í Ö º ¾

1

1 (¶ , w ) = Ê Ô Ë − 2 Ì Í Ö º å

1

( − 1) ááá ( − â )ã ( µ ( − â )â !

Þàß

=1

1 2

¹

´

2

¹

ß

ß

2 ä ß −´  ß ¼

2ä  ´ ¼

+É »

1 Ê −2Ì Í ¸

ä µ

Ê − Ì Í ¸ 1 2

ä µ

2

+É æ

1

2 ä −´  ¼

¼

1

where ¾ 1 ( ç ) and » 1 ( ç ) are the Bessel functions, å 1 ( ç ) and æ

w ) Ê ( − Ú )Ì Í ,

ä µ

2ä  ´ µ

1 Ê −2Ì Í ¸ ×

ä 1(ç

1 Ê −2Ì Í ¸ ×

for  < 0, for  > 0,

) are the modified Bessel functions.

2 È . A solution containing an arbitrary function of the space variable: ¹

´

( w )Ú Ú è ( ¶ )ã , ! ê

Û

(¶ , w ) = è (¶ ) + é

Ú =1

ê

≡ ë

ë ¶

Þ

Ì Í ¼ Ê

ë ¸ , ë ¶

ß where è (¶ ) is any infinitely differentiable function. This solution satisfies the initial condition ¹ (¶ , 0) = è (¶ ).

3 È . A solution containing an arbitrary function of time: ¹

Û

1 ´ [( − 1)!]2 ( µ 2 ) Ú

( ¶ , w ) = º + Ê −Ì Í ì ( w ) + é

Ú =2

−1

( Ú −1) Ê −Ì Ú Í ì (w ), Ë

ß ß ì where ì (w ) is any infinitely differentiable function. If (w ) is a polynomial, then the series has finitely many terms.

4 È . The transformation ( Ù ¹

1,

Ù 2 , and Ù 3 are any numbers)

(¶ , w ) = Ï (¿ , í ) exp Ö −

Ê −Ì Í ´ µ

2 (w

+Ù

1)

¿ =¶ −

× ,

1 µ

ln

Ù 2 , ( w + Ù 1 )2

í =Ù 3−

Ù 2 , w +Ù 1

leads to the same equation, up to the notation, ´

Ð Ï

=

Ð í

Ð Ð ¿

Ì î Ð Ï . ¼ Ê Ð ¿ ¸

5 È . The substitution ç = Ê −Ì Í leads to an equation of the form 1.3.4.1: Ð

¹

´

= µ Ð w

2

ç

¹ Ð 2

. Ð ç 2

6 È . A series solution of the original equation (under constant values of ¹ at the boundary and at the initial instant) can be found in Lykov (1967).

© 2002 by Chapman & Hall/CRC Page 116

5.

À Á

2

= Ã Ä 2Å  À

À Â

Á

+ ï Ã Ä Å  ðñï Ã Ø Ä Å  + Ç · À Á

2

À Æ

.

À Æ

This is a special case of equation 1.8.5.2 with è (w ) = Î and ì (w ) = 0. The substitution ¿ = leads to a constant coefficient equation of the form 1.1.4 with ò ≡ 0: ¹ Ð

Ð w

6.

À Á

2

= Ã Ä 2Å  À

À Â

Á

Ð ¿ 2

Ð

+Î

¹ Ð ¿

µ

(1− Ê −Ì Í )

.

+ï Ã Ä Å  ðï Ã Ø Ä Å  +Ç Ä Ñ Ò · À Á

2

À Æ

¹ Ð 2

=

1´ ä

+Ó Ä ó Ò

À Æ

Á

.

This is a special case of equation 1.8.5.2 with è (w ) = ÎÊ Ô Ë and ì (w ) = Õ1Ê ô Ë .

1.4.6. Other Equations 1.

À Á

2

=Ã Ä Å Ò À

À Â

Á

+Ç Ä Ñ Ò À Á

2

À Æ

+Ó Ä ó Ò

. Á

À Æ

´

This is a special case of equation 1.8.7.3 with è (w ) = Ê Ì Ë , ì (w ) = ÎÊ Ô Ë , and õ (w ) = Õ1Ê ô Ë . 2.

À Á

=Ã Ä Å Ò À

2

Á

+Ç

2

+Ó

Ä Ñ Ò À Á Æ

Ä ó Ò Æ

. Á

´

À Â À Æ À Æ This is a special case of equation 1.8.7.4 with (w ) = Ê Ì Ë , è (w ) = ÎÊ Ô Ë , ì (w ) = 0, õ (w ) = Õ1Ê ô Ë , and ö (w ) = 0. ß

3.

À Á

2

= Ã Ä Å  +Ñ Ò À

À Â

The transformation

Á À Æ

2

+Ç Ä ó Ò ¹

. Á

´ Î

(¶ , w ) = exp ¼ ÷ Ê ô Ë ¸ Ï (¶ , í ),

leads to an equation of the form 1.4.5.1: Ð

‹

í = 

Ê Ô Ë

Ï =Ê Ì Í Ð Í Í Ï .

1.5. Equations Containing Hyperbolic Functions and Arbitrary Parameters 1.5.1. Equations Containing a Hyperbolic Cosine 1.

À Á

=Ã À

À Â

2

Á À Æ

2

+ ( Ç coshø ù

+Ó ) . Á

Â

Ü ú

This is a special case of equation 1.8.1.1 with è (w ) = Î cosh 2.

À Á

=Ã À

À Â

2

Á À Æ

2

+ ( Ç coshø ù Â

+Ó Æ

w +Õ . Ü ú

) . Á

This is a special case of equation 1.8.1.6 with è (w ) = Õ and ì (w ) = Î cosh 3.

À Á À Â

=Ã À

2

Á À Æ

2

+ ( Ç cosh ø ù Æ

Â

+Ó ) . Á

This is a special case of equation 1.8.1.3 with è (w ) = Î cosh

w.

Ü ú w +Õ .

© 2002 by Chapman & Hall/CRC Page 117

4.

2

=Ã À

À Á À Â

Á

+ ( Ó cosh ø ù

2

À Æ

2

–Ç Â

Ü ú

) . Á

Æ

This is a special case of equation 1.8.1.7 with è (w ) = Õ cosh 5.

2

=Ã À

À Á À Â

Á

+ ( Ç coshø ù

2

À Æ

+ Ó )À Á Â

w. Ü ú

.

À Æ

This is a special case of equation 1.8.2.1 with è (w ) = Î cosh 6.

2

=Ã À

À Á À Â

Á

+ ( Ç cosh ø ù

2

À Æ

Æ

+Ó )À Á Â

w +Õ . Ü ú

.

À Æ

This is a special case of equation 1.8.2.3 with è (w ) = Î cosh 7.

2

=Ã À

À Á À Â

Á

+Ç À Á

2

À Æ

+ ( Ó cosh ø ù

Ü ú

+ s) . Á

Â

À Æ

w +Õ .

ö w + .

This is a special case of equation 1.8.3.1 with è (w ) = Î and ì (w ) = Õ cosh 8.

2

=Ã À

À Á À Â

Á

+ ¼ Ó

2

À Æ

cosh ø ù Æ

Ç

+ Â

À Á ¸

Æ

Ü ú

.

À Æ

This is a special case of equation 1.8.2.6 with è (w ) = Õ cosh 9.

=Ã

À Á À Â

2

À Æ

Á

+ (Ç

À Æ

2

cosh ø ù Æ

w. Ü ú

+Ó ) . Á

Â

w and ì (w ) = Õ .

This is a special case of equation 1.8.4.1 with è (w ) = Î cosh 10.

À Á

2

2

=Ã

À

Á

Æ

À Â

2

À Æ

+ ( Ç cosh ø ù

Ü ú

+Ó ) . Á

Â

This is a special case of equation 1.8.4.2 with è (w ) = Î cosh 11.

À Á

=Ã

À Â

4

Æ

2

À

Á À Æ

2

+ ( Ç cosh ø ù

Ü ú

+Ó ) . Á

Â

´

This is a special case of equation 1.8.8.4 with è (w ) = 12.

À Á

=Ã

À Â

2

À

Æ û

Á À Æ

2

+ ( Ç cosh ø ù Æ

+ Ó )À Á Â

w +Õ .

and ì (w ) = Î cosh

w +Õ .

Ü ú

.

À Æ

This is a special case of equation 1.8.4.5 with è (w ) = Î cosh

w +Õ .

1.5.2. Equations Containing a Hyperbolic Sine 1.

À Á

=Ã À

À Â

2

Á À Æ

2

+ ( Ç sinh ø ù

+Ó ) .

Ü ú

Á

Â

This is a special case of equation 1.8.1.1 with è (w ) = Î sinh 2.

À Á

=Ã À

À Â

2

Á À Æ

2

+ ( Ç sinh ø ù Â

+Ó Æ

w +Õ . Ü ú

) . Á

This is a special case of equation 1.8.1.6 with è (w ) = Õ and ì (w ) = Î sinh 3.

À Á À Â

=Ã À

2

Á À Æ

2

+ ( Ç sinh ø ù Æ

Â

+Ó ) .

w.

Ü ú

Á

This is a special case of equation 1.8.1.3 with è (w ) = Î sinh

w +Õ .

© 2002 by Chapman & Hall/CRC Page 118

4.

2

=Ã À

À Á À Â

Á

+ ( Ó sinh ø ù

2

À Æ

2

–Ç Â

Ü ú

) . Á

Æ

This is a special case of equation 1.8.1.7 with è (w ) = Õ sinh 5.

2

=Ã À

À Á À Â

Á

+ ( Ç sinh ø ù

2

À Æ

+ Ó )À Á Â

w. Ü ú

.

À Æ

This is a special case of equation 1.8.2.1 with è (w ) = Î sinh 6.

2

=Ã À

À Á À Â

Á

+ ( Ç sinh ø ù

2

À Æ

Æ

+Ó )À Á Â

w +Õ . Ü ú

.

À Æ

This is a special case of equation 1.8.2.3 with è (w ) = Î sinh 7.

2

=Ã À

À Á À Â

Á

+Ç À Á

2

À Æ

+ ( Ó sinh ø ù

Ü ú

+ s) . Á

Â

À Æ

w +Õ .

ö w + .

This is a special case of equation 1.8.3.1 with è (w ) = Î and ì (w ) = Õ sinh 8.

2

=Ã À

À Á À Â

Á

+ ¼ Ó

2

À Æ

sinh ø ù Æ

Ç

+ Â

À Á ¸

Æ

Ü ú

.

À Æ

This is a special case of equation 1.8.2.6 with è (w ) = Õ sinh 9.

=Ã

À Á À Â

2

À Æ

Á

+ (Ç

À Æ

2

sinh ø ù Æ

w. Ü ú

+Ó ) . Á

Â

w and ì (w ) = Õ .

This is a special case of equation 1.8.4.1 with è (w ) = Î sinh 10.

À Á

2

2

=Ã

À

Á

Æ

À Â

2

À Æ

+ ( Ç sinh ø ù

Ü ú

+Ó ) . Á

Â

This is a special case of equation 1.8.4.2 with è (w ) = Î sinh 11.

À Á

=Ã

À Â

4

Æ

2

À

Á À Æ

2

+ ( Ç sinh ø ù

w +Õ . Ü ú

+Ó ) . Á

Â

This is a special case of equation 1.8.8.4 with è (w ) = ü and ì (w ) = Î sinh 12.

À Á

=Ã

À Â

2

À

Æ û

Á À Æ

2

+ ( Ç sinh ø ù Æ

+ Ó )À Á Â

.

w +Õ .

Ü ú

À Æ

This is a special case of equation 1.8.4.5 with è (w ) = Î sinh

w +Õ .

1.5.3. Equations Containing a Hyperbolic Tangent 1.

À Á

=Ã À

À Â

2

Á À Æ

2

+ ( Ç tanh ø ù

+Ó ) .

Ü ú

Á

Â

This is a special case of equation 1.8.1.1 with è (w ) = Î tanh 2.

À Á

=Ã À

À Â

2

Á À Æ

2

+ ( Ç tanh ø ù Â

+Ó Æ

w +Õ . Ü ú

) . Á

This is a special case of equation 1.8.1.6 with è (w ) = Õ and ì (w ) = Î tanh 3.

À Á À Â

=Ã À

2

Á À Æ

2

+ ( Ç tanh ø ù Æ

Â

+Ó ) .

w.

Ü ú

Á

This is a special case of equation 1.8.1.3 with è (w ) = Î tanh

w +Õ .

© 2002 by Chapman & Hall/CRC Page 119

4.

2

=Ã À

À Á À Â

Á

+ ( Ó tanh ø ù

2

À Æ

2

–Ç Â

Ü ú

) . Á

Æ

This is a special case of equation 1.8.1.7 with è (w ) = Õ tanh 5.

2

=Ã À

À Á À Â

Á

+ ( Ç tanh ø ù

2

À Æ

+Ó )À Á Â

w. Ü ú

.

À Æ

This is a special case of equation 1.8.2.1 with è (w ) = Î tanh 6.

2

=Ã À

À Á À Â

Á

+ ( Ç tanh ø ù

2

À Æ

Æ

+Ó )À Á Â

w +Õ . Ü ú

.

À Æ

This is a special case of equation 1.8.2.3 with è (w ) = Î tanh 7.

2

=Ã À

À Á À Â

Á

+Ç À Á

2

À Æ

+ ( Ó tanh ø ù

Ü ú

+ s) . Á

Â

À Æ

w +Õ .

ö w + .

This is a special case of equation 1.8.3.1 with è (w ) = Î and ì (w ) = Õ tanh 8.

2

=Ã À

À Á À Â

Á

+ ¼ Ó

2

À Æ

tanh ø ù Æ

Ç

+ Â

À Á ¸

Æ

Ü ú

.

À Æ

This is a special case of equation 1.8.2.6 with è (w ) = Õ tanh 9.

=Ã

À Á À Â

2

À Æ

Á

+ (Ç

À Æ

2

tanh ø ù Æ

w. Ü ú

+Ó ) . Á

Â

w and ì (w ) = Õ .

This is a special case of equation 1.8.4.1 with è (w ) = Î tanh 10.

À Á

2

2

=Ã

À

Á

Æ

À Â

2

À Æ

+ ( Ç tanh ø ù

Ü ú

+Ó ) . Á

Â

This is a special case of equation 1.8.4.2 with è (w ) = Î tanh 11.

À Á

=Ã

À Â

4

Æ

2

À

Á À Æ

2

+ ( Ç tanh ø ù

w +Õ . Ü ú

+Ó ) . Á

Â

This is a special case of equation 1.8.8.4 with è (w ) = ü and ì (w ) = Î tanh 12.

À Á

=Ã

À Â

2

À

Æ û

Á À Æ

2

+ ( Ç tanh ø ù Æ

+Ó )À Á Â

.

w +Õ .

Ü ú

À Æ

This is a special case of equation 1.8.4.5 with è (w ) = Î tanh

w +Õ .

1.5.4. Equations Containing a Hyperbolic Cotangent 1.

À Á

=Ã À

À Â

2

Á À Æ

2

+ ( Ç cothø ù

+Ó ) .

Ü ú

Á

Â

This is a special case of equation 1.8.1.1 with è (w ) = Î coth 2.

À Á

=Ã À

À Â

2

Á À Æ

2

+ ( Ç cothø ù Â

+Ó Æ

w +Õ . Ü ú

) . Á

This is a special case of equation 1.8.1.6 with è (w ) = Õ and ì (w ) = Î coth 3.

À Á À Â

=Ã À

2

Á À Æ

2

+ ( Ç cothø ù Æ

Â

+Ó ) .

w.

Ü ú

Á

This is a special case of equation 1.8.1.3 with è (w ) = Î coth

w +Õ .

© 2002 by Chapman & Hall/CRC Page 120




4.

2


= 



2

+ (  coth

  

–

 2


) .

This is a special case of equation 1.8.1.7 with ( ) =  coth .


5.

2


= 



2

+ (  coth




  

+ )

. 

This is a special case of equation 1.8.2.1 with ( ) =  coth +  .


6.

2


= 





2

+ (  coth




  

+ )

. 

This is a special case of equation 1.8.2.3 with ( ) =  coth +  .


7.

= 




2




2

+

+ (  coth 

  




+ s) .

This is a special case of equation 1.8.3.1 with ( ) =  and  ( ) =  coth +  .


8.

2


= 



 2

+

 

coth

  




+  

. 

This is a special case of equation 1.8.2.6 with ( ) =  coth .


9.

= 

2






+ (

2



coth

  




+ ) .

This is a special case of equation 1.8.4.1 with ( ) =  coth and  ( ) =  .


10.

= 

2


 2



2

+ (  coth

  




+ ) .

This is a special case of equation 1.8.4.2 with ( ) =  coth +  .


11.

= 

2


 4 



2

+ (  coth

  




+ ) .

This is a special case of equation 1.8.8.4 with ( ) =  and  ( ) =  coth +  .


12. 

=

2


 





2

+ (  coth

  




+ ) 

.

This is a special case of equation 1.8.4.5 with ( ) =  coth +  .

1.6. Equations Containing Logarithmic Functions and Arbitrary Parameters 1.6.1. Equations of the Form


1. 

=

2






2

   

= 



2  

 2

+  ( ,  ) 

  

+  (  ,  )




+ (  ln +  ) .

This is a special case of equation 1.8.1.1 with ( ) =  ln +  . The! substitution  ( , ) =  ( , ) exp(  ln −  +  ) leads to the constant coefficient equation ! "  =  #2 #  , which is considered in Subsection 1.1.1.

© 2002 by Chapman & Hall/CRC Page 121




2.

2


= 



+ (

2






+  ln ) .

This is a special case of equation 1.8.1.6 with ( ) =  and  ( ) =  ln .


3.

2


= 





+ (  ln

2

 




+ ) .

This is a special case of equation 1.8.1.3 with ( ) =  ln +  .


4.

2


= 



 2

+ (– 

2

+  ln

 


) .

This is a special case of equation 1.8.1.7 with ( ) =  ln .


5.

2


= 






+ (  ln +  )

2

. 

The change of variable $ =  +  ln −  +  leads to the constant coefficient equation that is considered in Subsection 1.1.1.


6.

2


= 







=

! 2 %% 




+ (  ln +  )

2

! "

. 

This is a special case of equation 1.8.2.3 with ( ) =  ln +  . On passing from ,  to the new variables ( & and ' are any numbers) (

=)

* 2 (

)+ +& , $

=  * ( ),

we arrive at the constant coefficient equation in Subsection 1.1.1.

1.6.2. Equations of the Form


1.

2




= 





! ,

= 

=

   



!

( ) = ' exp(  ln −  +  ), *

for  (( , $ ); this equation is considered

%% 

 -



2  

 2

+  ( ,  ) 

  

+  (  ,  )




+ (

2

where

+  ln ) .

This is a special case of equation 1.8.8.1 with ( ) =  ,  ( ) = 0, . ( ) =  , and  ( ) =  ln .


2.

2






=





+ (

2






ln +  ) .

This is a special case of equation 1.8.8.1 with ( ) =  ,  ( ) = 0, . ( ) =  ln , and  ( ) =  .


3.

= 

2






+

2









+ (  ln + / ) . 

This is a special case of equation 1.8.8.1 with ( ) =  ,  ( ) =  , . ( ) = 0, and  ( ) =  ln + + .


4. 

=

 2 

2






2




+ (  ln +  ) .

This is a special case of equation 1.8.8.2 with ( ) =  ,  ( ) = 0, and . ( ) =  ln +  .


5. 

=

 2

2




2

+  ln






.

This is a special case of equation 1.8.8.3 with 0 ( ) =  , ( ) =  ( ) = . ( ) = 1 ( ) = 0, and  ( ) =  .

© 2002 by Chapman & Hall/CRC Page 122




6.

= 

2


 2



+

2

 



ln




.

This is a special case of equation 1.8.8.3 with 0 ( ) =  , ( ) =  ( ) = . ( ) = 1 ( ) = 0, and  ( ) =  .


7.

2


 2

= 



+  ln2

2






.

This is a special case of equation 1.8.8.3 with 0 ( ) =  , ( ) =  ( ) =  ( ) = 1 ( ) = 0, and . ( ) =  . See also equation 1.8.6.5.


8.

= 

2


 2 



+

2

 

ln2






.

This is a special case of equation 1.8.8.3 with 0 ( ) =  , ( ) =  ( ) =  ( ) = 1 ( ) = 0, and . ( ) =  .


1

9.

= 

2


 2





+ 23 ln2

2

+  ln








ln + / ln2 ) .

This is a special case of equation 1.8.8.3 with 0 ( ) =  , ( ) =  ( ) = 0, . ( ) =  ,  ( ) =  ln , and ( ) = + ln2 .


10.

= 

2


 4





2




+ (  ln +  ) .

This is a special case of equation 1.8.8.4 with ( ) =  and  ( ) =  ln +  .


11.

= 

2


 





2




+ (  ln +  ) .

This is a special case of equation 1.8.8.7 with ( ) =  ,  ( ) = 0, and . ( ) =  ln +  .


12. 

=

2


 





2




+ (  ln +  )

. 

This is a special case of equation 1.8.4.5 with ( ) =  ln +  .

1.7. Equations Containing Trigonometric Functions and Arbitrary Parameters 1.7.1. Equations Containing a Cosine


1. 

=

2




2

+ (  cos

  




+ ) .

This is a special case of equation 1.8.1.1 with ( ) =  cos +  .


2.

= 

2




2

+ (  cos

  

+






) .

This is a special case of equation 1.8.1.6 with ( ) =  and  ( ) =  cos .


3.

= 

2






2

+ (  cos

  




+ ) .

This is a special case of equation 1.8.1.3 with ( ) =  cos +  .


4. 

=

2




2

+ (  cos

  

–

 2
)

.

This is a special case of equation 1.8.1.7 with ( ) =  cos .

© 2002 by Chapman & Hall/CRC Page 123




5.

2


= 



2

+ (  cos




  

+ )

. 

This is a special case of equation 1.8.2.1 with ( ) =  cos +  .


6.

2


= 





2

+ (  cos




  

+ )

. 

This is a special case of equation 1.8.2.3 with ( ) =  cos +  .


7.

= 




2




2

+

+ (  cos 

  




+ s) .

This is a special case of equation 1.8.3.1 with ( ) =  and  ( ) =  cos +  .


8.

2


= 



 2

+

 

cos

  




+  

. 

This is a special case of equation 1.8.2.6 with ( ) =  cos .


9.

= 

2






+ (

2



cos

  




+ ) .

This is a special case of equation 1.8.4.1 with ( ) =  cos and  ( ) =  .


10.

= 

2


 2 



2

+ (  cos

  




+ ) .

This is a special case of equation 1.8.4.2 with ( ) =  cos +  .


11.

= 

2


 4



2

+ (  cos

  




+ ) .

This is a special case of equation 1.8.8.4 with ( ) =  and  ( ) =  cos +  .


12. 

=

2


 





2

+ (  cos




  

+ ) 

.

This is a special case of equation 1.8.4.5 with ( ) =  cos +  .

1.7.2. Equations Containing a Sine


1.

= 

2




2

+ (  sin

  




+ ) .

This is a special case of equation 1.8.1.1 with ( ) =  sin +  .


2.

= 

2




2

+ (  sin

  

+






) .

This is a special case of equation 1.8.1.6 with ( ) =  and  ( ) =  sin .


3.

= 

2






2

+ (  sin

  




+ ) .

This is a special case of equation 1.8.1.3 with ( ) =  sin +  .


4. 

=

2




2

+ (  sin

  

–

 2
)

.

This is a special case of equation 1.8.1.7 with ( ) =  sin .

© 2002 by Chapman & Hall/CRC Page 124




5.

2


= 



2

+ (  sin




  

+ )

. 

This is a special case of equation 1.8.2.1 with ( ) =  sin +  .


6.

2


= 





2

+ (  sin




  

+ )

. 

This is a special case of equation 1.8.2.3 with ( ) =  sin +  .


7.

= 




2




2

+

+ (  sin 

  




+ s) .

This is a special case of equation 1.8.3.1 with ( ) =  and  ( ) =  sin +  .


8.

2


= 



 2

+

 

sin

  




+  

. 

This is a special case of equation 1.8.2.6 with ( ) =  sin .


9.

= 

2






+ (

2



sin

  




+ ) .

This is a special case of equation 1.8.4.1 with ( ) =  sin and  ( ) =  .


10.

= 

2


 2 



2

+ (  sin

  




+ ) .

This is a special case of equation 1.8.4.2 with ( ) =  sin +  .


11.

= 

2


 4



2

+ (  sin

  




+ ) .

This is a special case of equation 1.8.8.4 with ( ) =  and  ( ) =  sin +  .


12. 

=

2


 





2

+ (  sin




  

+ ) 

.

This is a special case of equation 1.8.4.5 with ( ) =  sin +  .

1.7.3. Equations Containing a Tangent


1.

= 

2




2

+ (  tan

  




+ ) .

This is a special case of equation 1.8.1.1 with ( ) =  tan +  .


2.

= 

2




2

+ (  tan

  

+






) .

This is a special case of equation 1.8.1.6 with ( ) =  and  ( ) =  tan .


3.

= 

2






2

+ (  tan

  




+ ) .

This is a special case of equation 1.8.1.3 with ( ) =  tan +  .


4. 

=

2




2

+ (  tan

  

–

 2
)

.

This is a special case of equation 1.8.1.7 with ( ) =  tan .

© 2002 by Chapman & Hall/CRC Page 125




5.

2


= 



2

+ (  tan




  

+ )

. 

This is a special case of equation 1.8.2.1 with ( ) =  tan +  .


6.

2


= 





2

+ (  tan




  

+ )

. 

This is a special case of equation 1.8.2.3 with ( ) =  tan +  .


7.

= 




2




2

+

+ (  tan 

  




+ s) .

This is a special case of equation 1.8.3.1 with ( ) =  and  ( ) =  tan +  .


8.

2


= 



 2

+

 

tan

  




+  

. 

This is a special case of equation 1.8.2.6 with ( ) =  tan .


9.

= 

2






+ (

2



tan

  




+ ) .

This is a special case of equation 1.8.4.1 with ( ) =  tan and  ( ) =  .


10.

= 

2


 2 



2

+ (  tan

  




+ ) .

This is a special case of equation 1.8.4.2 with ( ) =  tan +  .


11.

= 

2


 4



2

+ (  tan

  




+ ) .

This is a special case of equation 1.8.8.4 with ( ) =  and  ( ) =  tan +  .


12. 

=

2


 





2

+ (  tan




  

+ ) 

.

This is a special case of equation 1.8.4.5 with ( ) =  tan +  .

1.7.4. Equations Containing a Cotangent


1.

= 

2




2

+ (  cot

  




+ ) .

This is a special case of equation 1.8.1.1 with ( ) =  cot +  .


2.

= 

2




2

+ (  cot

  

+






) .

This is a special case of equation 1.8.1.6 with ( ) =  and  ( ) =  cot .


3.

= 

2






2

+ (  cot

  




+ ) .

This is a special case of equation 1.8.1.3 with ( ) =  cot +  .


4. 

=

2




2

+ (  cot

  

–

 2
)

.

This is a special case of equation 1.8.1.7 with ( ) =  cot .

© 2002 by Chapman & Hall/CRC Page 126




5.

2


= 



2

+ (  cot

  




+ )

. 

This is a special case of equation 1.8.2.1 with ( ) =  cot +  .


6.

2


= 





2

+ (  cot




  

+ )

. 

This is a special case of equation 1.8.2.3 with ( ) =  cot +  .


7.




2


= 



2

+

+ (  cot 

  




+ s) .

This is a special case of equation 1.8.3.1 with ( ) =  and  ( ) =  cot +  .


8.

2


= 



 2

+

 

cot

  




+  

. 

This is a special case of equation 1.8.2.6 with ( ) =  cot .


9.

= 

2








+ (

2

cot

  




+ ) .

This is a special case of equation 1.8.4.1 with ( ) =  cot and  ( ) =  .


10.

= 

2


 2 



2

+ (  cot

  




+ ) .

This is a special case of equation 1.8.4.2 with ( ) =  cot +  .


11.

= 

2


 4



2

+ (  cot

  




+ ) .

This is a special case of equation 1.8.8.4 with ( ) =  and  ( ) =  cot +  .


12. 

=

2


 





2

+ (  cot

  




+ ) 

.

This is a special case of equation 1.8.4.5 with ( ) =  cot +  .

1.8. Equations Containing Arbitrary Functions 1.8.1. Equations of the Form


1. 

=

2




   

= 

2 

 

+  (  ,  )

 2




2

+4 ( ) .

1 5 . Particular solutions ( & , ' , and 6 are arbitrary constants): 

( , ) = ( &  + ' ) exp 798 

( , ) = & ( 

( , ) = & exp 7

( ) = ) *

( ) +  .

More sophisticated solutions are specified below. 2 5 . The original equation admits particular solutions of Z the form  X

( , ) =

Y[Z

X

Z

D X , (

)

(1)

=0 \

for Z any ] (Z  ). Substituting the expression of (1) into the original equation and matching the coefficients of like powers of , we arrive at the following system of ordinary differential equations for D X , = D X , (  ): \ Z  D XEFE , ZX + ] ( ) D EX ,Z X = 0,  D XEFE , + ] ( ) D EX , = (^ + 1) D X , +1 ; ^ = 0, 1, _G_G_ , 0 − 1,

© 2002 by Chapman & Hall/CRC Page 130

where the prime denotes the derivative with respect to  . Integrating these equations successively in order of decreasing number ^ , we obtain the solution ( & and ' are any numbers): 1 D X , X Z (  ) = & + ' ) * (  ) +  , * Z (  ) = exp H − ) ] (  ) +  K ,  (2) Xa − D X , (  ) = 0 ( 0 − 1) _G_G_ (^ + 1) ` AbD X , X (  )C ; ^ = 0, 1, _G_G_ , 0 − 1. Here, the integral operator ` a is introduced as follows: c ( g )Z h g 1d Z h g . ` a [ c ( )] ≡ i ) * ( ) e f (g ) j

(3)

The powers of the operator are defined by ` a [c (g )] = ` a k ` a −1 [c (g )] l . Formulas (1)–(3) give an exact analytical solution of the original equation for arbitrary m (g ). A linear combination of particular solutions r r r qsr

n

n

(g , o ) = p

(g , o )

(

=0 t

are arbitrary constants) t

is also a particular solution of the original equation. 3.

u v

=x u

2

v

2

+

( )u v

y z

.

w

u w u y c equation.u The y Generalized Ilkoviˇ equation describes mass transfer to the surface of a growing drop that flows out of a thin capillary into a fluid solution (the mass rate of flow of the fluid moving through the capillary is an arbitrary function of time).

1 { . Particular solutions ( | , } , and ~ are arbitrary constants): i

n

(g , o ) = | g n

(g , o ) = | g n

(g , o ) = | exp H‚~ g

(o ) + } ,

2i

2

i

(o ) = exp Hf

(o ) +  ~

(o ) h o;€ ,

i 2 (o

)h o +} ,

2

i 2 (o

(o ) + 2 |  f i

m

f

) h o;€ + } .

2 { . On passing from o , g to the new variables ( | is any number) ƒ

i 2 (o

=f

)h o +| , $

i

=g

(o ),

i

where

(o ) = exp Hf

for the function n (ƒ , $ ) we obtain a constant coefficient equation, in Subsection 1.1.1.

! „ n

=

! …… n

(o ) h o;€ , m

, which is considered

3 { . Consider the special case where the heat exchange occurs with a semiinfinite medium; the medium has a uniform temperature n 0 at the initial instant o = 0 and the boundary g = 0 is maintained at a constant temperature n 1 all the time. In this case, the original equation subject to the initial and boundary conditions n

=n n

= n

has the solution

0

n

†

n

1 0

at o

=0

at at g

=0 g

†

(initial condition),

$ −n 1 = erf ˆ , n n 2‰  ƒ j 0− " 1 ƒ

n

=f

i 2 0

( ) h  , $

=g

‡

(boundary condition), (boundary condition)

i

erf S = (o ),

i

2 ‰ Š

f

0

‹

exp Œ −  2 Ž h  ,

(o ) = exp Hf m

(o ) h o;€ ,

where erf S is the error function.

© 2002 by Chapman & Hall/CRC Page 131

4.

u v

2

=x u

v

2

+ ( – )u v z

y

.

w

u w u y y 1 { . Particular solutions ( | and } u are arbitrary constants): n

(g , o ) = | + }

i f

i

($ ) h $ ,

n

(g , o ) = |  o + | f n

(g , o ) = |  oW‘ ( $ ) + | f

ˆ f

( $ ) = exp H − i

h $ i

($ ) j ‘

ˆ f

1 f



m

($ ) h $ −  $ € , 

($ ) h $ ,

($ ) h $ i ($ ) h g , ($ ) j i

‘

i

($ ) = f

($ ) h $ ,

where $ = g − o . 

2 { . On passing from o , g to the new variables o , $ = g − o , we obtain a separable equation of the  form 1.8.2.2: ! n ! 2n ! n k ! ! = + m ($ ) + l ! . 2 o

5.

u v

2

=x u

u w

v

2

1

+ ’

u y

ˆ ’y z

w

$

.

u v j

w



$

u y

1 { . On passing from o , g to the new variables ƒ = ln o , “ = g ” ‰ o , we obtain a separable equation of the form 1.8.2.2: ! 2n ! n ! n 1 k ! ƒ = ! + m (“ ) + 2 “ l ! . 2 “

“

2 { . Consider the special case where the heat exchange occurs with a semiinfinite medium; the medium has a uniform temperature n 0 at the initial instant o = 0 and the boundary g = 0 is maintained at a constant temperature n 1 all the time. In this case, the original equation subject to the initial and boundary conditions n =n 0 at o = 0 (initial condition), n n = 1 at g = 0 (boundary condition), n

has the solution n

−n n 1−

n

0 0

†

= • – ‹

0

• –

where “ = š ” ‰ › . 6. 1{ .

u v

=x u

2

v

2

+ H

y z u w u y Particular solutions ( | r œ

(š , › ) = | š œ

n

0

and }

g

†

‡

exp k − ‘ (“ )l — “ exp k − ‘ 

( )+ w

at

exp H (! ),

(! , ) = '

2

(! , ) = '

2

 ! − =  *  , ( )  0 )  ! − = ( ) *  , ( ) 0

> (! ) =
( ) *  . ( ) 0

< 

More sophisticated solutions are specified below in Item 3 { . =

3 { . For any function (! ), the original equation admits ADexact analytical solutions of the form C  ?

(! , ) =

?

+

? @BA

A

−1

?

,

(! ).

(3)

=0

C A

C A

Substituting expression (3) into the original equation and matching the coefficients of like powers ? ? C C A A of , we arrive at the following system of ordinary differential equations for , = , (! ): ?%&%

=

?

(! ) , = (E + 1) E = 0, 1, FFF , G − 1;

C

, +1 ,

? ?

≡ 1,

,

where the prime stands for the C differentiation with respect to ! .A Integrating these equation succesA E , we obtain sively in order of decreasing number ?

?

,

(! ) = G (G − 1) FFF (E + 1) H I

−

[1].

(4)

Here, the integral operator H I is introduced as follows: H I [J (! )] ≡
0,  > 0, and 1 ≤ ≤ 2 . 1.8.9-1. General formulas for solving linear nonhomogeneous boundary value problems. The solution of the equation in question under the initial condition ÷  = ( ) at = 0 and the arbitrary linear nonhomogeneous boundary conditions  ß 1 +  1  = ú 1 ( ) at = 1 , ï



2 ï

can be represented as the sum ( , ) = 

é

0

2

ï

 é

2

= ú 2 ( ) at 

( ,  )  ( ,  , −  ) ø

+  ( 1 ) 



( ,  ,  ) =

0

1 ( , −  ) 

1 ( )  





( )  ( )  ( ,  , )  



(3)



+  ( 2 )  

0



( )



=1

 1

  Here, the modified Green’s function is given by



2

+

ø  

(2)

= 2,



1

ï

+ 

(1)

( ) 2





exp(− !

2 ( , −  ) 

2 ( )  

.



 

2

),



= 

2

 

( )

2

( ) 

,

(4)

1



   where the ! and ( ) are the  eigenvalues and corresponding eigenfunctions of the following Sturm–Liouville problem for a second-order linear ordinary differential equation:  [ ( )  ]  + [ !  ( ) − ( )] = 0, 



1



+  

= 0 at

1

= 1,



(5)

+  2  = 0 at = 2 . 2  occur in the integrands of the last two terms in solution (3) The functions  1 ( , ) and  2 ( , ) that   are expressed via the Green’s function (4). Appropriate formulas will be given below in Paragraphs 1.8.9-3–1.8.9-7 when considering specific boundary value problems. 



1.8.9-2. General properties of the Sturm–Liouville problem (5). 

%

&

%

&

1 " . There are infinitely many eigenvalues. All eigenvalues are real and different and can be ordered so that ! 1 < ! 2 < ! 3 < #$#$# , with ! as ' (therefore, there can exist only finitely many  negative eigenvalues). Each eigenvalue is of multiplicity 1. 2 " . The eigenfunctions are determined up to a constant multiplier. Each eigenfunction exactly ' − 1 zeros in  the open interval ( 1 , 2 ). 3 " . Eigenfunctions

1 ≤ ≤ 2:

( ) and

( ), '

≠ )

( ) has 

, are orthogonal with weight  ( ) on the interval

 2 (

 

 

( )

( )

( ) 

= 0 for

'

≠) .

1

  ( 4 " . An arbitrary function * (+  ) that has a continuous derivative and satisfies the boundary conditions of the Sturm–Liouville problem can be expanded into an absolutely and uniformly convergent series in eigenfunctions:    



*

(+ ) =

(+ ), *



=1 2

where the norm







is defined in (4).

*

1

=

2



2



(+ ) * (+ )

 

1

(+ )  +

,

, 



© 2002 by Chapman & Hall/CRC Page 151

5 " . If the conditions -

(+ ) ≥ 0, 

≤- 0,

1 1



≥0

2 2

(6)

are satisfied, there are no negative eigenvalues. If ≡ 0 and  1 =  2 = 0, then ! 1 = 0 is the least eigenvalue, to which there corresponds the eigenfunction . 1 = const. Otherwise, all eigenvalues are % & positive, provided that conditions (6) are satisfied. 6 " . The following asymptotic relation holds for large eigenvalues as '  2

2 2

= !

'

/ 0

:

0

+ 1 (1),

2

(+ ) (+ )

2

= 



3,

1

.

 +

(7)



Special, boundary value condition-dependent properties of the Sturm–Liouville problem are presented in Paragraphs 1.8.9-3 through 1.8.9-7. 4

576

8 9;:




( )= ?

3

1A 4

( + ) ( + )@

(+ ).

, 

The boundary conditions transform into boundary conditions of the same type. 1.8.9-3. First boundary value problem: the case of



=

1



2

= 0 and 

1

= 

= 1.

2

The solution of the first boundary value problem with the initial condition (1) and the boundary conditions B = 1 (C ) at + = + 1 , B 

= is given by formulas (3)–(4) with 

1 (+

,C ) = D D



2 (C

) at

=+ +

2



(+ ,  , C ) F

,

=

F

E

FHG

,C ) = −

2 (+ 

1

D D





(+ ,  , C ) F F

E

FHG

.

=

2



% & Some special properties of the Sturm–Liouville problem are worth mentioning. 



1 " . For ' , the asymptotic relation (7) can be used to estimate eigenvalues sponding eigenfunctions (+ ) satisfy the asymptotic relation

. The corre!





(+ ) 

=

I 0



4 23

1A 4

(+ ) (+ ) J

sin I

'

/0



 K

1

,

(+ ) (+ )

3,

L

+ J

+1

1 M '

N

,

0

=

2

O

P 1

K

3 Q

(R ) (R ) L

R

.

P



2 S . For T ≥ 0, the following upper estimate (Rayleigh principle) holds for the least eigenvalue: 2

U

1

≤ V

P

1

WYX

(R )( Z [ )2 + T (R ) Z 2 P

P V

P P

1

(R ) Z

2 L

2\ L

R

,

(8)

R

Q

where Z = Z (R ) is any twice differentiable function that satisfies the conditions Z (R 1 ) = Z (R 2 ) = 0. The equality in (8) is attained for Z = ] 1 (R ), where ] 1 ( R ) is the eigenfunction of the Sturm– U Liouville problem corresponding to the eigenvalue 1 . To obtain particular estimates, one may set Z = ( R − R 1 )( R 2 − R ) or Z = sin[ ^ ( R − R 1 ) _ ( R 2 − R 1 )] in (8).

© 2002 by Chapman & Hall/CRC Page 152

3 S . Suppose 0
0.

3 “ . The equation of thermal boundary layer on a flat plate corresponds to £

‰

(• ) = Pr ­

£

(• ), ¬

o

(• ) =

1 2

Pr › •

( • ) − ­ ( • ) , ­ ¬

where ­ (• ) is Blasius’ solution in the problem of translational flow past a flat plate and Pr is the Prandtl number (ˆ is the coordinate measured along the plate and ” is the transverse coordinate to the plate surface). In this case the formulas in Item 2 “ transform into Polhausen’s solution. See Schlichting (1981) for details.

®}¯

Reference: A. D. Polyanin, A. V. Vyazmin, A. I. Zhurov, and D. A. Kazenin (1998).

© 2002 by Chapman & Hall/CRC Page 156

5.

(± ) ² °

+

³ ±

‹}´

( ± )µ –

Œ

²

¶

µ µ

²

2

=

³

²

³

²

.

2

µ ²

For · = 1, equations of this sort govern the concentration distribution in the internal region of the diffusion wake behind a moving particle or drop. The transformation ( ¸ and ¹ are any numbers) 2

= š

»‰ º

(ˆ ) (ˆ ) Š

+¸ , ˆ

=” ¼

(ˆ ), »

where »

(ˆ ) =

exp ½ − º ¹

‰

o

(ˆ ) (ˆ ) Š

ˆ

¾

,

leads to an equation of the form 1.2.5: ˜ ˜

®}¯

2

= ™

˜

š

¼

™

˜

+

2

· ˜

¼

™ ¼

˜

.

References: Yu. P. Gupalo, A. D. Polyanin, and Yu. S. Ryazantsev (1985), A. D. Polyanin, A. V. Vyazmin, A. I. Zhurov, and D. A. Kazenin (1998).

6.

(± ) ² °

+

³ ±

½}´

( ± )µ –

¾ ¶

²

µ µ

²

2

=

³

²

²

The substitution ™

+ ¿ (± ) .

2

³

²

(ˆ , ” ) = À (ˆ , ” ) exp ½Áº ‰

(ˆ ) (ˆ ) Š

‰»

À

(ˆ ) ˜

+ ˆ

˜

®}¯

³ µ

½o

ˆ

leads to an equation of the form 1.9.1.5: ¾

·

(ˆ )” − ”

¾

À ˜ ˜

”

2

= ˜

À

2 ”

˜

.

Reference: A. D. Polyanin, A. V. Vyazmin, A. I. Zhurov, and D. A. Kazenin (1998).

7. °

( ± )µ Â

–1 ²

³ ±

+

( ± )µ

½´

– Â

²

¾ ¶

µ µ

»

exp ½ − º ¹

=

‰

o

²

³ µ

.

2

²

are any numbers) a

where (ˆ ) =

2

=

³ ²

The transformation ( ¸ and ¹

š

²

º

˜ ˜

ˆ

š

ˆ

+¸ ,

= •

Ã

2 +1

a ›Ä” »

( ˆ )

+1 2

,

, leads to the equation ¾

= ™

(ˆ ) (ˆ ) Š

‰ »

(ˆ ) (ˆ ) Š

+1

2 ˜ ˜

•

™

2

+

1 − 2¦ •

˜ ˜

™ •

, ¦

= Ã

1−· , +1

which is considered in Subsection 1.2.5 (see also equations in Subsections 1.2.1 and 1.2.3).

¨ 1.9.2. One-Dimensional Schrodinger Equation Å ÆÇ

È È

É Ê

=–

ÆÇ

2Ë

2

2É È È

Ì

2

+ Í

( Ì )É

1.9.2-1. Eigenvalue problem. Cauchy problem. Schr¨odinger’s equation is the basic equation of quantum mechanics; is the wave function, Î 2 = −1, Ï ™ is Planck’s constant, Ð is the mass of the particle, and Ñ (ˆ ) is the potential energy of the particle » in the force field.

© 2002 by Chapman & Hall/CRC Page 157

1 Ò . In discrete spectrum problems, the particular solutions are sought in the form (ˆ , ) = exp Ó − ™ a

where the eigenfunctions eigenvalue problem Ø

a

ÎÕÔ

a

»

ÖÕ×

and the respective energies Ú

a

2 Ú

ÙØ a

2Ð

+

2

Ï

a Ô

Û

Ø a Ô

have to be determined by solving the a

− Ñ ( Ù )Ü Ø

»

0 at Ù §

2

(Ù ),

Ï

š

§

Ý

, Þ

º

Ø

− a

The last relation is the normalizing condition for

.

a

| ß

= 0, |2

(1) Ú

= 1. Ù

Ø ß

a

2 Ò . In the cases where the eigenfunctions (Ù ) Ø form an orthonormal basis in of the Cauchy problem for Schro¨ dinger’s equation with the initial condition Ø = ã (Ù ) at â

), the solution

=0

(2)

Ö

is given by â

2 (á à

(Ù , ) =

Ú

(Ù , æ , ) ã (æ ) ß

ä

−

Ö

ß

(Ù , æ , ) =

, æ

Ö å

ç a

a ß

=0

Ö å

a

(Ù )

Ø

a Ô

(æ ) exp Ó − è

. é

Ø »

ÖÕ×

Various potentials ê (Ù ) are considered below and particular solutions of the boundary value problem (1) or the Cauchy problem for Schro¨ dinger’s equation are presented. In some cases, a a nonnormalized eigenfunctions ë (Ù ) are given instead of normalized eigenfunctions (Ù ); the former differ from the latter by a constant multiplier. Ø 1.9.2-2. Free particle:

(Ù ) = 0. ê

The solution of the Cauchy problem with the initial condition (2) is given by â ú}û

(Ù , ) = Ö

1 2ì

èîí

ä ï

( Ù − æ )2 exp ½ − 4 ß

−

(æ ) ã

èîï

ß (1977). Reference: W. Miller, Jr.

é Ú

, æ

= ï

ð

»

2ò

, ñ

ì èÕó

1.9.2-3. Linear potential (motion in a uniform external field):

ó

Solution of the Cauchy problem with the initial condition (2): â

1

(Ù , ) =

exp ü −

è¢ý7ï 2ì èîí ï See also Miller, Jr. (1977). ñ

2 3þ

− 13 Ù

è¢ý

ä

ï

Eigenvalues: a

=

Normalized eigenfunctions:

a

Ø

(Ù ) =

1 4 í

ø

−

exp ½ −

(Ù +

4

ì

1a 2


!Ù

0

(Ù ) = ê

é »

ÿ

ü


exp ü

+

1 2ò 1 2 þ

2 2 ÿ

,

Ù

− 21 æ 2 þ

a

− æ )2

õ ö ÷ùø

−

Ù

ö ÷ùø

èîï

ð

= 0, 1,



ì

ó

ó ó

é æ

> 0, < 0.

.

Ú

(æ ) ã

| | if ó | | if

ì

4

, ï

=

»

2ò

, ñ

ý

=

2 éó

ò

2

.

»

.




(æ ),

where  (æ ) are the Hermite polynomials. a The functions ( ) form an orthonormal basis in

ú}û

2 ý7ï

ß

1.9.2-4. Linear harmonic oscillator:

a

ß

ô õ

(Ù ) = ê

4

=

æ

2(

=







0

,



0

=

é ò

ÿ

,

).

References: S. G. Krein (1964), W. Miller, Jr. (1977), A. N. Tikhonov and A. A. Samarskii (1990).

© 2002 by Chapman & Hall/CRC Page 158

1.9.2-5. Isotropic free particle:

 2

( ) = ê



ó

.

Here, the variable  ≥ 0 plays the role of the radial coordinate, and > 0. The equation with ó  2  ê ( ) = results from Schr¨odinger’s equation for a free particle with
space coordinates if one ó passes to spherical (cylindrical) coordinates and separates the angular variables. The solution of Schr¨odinger’s equation satisfying the initial condition (2) has the form

exp  − 12

( , ) =

èîí

ñ

( + 1) sign ñ 2| | ï

where

0

é

= ï

ú}û

ä

2ò

, ñ



  ì



2

= 

 2

exp  è

2

+ 4



ï

1 + ≥ 1, 4

ò

 éó 2

  

2| |

 

ï



( )  

,



( ) is the Bessel function.

  Reference: W. Miller, Jr. (1977).

1.9.2-6. Isotropic harmonic oscillator:

( ) = ê

1 2ò ÿ

2 2

+



 −2 ó

.

Here, the variable ≥ 0 plays the role of the radial coordinate, and > 0. The equation with this ó  ê ( ) results from Schr¨ odinger’s equation for a harmonic oscillator with
space coordinates if one passes to spherical (cylindrical) coordinates and separates the angular variables. Eigenvalues:  a

=−

é

(2
+  + 1), ÿ

2

= 

ò

1 ≥ 1, 4

+

 éó 2

= 0, 1,




Normalized eigenfunctions:

a



( )= 



2
! (
+ 1 +  )

0

2 +1 2  

exp ü

− 12  2 þ

a

ü

2þ



,

= 

 



0



,

0



= 

a

a

where (  ) is the
th generalized Laguerre polynomial with parameter  . The norm |   to the semiaxis ≥ 0. a The functions ( ) form an orthonormal basis in 2 (  + ).

"!

, ÿ

( )|2 refers

Reference: W. Miller, Jr. (1977).

1.9.2-7. Morse potential: #

( ) =

# 0($

−2 % &('

− 2$

− % &('

).

Eigenvalues: a

= −#

0 )

1− *

1

2

(
+ 12 )+ ,

*

=

, -

2 

# 0

.

0≤



9:

:

: 8 =1 ( C

2

+; 0

: , ) 1 2 2 exp(− C ) 0(C

0 (C 2)

:

-

; 4 5 / +

- ) erfc 3 C

- ) :

:

− exp(− C

2 4 5/ 6

4 5/ −

- ) erfc 3 C

: :

2 4 5/ 6 7

C

,

= D1

.

Reference: H. S. Carslaw and J. C. Jaeger (1984).

2.1.3-6. Domain: 0 ≤  ≤  , 0 ≤ ≤ E . First boundary value problem. A circular cylinder of finite length is considered. The following conditions are prescribed: 

= ( , )  

=







1(

at

, ) at

= 2 ( , )  = 3 ( , )

at at



Solution:  ( , , ) = 2



F 0





− 2 

0



 

0

0



(initial condition),

 =

(boundary condition),

=0

(boundary condition), (boundary condition).

=E



− 2   + 2 

=0

0

0



 0

0



( ,  )  ( , , ,  , )  

F   



& 1 (



, ) " & &

 

     ( , , ,  , −  ) # A = 

, ) " & &

2(





 ( , , ,  , −  ) # B 

=0

 

 

   .  ( , , ,  , −  ) 3( ,  ) " & # B =F      

© 2002 by Chapman & Hall/CRC Page 183

Here,  ( , , ,  , ) =  1 (

   1  1   1 ( , , ) =  0 2 2   ( )  =1 1

 2   2 ( ,  , ) = sin  G sin  E E  

, , )  

 





,   , ), 

  exp  −    

2

2 2



E

are positive zeros of the Bessel function,

 

0

  G

=1

where the 

2(

 



exp  − G E





2

 = H : where

=?>

D

0

9:

I 8 =0

1 (2 J + 1)H  (2J + 1)2 H sin 2 exp 2 − K K2 2J + 1 7

2/

:

are positive zeros of the Bessel function,

0(

; Reference: H. S. Carslaw and J. C. Jaeger (1984).

,

) = 0.

0 (

:

9:

7 L I 8 =1 D

) = 0.

D



, 

Example 6. The initial temperature is the same at every point of the cylinder, + (, , - ) =  end faces are maintained at zero temperature, . 1 ( - , / ) = . 2 ( , , / ) = . 3 ( , , / ) = 0. Solution: 8



2

1 1(

;

: 0

)

D

0.

The lateral surface and the

: : ,

3 D 1

6

;

exp 3 − D 1

2

5/ 2

6 L

,

2.1.3-7. Domain: 0 ≤  ≤  , 0 ≤ ≤ E . Second boundary value problem. A circular cylinder of finite length is considered. The following conditions are prescribed:  & '  & )  & ) 

= ( , ) at  = 1 ( , ) at

=0  =

(initial condition), (boundary condition),

= =



, )

at

=0

(boundary condition),

3 ( , )

at

=E

(boundary condition).

2 (

 

Solution:  ( , , ) = 2

F 0

 0



+ 2  



0



0



0

 0

( ,  )  ( , , ,  , )  

F 

0



− 2  + 2 

 



   

0



,  )  ( , ,  ,  , −  )    

1 (

,  )  ( , , , 0, −  ) 

2( 3(

  







  

,  )  ( , , , E , −  )  



  .

Here,

1 (



2 (



, , ) = 

, , ) =



where the 

  ( , , ,  , ) =  1 ( , ,  )  2 ( ,  , ),         1    2

1 + exp  − 2 ,  2 0 0  2         ) 0 (

1  

1 E

2

+

2  E

=1

cos  G =1

 E



cos  G

  E

2 2





exp  − G

are positive zeros of the first-order Bessel function,



1 (





E

2



,

) = 0.

© 2002 by Chapman & Hall/CRC Page 184


 

2

2.1.3-8. Domain: 0 ≤  ≤  , 0 ≤ ≤ E . Third boundary value problem. A circular cylinder of finite length is considered. The following conditions are prescribed: 

& '  & ) 

+(

1

= ( , ) 



= 

1(



at

, ) at

− ( 2 = 2 ( , )  + ( 3  = 3 ( , )

& ) 

at at



=0

(initial condition),

 =

(boundary condition),

=0

(boundary condition), (boundary condition).

=E

The solution  ( , , ) is determined by the formula in Paragraph 2.1.3-7 where

, , ) =



and the 

P M

P

2

=

2

 



2 (





1

, , ) =

1 (





M

=1

(

2Q

=1

 ( , , ,   , ) =  1 ( , , )  2 ( ,  ,

 2         2 0 0 2 2 2   ( ( 1  +  ) 0 ( )

M ( O ) M ( ) P P exp(−  Q 2M R ), M 2 N N Q 2M N + ( Q 2M + (

3 2M

2 2 2 3

(

+

2Q

2 2M

+

E

2

), 



  exp  −     (

M ( O ) = cos( Q M O ) +

2

2



,

sin( Q M O ),

N

2 2 2M

(

 1+

Q M

2

Q

, 

N and Q M are positive roots of the transcendental equations   ( 2+( 3 tan( Q E )  1 ( ) − ( 1  0 ( ) = 0, = 2 . Q Q − ( 2( 3

2.1.3-9. Domain: 0 ≤ S ≤  , 0 ≤ O ≤ E . Mixed boundary value problems. 1 @ . A circular cylinder of finite length is considered. The following conditions are prescribed: T T & ) T & ) T

Solution:

= (S , O ) at  = 1 ( O , R ) at

R =0 S =

(initial condition), (boundary condition),

= =

  

T (S , O , R ) = 2U V

2 (S

,R )

at

O =0

(boundary condition),

3 (S

,R )

at

O =E

(boundary condition).

F 0

V

0

W

X Y

− 2U [ \ V ] V 0

− 2U [ V ] V W 0

0

+ 2U [ V ] V W 0

0

Here,

d

where the f

0

( ,  )  (S , O , ,  , R ) Z X

F ^ X ^ X ^

& X

, _ )  (S , O , , 0, R − _ ) Z

2( 3(

Z  X

 ( S , O , ,  , R − _ )a A X

1 ( , _ ) ` &

X

X

X

, _ )  (S , O , , E , R − _ ) Z

 (S , O , ,  , R ) =  1 (S X d f 1 e 1 cd  1 (S , , R ) = e 0g 2 2 b U \ (f ) X =1 1 j m k 1m 2m cd  2 (k , l , ) = + b cos g n o i =1

X

X

,d h , R )  X

\

e i

g

are positive zeros of the Bessel function,

f \

m l

cos g n o d e

0 (f

X

W

Z _ Z _ .

, d  , R ),

2(O 0

X

Z  Z _

=

X i

d j

exp g − [

i

[ f

2

2 2

m

exp g − n 2o

,

\ j2 i i

,

) = 0.

© 2002 by Chapman & Hall/CRC Page 185

h j 2 p . A circular cylinder of finite length is considered. The following conditions are prescribed: q r ' q

^

q

h

= 2( , j ) q =^ ( , ) 3 Solution: h

h

= ( , k j ) at Y = 1 (hk , j ) at at at

^

j

q ( ,k , ) = 2

+2 −2

0t

s

0

W

1(



 d

where the f

2 (k

j

0z

s

o x s

0z

s

0

o x s

0z

s

0

u

W W

2(

3(

u {

2

2m c d

,l , ) =

€

w l

yh

r

u {

ju

, | ) ( , k , , l , − | ) w l w |

1 (l

{

u

h

l

 ( , k , , l , − | )~  u h j l

 ( , k , , l , − | )~  u j j

,| ) } r u

r u

,| ) } r h

j

j

m k

sin g n o

sin g n o i

=1

2m 2

m l

exp g − x n 2o i

are positive zeros of the first-order Bessel function, h

2.1.3-10. Domain:

1

y

w

=0

w

u

=

w | w | .

u

t d j  ( , k , , l , ) =  1 ( , , d )  h 2 ( k , l , ),d ucd u d e f f f 2 1 e 1 + exp − , e 0 g 0 g g x u 2 2 € 2 i i i (f ) =1 0 o y y y j y

o y j

j

u

0t

s

1

, , )=

(boundary condition), (boundary condition).

k =

h

h

k =0 m

( , l ) ( , k , , l , ) w

u v

o x y

Here,

(initial condition), (boundary condition),

h

o s

+2

=0 =\

e

id 1 (f

, ) = 0.

m

≤

≤

2 , 0 ≤ k ≤ . First boundary value problem.

y

h A hollow circular cylinder of finite length isj considered. The following conditions are prescribed: h

q

= ( , k j ) at q = v ( k , j ) at 1 {

q q q

Solution: h

h = 2(k , j ) { = 3 (h , j ) {

=

j

q ( ,k , ) = 2

{

4(

, )

+2 −2 +2 −2

=0 =

1

=

2

y

at at

y k =0 m

at

k =

(initial condition), (boundary condition), (boundary condition), (boundary condition), (boundary condition).

h 2

o s

h

0t

W s W

o x y o x y o x s

1

2

s

0z

0t

W

s

0z

0z

u

r

u

h

u

w l j

 ( , k , , l , − | )~  u = h j ru W  ( , k , , l , − | )~  2 (l , | ) } r s 0z s 0t { u = h j r u W 2 s

u {

1

W o x s

( , l ) ( , k , , l , ) w

u v

1

j

2

W

s W

1

{

1 (l

3(

u

,| ) } r

,| ) } r

r

l

 ( , k , , l , − | )~  u h j

 ( , k , , l , − | )~  4( , | ) } r l u { u u

w l w | 1

w l w | 2

w

=0

w

=

u

u

w | w | .

t

© 2002 by Chapman & Hall/CRC Page 186

2‡

‚„ h

Here,

1 (k



,l , h j

2(



, , u

d h

(

e

Š

j

 ( , k , , l , ) =  1(k , l u  c d m k m 2m )= sin sin g n o g n o € i d d =1 e d d cd f 2 02 ( ˆ‰f ) )= o 2 € e 2 e 4 1 =1 0 (d f h ) − 02 ( ˆ‰f d y d e e f ) = ‹ 0 (f ) 0 g − 0 (f 1 i y

j

h

j

, )

2(

l i

j j

, , ), u

m2

2

exp g − x n 2o d h

d d Šh

)‹ d0 g

2 1

y 2

ˆ = y

, i

1

2

f

u

f y

d j

( ) exp g − x

( )

)Š

, i

y

1

i

,

,

where 0 (f ) and ‹ 0 (f ) are the Bessel functions, the f are positive roots of the transcendental equation e e 0 ( f ) ‹ 0 ( ˆ‰f ) − 0 ( ˆ‰f ) ‹ 0 ( f ) = 0. h

2.1.3-11. Domain:

1

y

m

≤

≤ y

2 , 0 ≤ k ≤ . Second boundary value problem.

h length is j considered. The following conditions are prescribed: A hollow circular cylinder of finite q r Πq r Πq

= ( ,k j)

= 1(k , j ) { = 2 ( hk , j ) { h j

r  q r  q

Solution:

h

q ( ,k , ) = 2

(initial condition), 1 2

(boundary condition), (boundary condition), (boundary condition), (boundary condition).

k =

h 2

o s

−2 +2 −2 +2 Here,

= y =

y k =0 m

at at

{

=0 h

at at

= 3( , ) { = 4( , ) j

h

at

v

0t

W s W

o x y o x y

1

2

( , l ) ( , k , , l , ) w

u v

1

u

s

0z

s

0t

{

s

0z

s

0t

{

2

o x s

0z

W s W

o x s h

0z

u {

1

s

2

W 1

W j

j

u {

h u

, | ) ( , k ,

2 (l

, | ) ( , k ,

3(

, | )  ( , k , , 0, − | ) w

4(

u

2, l

y

h

u

h

u

1, l

y

h

1 (f

) ‹ 1 ( ˆ‰f ) −

e

j

, −| )w l w | j

, | ) ( , k , , , − | ) w j

u

h

1 ( ˆ‰f

w | u

j

m

 ( , k , , l , ) =  1 (k , l , ) 2( u  c d m k m l 1m 2m  1 (k , l , ) = + € cos g n o i cos g n o i exp g d d =1 h j e d d cŽd f 2 12 ( ˆ‰f ) 1  2( , , ) = + e e o ( 22 − 12 ) d h 4 12 € =1 12 (f ) −d h 12 ( ˆ‰f u d h d o y d y y e e f f ( ) = ‹ 1 (f ) 0 g − 1 (f ) ‹ 0 g , i Š 1 1 i  e  y y where (f ) and ‹ (f ) are the Bessel functions of order  = e

, −| )w l w |

1 (l

j

the transcendental equation

w l

uj

w | . u

j

, , ), j u

2m 2

− x n 2o d h

)Š

d

( )

ˆ = y y

1

,

d j f

( ) exp g − x Š

2

i

u

,

0, 1 and the f

y

2 2 1

i

,

d

are positive roots of

) ‹ 1 (f ) = 0.

© 2002 by Chapman & Hall/CRC Page 187

h

2.1.3-12. Domain:

≤

1

y

m

≤

2 , 0 ≤ k ≤ . Third boundary value problem.

y

h j A hollow circular cylinder of finite length is considered. The following conditions are prescribed: h

q

= ( , k j ) at v q −  1 = 1 ( k , j ) at

r Πq r Πq

{ h +  2 q = 2(k r  q { −  3 q = 3 (h r  q

at

{

1

(initial condition), (boundary condition),

= 2 y k =0 m

(boundary condition), (boundary condition),

k =

(boundary condition).

y

, j ) at , j ) at

{ +  4 q = 4( , )

=0 = h

For the solution of this problem, see Subsection 2.2.3-4 with ‘ ≡ 0.

2.2. Heat Equation with a Source ’

= “

2.2.1. Problems in Cartesian Coordinates

’

•

–

”

2“

+ —

(˜ , ™ , ” )

In the rectangular Cartesian coordinate system, the heat equation has the form š

š ›

=

š œ

2›

š

š ž

x 

2

2›

+ š Ÿ

2

ž

Ÿ

œ

+ ‘ ( , , ).  

It governs two-dimensional unsteady thermal processes in quiescent media or solids with constant thermal diffusivity in the cases where there are volume thermal sources or sinks. 2.2.1-1. Domain: − ¡

ž


?; + @A< + B ) +

1 3

leads to the two-dimensional heat equation = = See also Niederer (1973) and Boyer (1974). % 2.

1 2

=

2

4

1 3

. 1

2

2

+

2

1

1 5

2

–

2 /

C 7

1 6

2

5

+ 7

2

6

+

( > 2 + @ 2 )  C

9 F 2



,

, "E"

+ =

DD

3

7

=; + #

>?

2

, 

=< +

@A

2

.

= F

> 0.

The transformation ( G is an arbitrary constant) :

(; , < ,  ) = = (# ,  , $ ) exp H #

= ; exp C 2 J

 > F

, 

1 2I

= < exp C 2 J

>   > F

= = leads to the two-dimensional heat equation M See also Niederer (1973) and Boyer (1974).

 C

2

(; ,

+ < 2) + C 2J $

DD

=

= +

1 4J "E"

 > = F

 >

−B

exp C 4 J

F LK

,

 > F

+G

.

© 2002 by Chapman & Hall/CRC Page 198

3.

1 2

2

= 4

. 1

1 3

2

2

2

+ 1

1 5

2

2

CL7

1 cos C 2 J

(; , < ,  ) =

cos C 2 J

,

 > F

2

7

9 F

> J

2J

, 2

tan C 2 J 

cos C 2 J

4.

1 2

2

= 1

1 3

2

+

2

1 5

2

1

2

+

2

C 4

1 6

–2

+

5

–2

7

 >  F C ;

6

F

,

 > F

= = leads to the two-dimensional heat equation M See also Niederer (1973) and Boyer (1974).

= $ "E"

+ =

DD

> 0. 7


 F

;

=

+

5

1 6

The transformation :

#

+

2 /

=

2

+


(# ,  , $ ),

LK =

 >  F

.

. 2

This is a special case of equation 2.3.2.7. Boyer (1976) showed that this equation admits the separation of variables into 25 systems of coordinates for  > = 0 and 15 systems of coordinates for  > ≠ 0. 5.

1 2

2

= 4

. 1

1 3

2

2

2

+

2

1 5

1

2 /

+ (7

+8

3ON 5

1 6

3?P

+s ) . 3RQ

6

This is a special case of equation 2.3.2.2 with S (T ) = 6.

1 2

2

= 4

. 1

1 3

2

2

2

+

2

1 5

1

2 /

+ \ –7 (

1 6

2

5

2

+ 6

)+8

1 2

2

= 4

. 1

1 3

2

2

2

+

2

1 5

1

2 /

+

+

7 11 2

1

1 6

1 5

+8

1

3?N

5 1

@ 1 T]U

+8

7 21 2

, W (T ) =

>?TVU

This is a special case of equation 2.3.2.3 with S (T ) = 7.

2

, and X (T ) = +s

2

3?N

6

, W (T ) =

@ 2 T]U

2

3RQ

 2

YAT[Z

.

.

, and X (T ) =

YAT[Z

.

. 2

1 6

2

*

@AT

This equation describes an unsteady temperature (concentration) field in a medium moving with a constant velocity, provided there is volume release (absorption) of heat proportional to temperature. : The substitution (; , < , T ) = exp C_^ ^ 1

=−

> 1

,

8.

1 2

2

=

. 1 4

1 3

2

2

2

1 5

1

2 /

+

+

7 11 2

1 6

: The transformation

c

1;

#

1 3

=

2

4

. 1

2

2

+

1 5

2

1

2 1 6

`

b d

2a

(; , < , T ),

T F a

+ > 22 , 4b that is considered in Subsection 2.1.1.

=@ −

+@

)T +

2


leads to the two-dimensional heat equation tion 2.1.1. 1 2

,

7 21 2

1 5

(; , < , T ) = exp \ ( @

9.

a

> 2

2b =

+`

2
21

%

2

+

=−

^ 2

2b leads to the two-dimensional heat equation

+^

1;

1 2 1 5

2 1

+ @ 22 )T 3 + 12 ( > ,

1T

c

=< +b e

= b (c a

%

+

+8 5

7 2

2

3?N

1 2 1 6

DD

a

2

1@ 1

@ 2T

2

+>

2

1

2 @ 2 )T

+>

"E"

+c

+ (8

+9 ) . 6

+B

T  a

(# , e , T ),

2T

) that is considered in Subseca

1

3?P

2

5

+8

2

2

3?P

6

+s ) . 3RQ

2

This is a special case of equation 2.3.2.5. The equation can be reduced to the two-dimensional heat equation treated in Subsection 2.1.1.

© 2002 by Chapman & Hall/CRC Page 199

h f g 1 2

10.

h 2 g

+

2

2

2 = 0. 2 2 / 2i 1 3 1 5 1 6 Two-dimensional Schro¨ dinger equation, j 2 = −1. Fundamental solution:k . 1

+

2

1

(; , < , T ) = −

mon

jL,

exp H

2

2l

jL,

2 Xg

(;

T

2

l

+ < 2) − j

2

Xg T Reference: V. S. Vladimirov, V. P. Mikhailov, A. A. Vasharin, et al. (1974).

. K

2.3.2. Equations Containing Arbitrary Functions 1.

2

=

1 2

4

. 1

1 3

2

2

2

+ 1

2

1 5

+p ( ) .

2 /

3 2

1 6

This equation describes two-dimensional thermal phenomena in quiescent media or solids with : constant thermal diffusivities in the case of unsteady volume heat release proportional to temperature. The substitution (; , < , T ) = exp \rq S (T ) s T ra (; , < , T ) leads to the two-dimensional heat equation c t a = b ( c u u a + c vEv a ) treated in Subsection 2.1.1. 2.

2y

=

1 2

4

w x

1 3

2

2y

+ x

+[

2 |

x z

x {

The transformation 

z

(€ ,  , T ) = ‚ (ƒ , e , T ) exp „…€ = ƒ

(} ) + p

{ ~

(T ) +  †

+ 2b €

h

Š

(} ) + (} )]y .

(T ) + ˆ (‰ ) + b ‡

(‰ ) ‹ ‰ , †

= e

Š

+ 2b 

2

† Š

‡

(‰ ) ‹

+b ‰

Š

2

‡

(‰ ) ‹

‰Œ

,

1 , 2 ¯

(‰ ) ‹ ‰ ,

where †

(‰ ) = Š

Ž

(‰ ) ‹ ‰ ,

(‰ ) = ‡

leads to the two-dimensional heat equation 3.

y x x

}

=

2y

—

w x

2

2y

+ x

x z

2 |

+ [– ˜ (

x {

2

z

Š

=

c ‘’‚

+

2

{

(‰ ) ‹ ‰ , 

z

(€ ,  , ‰ ) = ‚ (ƒ , e ,  ) exp „ = Ÿ exp “ 2 ¢

,

b œ £–

=   exp “ 2 ¢ e

leads to an equation of the form 2.3.2.2: ¥ ¦ ¥ £

=

¥ 2¦

¤ § ¥ ¡

2

+

¥ 2¦ ¥ ¨

ln( ¯A ) 1 1 Ž § © , ¬ ( ) = ( ¯A )3 ° ¯ ( ¯A )3 ° 2 2 › . Case œ < 0. The transformation ± †

( ) =

2©

2

+

¯=

Ÿ

¨



(‰ ) ‹ ‰ ,

.

 

{

œ

(Ÿ

b

, 

2

+   2 )Œ , 1

=

4¢

exp “ 4 ¢

¤ œ

¦

¨

¡

¤ œ £–

( ) + ¬ ( ) + ­ ( )® ,

ª«†

, ©

¢

−œ

2¢ ¤

( ) = ­

tan “ 2 ¢ − ¤ ,

¯

=

œ £–

1 ¯A

(Ÿ

 §

2

ln( ¯A ) ¯

©

+

= 4¢

¤ œ

.

+   2 )Œ ,

1

tan “ 2 ¢ − ¤ œ £ – cos “ 2 ¢ − ¤ œ £– cos “ 2 ¢ − ¤ œ £– 2 ¢ −¤ œ also leads to an equation of the form 2.3.2.2 (the transformed equation is not written out here). ,

¯=

1 2ž

¤ œ £–

¯

(Ÿ ,   , £ ) = ² ( ¯, ¯ ,  ¯ ) exp ³ ¡

~

ln( ¯A )

 §

¨

¡

•E• ‚ –

Š

) + ™ (} ) + (} ) + š (} )]y .

1 › . Case œ > 0. The transformation 

¡

+c

b “_c ””E‚

(‰ ) = ˆ

© 2002 by Chapman & Hall/CRC Page 200




4. 

2




=  1( )

2




    2 +  ( , , ).

 2 +  2( )

This is a special case of equation 2.3.2.8. Let 0 <  1 ( ) < and 0 <  2 ( ) < . For the first, second, third, and mixed boundary value problems treated in rectangular, finite, or infinite domains ( 1 ≤ ≤ 2 ,  1 ≤  ≤  2 ), the Green’s function can be represented in the product form

( ,  , ,  ,  ,  ) = 



 1=



1 (

 1 ( )   , 

, ,  1) 



2 (



 2=

,  ,  2),

 2 ( )   . 



Here, 1 = 1 ( , ,  ) is the auxiliary Green’s function that corresponds to the one-dimensional heat equation for  1 ( ) = 1,  2 ( ) = 0, and  ( ,  ,  ) = 0 with homogeneous boundary conditions at = 1 and = 2 (the 1 ’s for various boundary value problems can be found in Subsections 1.1.1 and 1.1.2). Similarly, 2 = 2 ( ,  ,  ) is the auxiliary Green’s function that corresponds to the one-dimensional heat equation for  1 ( ) = 0,  2 ( ) = 1, and  ( ,  ,  ) = 0 with homogeneous boundary conditions at  =  1 and  =  2 . Note that the Green’s functions 1 and 2 are introduced for  = 0. See Subsection 0.8.1 for solution of various boundary value problems with the help of the Green’s function. Example 1. Domain: −  <  <  An initial condition is prescribed:

, −

6 ? 2 XZY 0

U1

sin [

2 XZY U2

=1

0

sin [ =1

3.1.1-20. Domain: 0 ≤ B ≤ U 1 , 0 ≤ E ≤ U 2 , − V

2

(F − 4 ) : D 46 ?

> \ B U1 ] > \ E U2 ]

sin [ sin [

1 (B

> \ 2 U1 ] > \ 3 U2 ]

, 2 , ? )D

2 (E

exp [ − exp [ −

, 3 , ? ),

> 2\ 26 ? U 12

, ]

> 2\ 26 ? U 22 ]

.

< F < V . Second boundary value problem.

An infinite cylindrical domain of a rectangular cross-section is considered. The following conditions are prescribed: H = I (B , E , F ) at ? = 0 (initial condition), K ^ H K ^ H K _ H K _ H

= J 1 (E , F , ? ) = J 2 (E , F , ? )

at at

B =0

= J 3 (B , F , ? ) at = J 4 (B , F , ? ) at

E =0

B =U1 E =U2

(boundary condition), (boundary condition), (boundary condition), (boundary condition).

© 2002 by Chapman & Hall/CRC Page 215

Solution: H

2

1

(B , E , F , ? ) = / 0 / W / W I (2 , 3 , 4 ) (B , E , F , 2 , 3 , 4 , ? ) 5 2 5 3 5 4 −

0

1

0

0

2

− 6 / 7 / 0 / W J 1 (3 , 4 , 8 ) (B , E , F , 0, 3 , 4 , ? − 8 ) 5 3 5 4 5 8 0

−

1

0

0

2

+ 6 / 7 / 0 / W J 2 (3 , 4 , 8 ) (B , E , F , U 1, 3 , 4 , ? − 8 ) 5 3 5 4 5 8 0

−

1

0

0

1

− 6 / 7 / 0 / W J 3 (2 , 4 , 8 ) (B , E , F , 2 , 0, 4 , ? − 8 ) 5 2 5 4 5 8 0

−

1

0

0

1

+ 6 / 7 / 0 / W J 4 ( 2 , 4 , 8 ) ( B , E , F , 2 , U 2, 4 , ? − 8 ) 5 2 5 4 5 8 , 0

−

1

0

0

where 1

(B , E , F , 2 , 3 , 4 , ? ) = D 1 (B , 2 , ? ) = D 2 (E , 3 , ? ) =

1 2= > 6 ? 1 U1

1 U2

exp 9 −

XZY 91+2 0

( F − 4 )2 : D 46 ?

cos [ =1

XZY 91+2 0

cos [ =1

3.1.1-21. Domain: 0 ≤ B ≤ U 1 , 0 ≤ E ≤ U 2 , − V

> \ B

1 (B

, 2 , ? )D

U1 ]

cos [

> \ E U2 ]

cos [

2 (E

> \ 2 U1 ] > \ 3 U2 ]

, 3 , ? ),

exp [ − exp [ −

> 2\ 26 ?

: ,

U 12 > 2\ 26 ?

]

U 22 ]

: .

< F < V . Third boundary value problem.

An infinite cylindrical domain of a rectangular cross-section is considered. The following conditions are prescribed: H K ^ H K ^ H

−`

= I (B , E , F ) = J 1 (E , F , ? )

1

H H

at at

+ ` 2 = J 2 (E , F , ? ) at − ` 3 H = J 3 (B , F , ? ) at

K _ H K _ H

+ ` 4 H = J 4 (B , F , ? ) at

? =0 B =0

(initial condition), (boundary condition),

B =U1 E =0

(boundary condition), (boundary condition),

E =U2

(boundary condition).

The solution H (B , E , F , ? ) is determined by the formula in Paragraph 3.1.1-20 where 1 ( F − 4 )2 : D 1 (B , 2 , ? ) D 2 (E , 3 , ? ), (B , E , F Y , 2 , 3 ,Y 4 , ? ) = exp 9 − 46 ? 2= > 6 Y ? 1 Y XZY X d (E ) d (3 ) (B ) (2 ) b b b b D 1 (B , 2 , ? ) = 0 exp(− 6 c 2 ? ), D 2 (E , 3 , ? ) = 0 exp(− 6 f 2d ? ). 2 2 =1

Here, Y

Y

the c

e

a B )+

d (E ) = cos( f d E ) +

d

d =1 e

a

Y

(B ) = cos(c a

a

Y

Y Y

` 1 c

sin(c

Y B ),

` 3 sin( f d E ), f d

b b a d

Y

e

e

2Y

2 1 2 2

Y Y ` 1 U1 ` 12 + [ 1 + , 2c 2 2 c 2 ]

b 2 =

` 2 c +` 2 2 2c c + `

b 2 =

` 4 f 2d + ` 32 ` 3 U2 ` 2 + 2 + [ 1 + 23 ; 2 2 2 2f d f d + ` 4 2f d 2 f d ]

+

e and f d are positive roots of the transcendental equations

tan(c U 1 ) c

=

` 1+` 2 , c 2 − ` 1` 2

tan( f U 2 ) f

=

` 3+` 4 . f 2 − ` 3` 4

© 2002 by Chapman & Hall/CRC Page 216

NPQ

3.1.1-22. Domain: 0 ≤ B ≤ U 1 , 0 ≤ E ≤ U 2 , − V

< F < V . Mixed boundary value problem.

An infinite cylindrical domain of a rectangular cross-section is considered. The following conditions are prescribed: H

= I (B , E , F )

at

? =0

(initial condition),

H

= J 1 (E , F , ? )

at

B =0

(boundary condition),

H

= J 2 (E , F , ? )

at

B =U1

(boundary condition),

= J 3 (B , F , ? ) at

E =0

(boundary condition),

= J 4 (B , F , ? ) at

E =U2

(boundary condition).

K _ H K _ H

Solution: H

2

1

(B , E , F , ? ) = g h g W g W I (i , j , k ) l (B , E , F , i , j , k , ? ) m i m j m k −

0

0

h

K

2

+ n g 7 g h g W J 1 (j , k , 8 ) 9 K 0

−

h −

0

h

=0

m j m k m 8

K

2

− n g 7 g h g W J 2 (j , k , 8 ) 9 K 0

l (B , E , F , i , j , k , ? − 8 )o p i

0

l (B , E , F , i , j , k , ? − 8 )o p i

m j m k m 8

=

1

W

1

− n g 7 g h g W J 3 (i , k , 8 ) l (B , E , F , i , 0, k , ? − 8 ) m i m k m 8 0

−

0

h

1

+ n g 7 g h g W J 4 (i , k , 8 ) l (B , E , F , i , U 2, k , ? − 8 ) m i m k m 8 , 0

−

0

h

where l (B , E , F , i , j , k , ? ) =

2 U 1U 2 = > n ?

1 X + h 2 d

×9

exp 9 − cos [

( F − k )2 ZX Y o 9 h 4n ?

> q

=1

B U2 ]

cos [

sin [ =1

> q

i U2 ]

> \ B U1 ]

exp [ −

sin [

> \ i U1 ]

> 2q 2 n ? U 22 ]

exp [ −

> 2\ 2 n ? U 12 ]

o

o .

3.1.1-23. Domain: 0 ≤ B ≤ U 1 , 0 ≤ E ≤ U 2 , 0 ≤ F < V . First boundary value problem. A semiinfinite cylindrical domain of a rectangular cross-section is considered. The following conditions are prescribed: H

= I (B , E , F )

at

? =0

(initial condition),

H

= J 1 (E , F , ? )

at

B =0

(boundary condition),

H

= J 2 (E , F , ? )

at

B =U1

(boundary condition),

H

= J 3 (B , F , ? ) at

E =0

(boundary condition),

H

= J 4 (B , F , ? ) at

E =U2

(boundary condition),

H

= J 5 (B , E , ? ) at

F =0

(boundary condition).

© 2002 by Chapman & Hall/CRC Page 217

Solution: H

2

1

(B , E , F , ? ) = g h g W g W I (i , j , k ) l (B , E , F , i , j , k , ? ) m i m j m k 0

0

0

K

2

+ n g 7 g h g W J 1 (j , k , 8 ) 9 K 0

0

0

K

2

− n g 7 g h g W J 2 (j , k , 8 ) 9 K 0

0

0

i

0

K

1

+n g 7 g h g 0

0

j K

1

0

0

2

+ n g 7 g W g W J 5 (i , j , 8 ) 9 K 0

where

0

j K

1

0

l (B , E , F , i , j , k , ? − 8 )o p

=0

k

l (B , E , F , i , j , k , ? − 8 )o r

1

W =0

l (B , E , F , i , j , k , ? − 8 )o M

m j m k m 8 m j m k m 8

=

l (B , E , F , i , j , k , ? − 8 )o M

W J 3 (i , k , 8 ) 9 K

− n g 7 g h g W J 4 (i , k , 8 ) 9 K 0

l (B , E , F , i , j , k , ? − 8 )o p i

m i m k m 8 m i m k m 8

=

2

W =0

m i m j m 8 ,

l (B , E , F , i , j , k , ? ) = l 1 (B , i , ? ; U 1 ) l 1 (E , j , ? ; U 2 ) l 2 ( F , k , ? ), > \ B > \ i > 2\ 2n ? 2 XZY h sin [ l 1 (B , i , ? ; U ) = sin [ exp [ − , U U ] U ] U2 ] =1

1

l 2 (F , k , ? ) =

exp 9 −

2s t n u v

( w − k )2 ( w + k )2 o − exp 9 − o x . 4n u 4n u

3.1.1-24. Domain: 0 ≤ y ≤ z 1 , 0 ≤ { ≤ z 2 , 0 ≤ w < | . Second boundary value problem. A semiinfinite cylindrical domain of a rectangular cross-section is considered. The following conditions are prescribed:} }

= ~ (y , { , w )

at

u =0

(initial condition),

}

= € 1 ({ , w , u ) = € 2 ({ , w , u )

at at

y =0

(boundary condition), (boundary condition),

= € 3 (y , w , u ) at  _ } = € 4 (y , w , u ) at

{ =0 { =z2

(boundary condition), (boundary condition),

= € 5 (y , { , u ) at

w =0

(boundary condition).

 ^  ^  _

 

Solution:

} }

}

2

y =z1

1

(y , { , w , u ) = g h g ‚ g ‚ ~ (i , j , k ) l (y , { , w , i , j , k , u ) m i m j m k 0

0

0

2

− n g 7 g h g ‚ € 1 (j , k , 8 ) l (y , { , w , 0, j , k , u − 8 ) m j m k m 8 0

0

0

2

+ n g 7 g h g ‚ € 2 (j , k , 8 ) l (y , { , w , z 1, j , k , u − 8 ) m j m k m 8 0

0

0

1

− n g 7 g h g ‚ € 3 (i , k , 8 ) l (y , { , w , i , 0, k , u − 8 ) m i m k m 8 0

0

0

1

+ n g 7 g h g ‚ € 4 (i , k , 8 ) l (y , { , w , i , z 2, k , u − 8 ) m i m k m 8 0

0

0

2

1

− n g 7 g ‚ g ‚ € 5 (i , j , 8 ) l (y , { , w , i , j , 0, u − 8 ) m i m j m 8 , 0

0

0

© 2002 by Chapman & Hall/CRC Page 218

ƒP…

where l (y , { , w , i , j , k , u ) = l 1 (y , i , u ) = l 2 ({ , j , u ) =

1

exp 9 −

2s t n u v

‰ZŠ 91+2 h

1 z1

( w − k )2 ( w + k )2 o + exp 9 − o x l 4n u 4n u

cos ‹

t Œ y z1 

=1

‰ZŠ

1 z2 Ž

1+2 

cos ‹

cos ‹

t Π{

cos ‹

z2 

=1

t Œ i z1  t Œ  z2 

exp ‹ − exp ‹ −

1 (y

,i ,u )l

t 2Π2n u z 12

, j , u ),

o , 

t 2Œ 2‘ u z 22

2 ({

 ’

.

3.1.1-25. Domain: 0 ≤ y ≤ z 1 , 0 ≤ { ≤ z 2 , 0 ≤ w < | . Third boundary value problem. A semiinfinite cylindrical domain of a rectangular cross-section is considered. The following conditions are prescribed: }

“ =0

(initial condition),

− • 1 } = € 1 ({ , w , “ )

at

y =0

(boundary condition),

}

+ • 2 } = € 2 ({ , w , “ )

at

y =z1

(boundary condition),

}

− • 3 } = € 3 (y , w , “ ) at

{ =0

(boundary condition),

+ • 4 } = € 4 (y , w , “ ) at

{ =z2

(boundary condition),

−•

w =0

(boundary condition).

}

 –  

at

}

 ”  –

= ~ (y , { , w )

}

 ”

}

}

5

= € 5 (y , { , “ ) at

The solution (y , { , w , “ ) is determined by the formula in Paragraph 3.1.1-24 where —

(y , { , w , ˜ ,  , ™ , “ ) = š

1 (y

, ˜ , “ )š

2 ({

,  , “ )š

3(›

, ™ , “ ),

(› − ™ ) ( › + ™ )2 + exp − ‘ 4 “ ’ 4‘ “ ’ Ÿ 2œ  ‘ “ ž Ž Ž › +™ − • 5 exp  ¡• 52 ‘ “ + • 5 ( › + ™ )¢ erfc ‹ + • 5œ ‘ “ , 2œ ‘ “  1

š 3 (› , ™ , “ ) =

and the functions š

1 (£

, ˜ , “ ) and š

2 (¤

exp −

2

,  , “ ) can be found in Paragraph 3.1.1-21.

3.1.1-26. Domain: 0 ≤ £ ≤ ¥ 1 , 0 ≤ ¤ ≤ ¥ 2 , 0 ≤ › < ¦ . Mixed boundary value problems. 1 § . A semiinfinite cylindrical domain of a rectangular cross-section is considered. The following conditions are prescribed:¨ ¨ ¨ ¨ ¨ ¨ « ¬

= © (£ , ¤ , › )

at

“ =0

(initial condition),

= ª 1 (¤ , › , “ )

at

£ =0

(boundary condition),

= ª 2 (¤ , › , “ )

at

£ =¥1

(boundary condition),

= ª 3 (£ , › , “ ) at

¤ =0

(boundary condition),

= ª 4 (£ , › , “ ) at

¤ =¥2

(boundary condition),

= ª 5 (£ , ¤ , “ ) at

› =0

(boundary condition).

© 2002 by Chapman & Hall/CRC Page 219

Solution: ¨ 2

—

1

(£ , ¤ , › , “ ) = ­  ­ ® ­ ® © (˜ ,  , ™ ) (£ , ¤ , › , ˜ ,  , ™ , “ ) ¯ ˜ ¯  ¯ ™ 0

+‘ ­

0

0

«

2

0

° ­

0

—

® ª 1 ( , ™ , ± ) « Ž ˜ 0 « 2

 ­

− ‘ ­ ° ­  ­ ® ª 2 ( , ™ , ± ) «

—

(£ , ¤ , › , ˜ ,  , ™ , “ − ± )

’ ² =0

¯  ¯ ™ ¯ ±

(£ , ¤ , › , ˜ ,  , ™ , “ − ± )

¯  ¯ ™ ¯ ± ’ ² =1 « ® 1 — + ‘ ­ ° ­  ­ ® ª 3 (˜ , ™ , ± ) « (£ , ¤ , › , ˜ ,  , ™ , “ − ± ) ¯ ˜ ¯ ™ ¯ ± Ž  ’ ³ =0 0 0 0 « 1 — − ‘ ­ ° ­  ­ ® ª 4 (˜ , ™ , ± ) « (£ , ¤ , › , ˜ ,  , ™ , “ − ± ) ¯ ˜ ¯ ™ ¯ ± Ž  ’ ³ =2 0 0 0 ® 2 1 — − ‘ ­ ° ­ ® ­ ® ª 5 (˜ ,  , ± ) (£ , ¤ , › , ˜ ,  , 0, “ − ± ) ¯ ˜ ¯  ¯ ± , 0

0

0

0

˜

Ž

0

0

where —

(£ , ¤ , › , ˜ ,  , ™ , “ ) = š

(£ , ˜ , “ ; ¥ ) =

( › − ™ )2 ( › + ™ )2 š (£ , ˜ , “ ; ¥ 1 ) š (¤ ,  , “ ; ¥ 2), + exp − 4‘ “ ’ 4‘ “ ’ Ÿ Ž Ž

1

exp −

2œ  ‘ “ ž 2 ´Zµ ¥

sin ¶ 

 · £ ¥

=1

¸

sin ¶

 · ˜ ¥

¸

exp ¶ −

 2· 2‘ “ ¥2 ¸

.

2 § . A semiinfinite cylindrical domain of a rectangular cross-section is considered. The following ¨ conditions are prescribed: ¨

= © (£ , ¤ , › )

at

“ =0

(initial condition),

¨ = ª 1 (¤ , › , “ ) « ” ¨ = ª 2 (¤ , › , “ )

at at

£ =0

(boundary condition), (boundary condition),

= ª 3 (£ , › , “ ) at = ª 4 (£ , › , “ ) at

¤ =0 ¤ =¥2

(boundary condition), (boundary condition),

= ª 5 (£ , ¤ , “ ) at

› =0

(boundary condition).

« ”

¨

« –

« – ¨

£ =¥1

Solution: ¨ 2

—

1

(£ , ¤ , › , “ ) = ­  ­ ® ­ ® © (˜ ,  , ™ ) (£ , ¤ , › , ˜ ,  , ™ , “ ) ¯ ˜ ¯  ¯ ™ 0

0

0

2

—

2

—

− ‘ ­ ° ­  ­ ® ª 1 ( , ™ , ± ) (£ , ¤ , › , 0,  , ™ , “ − ± ) ¯  ¯ ™ ¯ ± 0

0

0

+ ‘ ­ ° ­  ­ ® ª 2 ( , ™ , ± ) (£ , ¤ , › , ¥ 1,  , ™ , “ − ± ) ¯  ¯ ™ ¯ ± 0

0

0

1

—

1

—

− ‘ ­ ° ­  ­ ® ª 3 (˜ , ™ , ± ) (£ , ¤ , › , ˜ , 0, ™ , “ − ± ) ¯ ˜ ¯ ™ ¯ ± 0

0

0

+ ‘ ­ ° ­  ­ ® ª 4 ( ˜ , ™ , ± ) ( £ , ¤ , › , ˜ , ¥ 2, ™ , “ − ± ) ¯ ˜ ¯ ™ ¯ ± 0

+‘ ­

0

0

2

1

«

° ­ ® ­ ® ª 5 (˜ ,  , ± ) « Ž ™ 0 0 0

—

(£ , ¤ , › , ˜ ,  , ™ , “ − ± )

’ ¹ =0

¯ ˜ ¯  ¯ ± ,

© 2002 by Chapman & Hall/CRC Page 220




where 





1

( , , ,  , , ,  ) =

exp  −

2     1+2 

1

( ,  ,  ;  ) =





( − )2 ( + )2 − exp  − ( ,  ,  ;  1 ) ( , ,  ;  2), 4   4     

cos 

  

=1



  

cos  





exp  −

2 2





 

2



.

 

3.1.1-27. Domain: 0 ≤  ≤  1 , 0 ≤ ≤  2 , 0 ≤ ≤  3 . First boundary value problem. A rectangular parallelepiped is considered. The following conditions are prescribed: 

= ( , , ) = ! 1 ( , ,  )

at at

 =0

= ! 2 ( , ,  ) at  = ! ( , ,  ) at 3

 =

= ! 4 ( , ,  ) at = ! 5 ( , ,  ) at

=

=0

(boundary condition), (boundary condition),

= ! 6 ( , ,  ) at

=

(boundary condition).

    

(initial condition), (boundary condition),

 =0

(boundary condition), (boundary condition),

1

=0 2

3

Solution:  ( , , ,  ) = "

0

3

2

# "

# "

0



1

0

( , , ) ( , , ,  , , ,  ) $  $ $

#

3



2

+  " % " # " # ! 1 ( , , & )  ' 0

0

0

3

'

2



−  " % " # " # ! 2 ( , , & )  ' 0

0

0

3



+  " % " # " # ! 3 ( , , & )  ' 0

0

0

3

1

0

0

0

2

'

1



0

0



2

1

0

where  

1 (





2 (

+-,

3(

0

0

'

( , , ,  , , ,  ) = , , ) = , , ) = , , ) =

2 

sin 

1



2 

=1

sin 

2



2  3

=1

sin   =1



1 (

   1    2   

3 

( , , ,  , , ,  − & )

'

−  " % " # " # ! 6 ( , , & )  '

( , , ,  , , ,  − & )



+  " % " # " # ! 5 ( , , & )  ' 0

( , , ,  , , ,  − & )



'

−  " % " # " # ! 4 ( , , & )  '

( , , ,  , , ,  − & )



'

1

( , , ,  , , ,  − & )





( , , ,  , , ,  − & )

 (

=0

 (

=

$ $ $ & $ $ $ &

1

#  )

=0

 )

=

$  $ $ & $  $ $ &

2

#  *

=0

 *

=

$  $ $ & $  $ $ & ,

3

#

, , ) sin  sin  sin 



2 (

, , )

   1    2    3 



3(

, ,  ), 2 2





exp  − 

exp  − exp  −

 

,

 12 2 2



  



 22 2 2





2 3

   

, .

Reference: H. S. Carslaw and J. C. Jaeger (1984).

© 2002 by Chapman & Hall/CRC Page 221

3.1.1-28. Domain: 0 ≤  ≤  1 , 0 ≤ ≤  2 , 0 ≤ ≤  3 . Second boundary value problem. A rectangular parallelepiped is considered. The following conditions are prescribed:  

' .  ' .  ' /  ' /  ' 0  ' 0

Solution:

= =! =! =! =! =! =!

 ( , , ,  ) = "

( , , ) , , ) 2 ( , ,  ) 3 ( , ,  ) 4 ( , ,  ) 5 ( , ,  ) 6 ( , ,  )

0

3

2

# "

# "

0

0

(initial condition), (boundary condition), (boundary condition), (boundary condition), (boundary condition), (boundary condition), (boundary condition).

 =0  =

1

=0 =

2

=0

=

3



1

( , , ) ( , , ,  , , ,  ) $  $ $

#

3

 =0

at at at at at at at

1 (

2



2



−  " % " # " # ! 1 ( , , & ) ( , , , 0, , ,  − & ) $ $ $ & 0

0

0

3

+  " % " # " # ! 2 ( , , & ) ( , , ,  1, , ,  − & ) $ $ $ & 0

0

0

3

1



1



−  " % " # " # ! 3 ( , , & ) ( , , ,  , 0, ,  − & ) $  $ $ & 0

0

0

3

+  " % " # " # ! 4 ( , , & ) ( , , ,  ,  2, ,  − & ) $  $ $ & 0

0

0

2

1



1



−  " % " # " # ! 5 ( , , & ) ( , , ,  , , 0,  − & ) $  $ $ & 0

0

0

2

+  " % " # " # ! 6 (  , , & ) (  , , ,  , ,  3,  − & ) $  $ $ & , 0

where

0

0









( , , ,  , , ,  ) = 1 ( ,  ,  ) 2 ( , ,  ) 3 ( , ,  ),         1  2 2   1 + 2  cos  cos  exp  − 1 ( ,  ,  ) = 2 1





2 (

3 (

, , ) = , , ) =

1 2

1 3

1 

=1

 1+2 

cos 

  2 

=1

 1+2 

cos 

 

3 

=1

1 

cos  cos 

  2    3 

1



exp  − exp  −

2 2







 

2 2

2 2



 

   

 32

 

, , .

3.1.1-29. Domain: 0 ≤  ≤  1 , 0 ≤ ≤  2 , 0 ≤ ≤  3 . Third boundary value problem. A rectangular parallelepiped is considered. The following conditions are prescribed:  

' .  ' .  ' /  ' /  ' 0  ' 0

−1 +1 −1 +1 −1 +1



1



2



3



4 5 6





= =! =! =! =! =! =!

( , , ) , , ) 2 ( , ,  ) 3 ( , ,  ) 4 ( , ,  ) 5 ( , ,  ) 6 ( , ,  ) 1 (

at at at at at at at

 =0  =0  =

1

=0 =

2

=0

=

3

(initial condition), (boundary condition), (boundary condition), (boundary condition), (boundary condition), (boundary condition), (boundary condition).

© 2002 by Chapman & Hall/CRC Page 222


 

The solution  ( , , ,  ) is determined by the formula in Paragraph 3.1.1-28 where 

( , , ,  , , ,  ) =

The functions 1 ( ,  ,  ) and   3 ( , ,  ) is given by

2 (

1 (



 

( ) = cos( 4 2

 )+



1



sin( 4

4

( )

, , ) =  

5

3

2 

=1

3 2

3

 ),

3 22

=

( )

3(



, ,  ).

2



exp(−  4 2  ), 

1

24



6 2

2 4 4

2

2 

where the 4

, , )

 



3(

2 (



, ,  ) can be found in Paragraph 3.1.1-21, and the function





, , )

are positive roots of the transcendental equation

+1 +1

2 5 2 6

+

1

24



5 2

tan( 4  3 ) 4

=

3

+

2 1

4

 1+ 4

1

2 5 2



,

+1 6 . − 1 51 6

5 2

3.1.1-30. Domain: 0 ≤  ≤  1 , 0 ≤ ≤  2 , 0 ≤ ≤  3 . Mixed boundary value problems. 1 5 . A rectangular parallelepiped is considered. The following conditions are prescribed: 

= ( , , )

at

 =0

(initial condition),

= ! 1 ( , ,  ) = ! 2 ( , ,  )

at at

 =0

(boundary condition), (boundary condition),

= ! 3 ( , ,  ) at  = ! ( , ,  ) at 4

=0 2

(boundary condition), (boundary condition),

= ! 5 ( , ,  ) at

=0

(boundary condition),

= ! 6 ( , ,  ) at

=

(boundary condition).

  



' 0  ' 0

 = =

1

3

Solution:  ( , , ,  ) = "

0

3

2

# "

# "

0



1

0

( , , ) ( , , ,  , , ,  ) $  $ $

#

3



2

+  " % " # " # ! 1 ( , , & )  ' 0

0

0

3

'

2



−  " % " # " # ! 2 ( , , & )  ' 0

0

0

3



+  " % " # " # ! 3 ( , , & )  ' 0

0

0

3

−  " % " # " # ! 4 ( , , & )  ' 0

0

0

2

1



1



'

( , , ,  , , ,  − & )



'

1

( , , ,  , , ,  − & )



'

1

( , , ,  , , ,  − & )







( , , ,  , , ,  − & )

 (

=0

 (

=

0

0

$ $ $ & 1

#  )

=0

 )

=

−  " % " # " # ! 5 ( , , & ) ( , , ,  , , 0,  − & ) $  $ $ &

$ $ $ &

$  $ $ & $  $ $ &

2

#

0

2

+  " % " # " # ! 6 (  , , & ) (  , , ,  , ,  3,  − & ) $  $ $ & , 0

0

0

© 2002 by Chapman & Hall/CRC Page 223

where 





( , , ,  , , ,  ) = , , ) =



1 (

, , ) =



2 (

, , ) =

3 (

2 

2

2 

cos  

3 8

, , )

 9

1   1

=1

2 

3



  

sin 

2 

=1

+

2 (

sin 

 1

sin  



1 

=1

2 76

3

, , )   

sin  

1

1

1 (

cos  

3(

, ,  ),

exp  −  9

3

2 2



exp  −





2 2



 

2 1

1

 

 22

, 

exp  − 

, 



2

2

9 

 

2 3

. 

2 5 . A rectangular parallelepiped is considered. The following conditions are prescribed: 

= ( , , )

at

 =0

(initial condition),

= ! 1 ( , ,  ) = ! 2 ( , ,  )

at at

 =0

(boundary condition), (boundary condition),

= ! 3 ( , ,  ) at ' /  = ! 4 ( , ,  ) at

=0

= ! 5 ( , ,  ) at = ! 6 ( , ,  ) at

=0

  

' /  ' 0  ' 0

Solution:

 ( , , ,  ) = "

0

3

2

# "

# "

0

=

=

1

2

3



2

+  " % " # " # ! 1 ( , , & )  ' 0

0

3

'

2

0

0

3





1



( , , ,  , , ,  − & )



'

2

( , , ,  , , ,  − & )



−  " % " # " # ! 2 ( , , & )  ' 0

(boundary condition), (boundary condition).

( , , ) ( , , ,  , , ,  ) $  $ $

#

3

0

(boundary condition), (boundary condition),



1

0

 =

 (

=0

 (

=

0

$ $ $ & 1

#

−  " % " # " # ! 3 ( , , & ) ( , , ,  , 0, ,  − & ) $  $ $ & 0

$ $ $ &

0

3

+  " % " # " # ! 4 ( , , & ) ( , , ,  ,  2, ,  − & ) $  $ $ & 0

0

0

2

1



1



−  " % " # " # ! 5 ( , , & ) ( , , ,  , , 0,  − & ) $  $ $ & 0

0

0

2

+  " % " # " # ! 6 (  , , & ) (  , , ,  , ,  3,  − & ) $  $ $ & , 0

where 

( , , ,  , , ,  ) = 

1 (

 

2 (

3(

, , ) = , , ) = , , ) =

0

0



1 (

2  1

1 2

1 3

, , )



sin   =1

+ +

2 76

3 8

, , )

  

=1

cos   =1



3(

1 

 1

cos 

2   9

3

, ,  ),   

sin 

1 

cos  

2

2 

2 (



cos 

exp  −  1

3

2 2



 

 12

exp  −

2   9





, 



exp  −

2 2

1





 

2 2 2

 2

9 

2 3

,

  

.

© 2002 by Chapman & Hall/CRC Page 224


 

3.1.2. Problems in Cylindrical Coordinates The three-dimensional sourceless heat equation in the cylindrical coordinate system has the form '

 '

=  

:

1: : 

'

 '

'

:

+ 

'

:1

2

'

2

+ '

2

' ;

:

2

'

2



=< 

,

2

+ 2.

It is used to describe nonsymmetric unsteady processes in moving media or solids with cylindrical or plane boundaries. A similar equation is used to study the corresponding three-dimensional unsteady : mass-exchange processes with constant diffusivity. One-dimensional problems with axial symmetry that have solutions of the form  =  ( ,  ) : : are discussed in Subsection 1.2.1. Two-dimensional problems whose solutions have the form  =  ( , ,  ) or  =  ( , ,  ) are considered in Subsections 2.1.2 and 2.1.3. ;

3.1.2-1. Remarks on the Green’s functions. For the three-dimensional problems dealt with in Subsection 3.1.2, the Green’s function can be represented in the product form : :    ( , ,

,  , ,

,  ) = ( 1 , ,  , ,  ) 2 ( , ,  ), : ;



;

where 1 ( , ,  , ,  ) is the Green’s function of the two-dimensional boundary value problem (such  ; functions are presented in Subsection 2.1.2), and 2 ( , ,  ) is the Green’s function of the corresponding one-dimensional boundary value problem (such functions can be found in Subsections 1.1.1 and 1.1.2). Example. The Green’s function of the first boundary value problem for a semiinfinite circular cylinder (0 ≤ = ≤ > , 0 ≤ ? ≤ 2 @ , 0 ≤ A < B ) of Paragraph 3.1.2-5 is the product of the two-dimensional Green’s function of the first boundary value problem of Paragraph 2.1.2-2 (0 ≤ = ≤ > , 0 ≤ ? ≤ 2@ ) and the one-dimensional Green’s function of the first boundary value problem of Paragraph 1.1.2-2 (0 ≤ A < B ), in which one should perform obvious renaming of variables.

General formulas that enable one to obtain solutions of basic boundary value problems with the help of the Green’s function can be found in Subsection 0.8.1. :

3.1.2-2. Domain: 0 ≤

≤C ,0≤

≤ 2 , − D ;

< < D . First boundary value problem.

An infinite circular cylinder is considered. The following conditions are prescribed: : 

= ( , , ) at ; = ! ( , ,  ) at 

;

Solution:

:  ( , , , ) = " ;

2E

 "

−

"

0

0

0

Here, 

1(

, , , , )= ;

1  C

2

1

;

2E

! ( , , & )  '

0

'

 :

 

−

( , , , , ,  , )=   =0

:

( , , ) ( , , ,  , , ,  ) $  $ $

F 

− C " % "  "

8

;

=1

[ H I (J G

8 C )]2

( − )2 , 4  

: 

( , , , , , , −& ) ;



$ $ $ & .

=



1 (

H

 (

F :









(initial condition), (boundary condition).

=C 



:

: = 0

,  ,  , ,  )  2( , ,  ), ; :

8

(J 

) H (J

8  ) cos[ ( − )] exp(−J ;

 28

  ),



1 for  = 0, 2 for  = 1, 2, KKK , 2      G where the H ( ) are the Bessel functions (the prime denotes the derivative with respect to the argument) and the J 8 are positive roots of the transcendental equation H (J C ) = 0. 

+-,

2( , ,  ) =

exp  −

=

Reference: B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

© 2002 by Chapman & Hall/CRC Page 225

:

3.1.2-3. Domain: 0 ≤

≤C ,0≤

≤ 2 , − D ;

< < D . Second boundary value problem.

An infinite circular cylinder is considered. The following conditions are prescribed: :



= ( , , ) at ; = ! ( , ,  ) at 

' L

: = 0

(initial condition), (boundary condition).

=C

;

Solution: :  ( , , , ) = " ;

−

 "

2E 0

"

0



0

−

0

 ! ( , , & ) ( , , , C , , ,  − & ) $ $ $ & . ;

:

 : 1(

, , , , )= ;



1

1 

+

2

 C

:

  ( , , ,  , ,  ,  ) =  1( , ,:  ,  ,  )  

;



 =0

8



 =1

G

28

;





2(

, ,  ),

(J 8 ) H (J 8  ) cos[ ( − )] exp(−J 2 2 8 ; (J C −  2)[ H (J 8 C )]2 J

( − )2 exp  − , 2( , ,  ) = 4   2    

H



1



= 

 28

  ),

1 for  = 0, 2 for  = 1, 2, KKK ,

G 8 are positive roots of the transcendental equation



where the H ( ) are the Bessel functions and the J H I (J C ) = 0.

+-,

;



Here,



:

2E

+ C " % "  "

: 

( , , ) ( , , ,  , , ,  ) $  $ $

F 

Reference: B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

:

3.1.2-4. Domain: 0 ≤

≤C ,0≤

≤ 2 , − D ;

< < D . Third boundary value problem.

An infinite circular cylinder is considered. The following conditions are prescribed: :



:

' L

= ( , , ) at ;   +1 = ! ( , ,  ) at

: = 0

;

=C

(initial condition), (boundary condition).

The solution  ( , , ,  ) is determined by the formula in Paragraph 3.1.2-3 where ;

:       ( , , ,  , , ,  ) = 1 ( : , ,  ,  ,  ) 2 ( , ,  ),  ;  ;  :    J 2 8 H (J 8 ) H (J 8  ) 1  28   cos[ ( − )] exp(−J   ), 1( , ,  , ,  ) = ; ;  (J 2 8 C 2 + 1 2 C 2 −  2 )[ H (J 8 C )]2 =0 8 =1 G   2  1 ( − ) 1 for  = 0, (

,

,  ) = exp  − , = 2 2 for  = 1, 2, KKK 4     2      G Here, the H ( ) are the Bessel functions and the J 8 are positive roots of the transcendental equation   

:

J H I (J C ) + 1 H +-,

(J C ) = 0.

Reference: B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

© 2002 by Chapman & Hall/CRC Page 226


 

:

3.1.2-5. Domain: 0 ≤

≤C ,0≤

≤ 2 , 0 ≤ < D . First boundary value problem. ;

A semiinfinite circular cylinder: is considered. The following conditions are prescribed:  = ( , , ) at  = 0 (initial condition), : ;



= ! 1 (: , ,  ) at  = ! ( ; , ,  ) at 2

=C

;

Solution: :

 ( , , , ) = " ;

0

2E

 "

"

0

0

0

2E

+ " % " 0

Here,

"

0

( , , ) ( , , ,  , , ,  ) $  $ $ ;

2E

0

: 

F 

− C " % "  "

(boundary condition), (boundary condition).

=0

0

0

:

;

: 

( , , , , , , −& ) ;

'

( , , , , , , −& )



'

F  ! 2 ( , , & )  '

: 

! 1 ( , , & )  '

 (

$ $ $ &

=

F  *

=0

$  $ $ & .

:    ( , , ,  , , ,  ) = 1 ( ,  ,  , ,  )  2 ( , ,  ),  ; ; : :     1   2   H (J 8 ) H (J 8  ) cos[ ( − )] exp(−J 8   ), 1( , ,  , ,  ) = ; ;  C 2 [ H I (J 8 C )]2 =0 8 =1 G     1 ( − )2 ( + )2 1 for  = 0, exp  − − exp  − , = 2 ( , ,   ) = 2 for  = 1, 2, KKK , 4   4    2      G derivative with respect to the where the H ( ) are the Bessel functions (the prime denotes the argument) and the J 8 are positive roots of the transcendental equation H (J C ) = 0. +-, 

Reference: B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

:

3.1.2-6. Domain: 0 ≤

≤C ,0≤

≤ 2 , 0 ≤ < D . Second boundary value problem. ;

A semiinfinite circular cylinder is : considered. The following conditions are prescribed:  = ( , , ) at  = 0 (initial condition), : ;  = ! 1 (: , ,  ) at =C (boundary condition), ' L  ' 0

Solution:

;

= ! 2( , ,  )

:  ( , , , ) = " ;

0

2E

 "

"

0

0

− " % " 0

Here, 

1(

, , , , ) = ;

1  C

2



0

: 

;

2E

:



! 1 ( , , & ) ( , , , C , , ,  − & ) $ $ $ & ; :  " F  ! 2 ( , , & ) ( , , ,  , , 0,  − & ) $  $ $ & . ; 0 0

:

 :

0 2E

(boundary condition).

( , , ) ( , , ,  , , ,  ) $  $ $

F 

+ C " % "  " 0

=0

at

;

:

  ( , , ,  , ,  ,  ) =  1 ( , ,:  ,  ,  ) 

+ 

; 1  



 =0

8



 =1

G

28

;



2(

, ,  ),

) H (J 8  ) cos[ ( − )] exp(−J 2 2 2 8 ; (J −  )[ H (J 8 C )]2 C J

H

(J

8





 28

  ),



( − )2 ( + )2 1 for  = 0, exp  − + exp   − , = 2 ( , ,  ) = 2 for  = 1, 2, KKK , 4   4    2     G where the H ( ) are the Bessel functions and the J 8 are positive roots of the transcendental equation H I (J C ) = 0. +-, 

1

Reference: B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

© 2002 by Chapman & Hall/CRC Page 227

:

3.1.2-7. Domain: 0 ≤

≤C ,0≤

≤ 2 , 0 ≤ < D . Third boundary value problem. ;

A semiinfinite circular cylinder is considered. The following conditions are prescribed: :

 

+1

' L  ': 0

1

= ( , , ) ; = ! ( : , , )



: = 0

at at

; − 1 2  = ! 2 ( , ,  ) at ;

=C

(initial condition), (boundary condition),

=0

(boundary condition).

The solution  ( , , ,  ) is determined by the formula in Paragraph 3.1.2-6 where ;

:

:       ( , , ,  , , ,  ) = 1( : , ,  ,  ,  ) 2( , ,  ),  ; ;  :     J 2 8 H (J 8 ) H (J 8  ) 1  28   cos[ ( − )] exp(−J   ), 1( , ,  , ,  ) = ;  ; (J 2G 8 C 2 + 1 12 C 2 −  2)[ H (J 8 C )]2 =0 8 =1   1 ( − )2 ( + )2 ( + + M )2 (

,

,  ) = exp  − + exp  − − 2 1 2 "  exp  − − 1 2M $ M . 2 4   4   4     2     0 

Here, 0 = 1 and = 2 for  = 1, 2, KKK ; the H ( ) are the Bessel functions and the J positive roots of the transcendental equation   G G J H I (J C ) + 1

8

are

(J C ) = 0.

1H

:

3.1.2-8. Domain: 0 ≤

≤C ,0≤

≤ 2 , 0 ≤ < D . Mixed boundary value problems. ;

1 5 . A semiinfinite circular cylinder is considered. The following conditions are prescribed: :



= ( , , )

' 0

at

: = 0

(initial condition),

= ! 1 (: , ,  ) at ; = ! 2 ( , ,  ) at

=C

=0

(boundary condition), (boundary condition).

;

 

;

Solution: :  ( , , , ) = " ;

0

2E

 "

0

"

0

2E

− " % " 0

Here,

0

0

"

2E 0

:

( , , ) ( , , ,  , , ,  ) $  $ $

F 

− C " % "  " 0



;

! 1 ( , , & )  ' ':





:

( , , , , , , −& ) ;

 (

=

$ $ $ &

F  F  ! 2 ( , , & ) ( , , ,  , , 0,  − & ) $  $ $ & . ; 0

:    ( , , ,  , , ,  ) = 1 ( ,  ,  , ,  )  2 ( , ,  ),  ; ; : :        1   H (J 8 ) H (J 8  ) cos[ ( − )] exp(−J 2 8   ), 1( , ,  , ,  ) = ; ;  C 2 [ H I (J 8 C )]2 8 =0 =1 G     1 ( − )2 ( + )2 1 for  = 0, exp  − + exp  − , = 2 ( , ,  ) = 2 for  = 1, 2, KKK , 4   4    2     G   where the H ( ) are the Bessel functions (the prime denotes the derivative with respect to the argument) and the J 8 are positive roots of the transcendental equation H (J C ) = 0. 

:

© 2002 by Chapman & Hall/CRC Page 228


 

2 5 . A semiinfinite circular cylinder is considered. The following conditions are prescribed: : 

' L 



= ( , , ) at ; = ! 1 (: , ,  ) at

: = 0

=C

(initial condition), (boundary condition),

= ! 2( , ,  )

=0

(boundary condition).

;

;

Solution:

:  ( , , , ) = " ;

2E

0

 "

0

"

0

at

: 

( , , ) ( , , ,  , , ,  ) $  $ $

F 

;

:  ! 1 ( , , & ) ( , , , C , , ,  − & ) $ $ $ & ; 0 0 0 : 2E  + " % " " F  ! 2 ( , , & )  ' ( , , , , , , −& ) $  $ $ & . ;

 * =0 0 0 0 ' Here, : :    ( , , ,  , ,  ,  ) =  1 ( , ,:  ,  ,  )  2 ( , ,  ),  ; ; :    2   J 8 H (J 8 ) H (J 8  ) 1 1    + cos[ ( − )] exp(−J 2 8   ), 1( , ,  , ,  ) = ; ;  C 2  (J 2 8 C 2 −  2 )[ H (J 8 C )]2 8 =0 =1 G    2 2  1 ( − ) ( + ) 1 for  = 0, exp  − − exp   − , = 2 ( , ,  ) = 2 for  = 1, 2, KKK , 4   4    2     G where the H ( ) are the Bessel functions and the J 8 are positive roots of the transcendental equation 2E

+ C " % "  "

H I (J C ) = 0.

:

3.1.2-9. Domain: 0 ≤

≤C ,0≤

≤ 2 , 0 ≤ ≤  . First boundary value problem. ;

A circular cylinder of finite length is considered. The following conditions are prescribed: : 

= ( , , )

at

: = 0

(initial condition),

= ! 1 (: , ,  ) at ; = ! 2 (: , ,  ) at

=C

=0

(boundary condition), (boundary condition),

= ! 3( , ,  )

=

(boundary condition).

;

 

;



;

Solution:

:  ( , , , ) = " ;

#"

0

2E 0

"

0

0

+ " % " 0

− " % " 0

0

"

2E 0

0

"

0

0

:

( , , ) ( , , ,  , , ,  ) $  $ $ 2E

0

2E



F 

− C " % " # "

at

;

! 1 ( , , & )  ' '

 (

( , , , , , , −& )



;

: 

' '

( , , , , , , −& )



F  ! 2 ( , , & )  ' F  ! 3 ( , , & )  '

: 

:

;

( , , , , , , −& ) ;

$ $ $ &

=

F  *

=0

 *

=

$  $ $ &

$  $ $ & .

# :    ( , , ,  , , ,  ) = 1 ( ,  ,  , ,  )  2 ( , ,  ),  ; ; : :        1   H (J 8 ) H (J 8  ) cos[ ( − )] exp(−J 2 8   ), 1( , ,  , ,  ) = ; ;  C 2 [ H I (J 8 C )]2 8 =0 =1 G     

    2 2 2  1 for  = 0,  (

,

,  ) = sin  sin  exp  − , = 2 2 for  = 1, 2, K KK ,      2  =1 G

Here,



:

© 2002 by Chapman & Hall/CRC Page 229

 



where the H ( ) are the Bessel functions (the prime denotes the derivative with respect to the argument) and the J 8 are positive roots of the transcendental equation H (J C ) = 0. :

3.1.2-10. Domain: 0 ≤

≤C ,0≤

≤ 2 , 0 ≤ ≤  . Second boundary value problem. ;

A circular cylinder of finite length : is considered. The following conditions are prescribed: 

' L  ' 0 

Solution:



= ( , , ) at ; = ! 1 (: , ,  ) at

: = 0

= ! 2 (: , ,  ) ; = ! 3( , ,  )

=0

;

' 0

:  ( , , , ) = " ;

0

2E

#"

"

0

0

=C

at at

;

(initial condition), (boundary condition),

= : 

( , , ) ( , , ,  , , ,  ) $  $ $

F 

;

2E

+ C " % " # "

(boundary condition), (boundary condition).



:

! 1 ( , , & ) ( , , , C , , ,  − & ) $ $ $ & ; :  − " % " " F  ! 2 ( , , & ) ( , , ,  , , 0,  − & ) $  $ $ & ; 0 0 0 : 2E  + " % " " F  ! 3 ( , , & ) ( , , ,  , ,  ,  − & ) $  $ $ & . ; 0 0 0 0

Here,  

1(

, , , , ) = ;

2( , ,  ) =





1 

+

2  

 C

:

  ( , , ,  , ,  ,  ) =  1 ( , ,:  ,  ,  )  ;

1

1 

+

2





 8

=0



 =1

  

cos  

0

:

 :

0 2E

=1



G

;

J

(J

H

     

where the H ( ) are the Bessel functions and the J H I (J C ) = 0.

8



2(



, ,  ),

) H (J 8  ) cos[ ( − )] exp(−J 2 2 2 ; (J 8 C −  )[ H (J 8 C )]2

cos  

28

exp  −

2 2

  



2



 

,



=

 28

  ),

1 for  = 0, 2 for  = 1, 2, KKK ,

G 8 are positive roots of the transcendental equation

:

3.1.2-11. Domain: 0 ≤

≤C ,0≤ ;

≤ 2 , 0 ≤ ≤  . Third boundary value problem.

A circular cylinder of finite length is: considered. The following conditions are prescribed: 



' L  ' 0  ': 0

= ( , , ) ; + 1 1 = ! ( : , ,  )

at at

; − 1 2  = ! 2 (: , ,  ) at ; + 1 3  = ! 3 ( , ,  ) at

: = 0

=C

=0

(initial condition), (boundary condition), (boundary condition), (boundary condition).

= ;  N The solution ( , , ,  ) is determined by the formula inN Paragraph 3.1.2-10 where N ; : : N ON   ( ) ( ) 2 3 3 ( , , ,  , , ,  ) = 1( , ,  , ,  )  exp(−  Q  ), 2      ; ; =1 P P :  :      J 2 8 H (J 8 ) H (J 8  P ) 1    cos[ ( − )] exp(−J 2 8   ), 1( , ,  , ,  ) = ; ;  ( J 2 8 C 2 + 1 12 C 2 −  2 )[ H (J N 8 C )]2 8 G N N N =0 =1 N N N 2N N N 2 3 3 2 1 2 1 3 Q +1 2 1 2  1 22 ( ) = cos( Q ) + sin( Q ), = + +  1 + . Q 2 Q 2 Q 2 + 1 32 2 Q 2 2 Q 2 P P

© 2002 by Chapman & Hall/CRC Page 230




  

 

N 8

Here, 0 = 1 and = 2 for  = 1, 2, KKK ; the H ( ) are the Bessel functions; and the J are positive roots of the transcendental  equations  G G 1 2+1 3 tan( Q  ) J H I (J C ) + 1 1 H (J C ) = 0, = 2 . Q Q − 1 21 3

and Q

:

3.1.2-12. Domain: 0 ≤

≤C ,0≤

≤ 2 , 0 ≤ ≤  . Mixed boundary value problems. ;

1 5 . A circular cylinder of finite : length is considered. The following conditions are prescribed: 

= =!  =! ' 0  =!

( , , ) ; 1 (: , ,  ) ; 2 (: , ,  ) ; 3( , ,  )



' 0

Solution:

:  ( , , , ) = " ;

0

;

2E

#"

"

0

0

0

2E

=C

=0

= : 

;

2E

0

(initial condition), (boundary condition), (boundary condition), (boundary condition).

( , , ) ( , , ,  , , ,  ) $  $ $

F 

− C " % " # "

: = 0

at at at at

0

: 

! 1 ( , , & )  ' ' :

( , , , , , , −& ) ;



 (

$ $ $ &

=

F F  ! 2 ( , , & ) ( , , ,  , , 0,  − & ) $  $ $ & ; 0 0 0 : 2E  + " % " " F  ! 3 ( , , & ) ( , , ,  , ,  ,  − & ) $  $ $ & . ; 0 0 0

− " % "

Here,



1(

2(

, , , , )=

1

;



=1

( , , , , ,  , )= ;



: 



:







1 ( ,  ,  , ,  )  2( , ,  ), ;

8  ) cos[ ( − )] exp(−J 2 8   ), ; =0 =1 G    

    2 2  1 for  = 0, cos  cos  exp  − , = 2 for  = 1, 2, K KK ,     2   G 2

 C

1 2    , , )= +  



"

:

 :









8

[ H I (J

8 C )]2

(J

H

8

) H (J

where the H ( ) are the Bessel functions (the prime denotes the derivative with respect to the argument) and the J 8 are positive roots of the transcendental equation H (J C ) = 0. 2 5 . A circular cylinder of finite : length is considered. The following conditions are prescribed: 

= ( , , )

' L 

at

: = 0

(initial condition),

= ! 1 (: , ,  ) at ; = ! 2 (: , ,  ) at

=C

=0

(boundary condition), (boundary condition),

= ! 3( , ,  )

=

(boundary condition).

;



;



;

Solution:

:  ( , , , ) = " ;

0

#"

+ C "

2E 0

"

0

at

: 

( , , ) ( , , ,  , , ,  ) $  $ $

F  2E



;

:

! 1 ( , , & ) ( , , , C , , ,  − & ) $ ; :  + " % " " F  ! 2 ( , , & )  ' ( , , , , , , −& ;

0 0 0 ' : 2E  − " % " " F  ! 3 ( , , & )  ' ( , , , , , , −& ;

0 0 0 ' 0

% "

0 2E

#"

0

$ $ &

) )

 *

=0

 *

=

$  $ $ & $  $ $ & .

#

© 2002 by Chapman & Hall/CRC Page 231

Here, :

 

:

, , , , ) =



1(

;

, , ) =

2 (



2 

1





 



8

=0

=1





 

sin  

; 1 

+

2

 C

:

  ( , , ,  , ,  ,  ) =  1 ( , ,:  ,  ,  )  G

=1

;



(J



8

2(



, ,  ),

8  )

) H (J cos[ ( − )] exp(−J ; (J 2 8 C 2 −  2 )[ H (J 8 C )]2 J

H

 

sin  

28

exp  − 



2 2

 







2







=

,

 28

  ),

1 for  = 0, 2 for  = 1, 2, KKK ,

G 8 are positive roots of the transcendental equation



where the H ( ) are the Bessel functions and the J H I (J C ) = 0. :

3.1.2-13. Domain: C

≤

1

≤C

0≤

2,

≤ 2 , − D ;

< < D . First boundary value problem.

An infinite hollow circular cylinder is considered. The following conditions are prescribed: :



= ( , , )

at

;



= ! 1 ( , ,  ) at ; = ! 2 ( , ,  ) at 

: = 0

(initial condition),

: =C

1

=C

2

;

(boundary condition), (boundary condition).

Solution: :  ( , , , ) = " ;

 "

−



+ C

2E 0

"

1

− C

0

"

2

0

"

( , , ) ( , , ,  , , ,  ) $  $ $

F

;

1

F

% "

 "

−



% "

−

 "

: 

2

2E 0

 

:

1

( , , , , , , ) = 





 (

( , , , , , , −& )

$ $ $ &

=

1

F  (

$ $ $ & .

=

2

F

exp  −

 R  2     

;

:

:

;

;



'



Here,



! 2 ( , , & )  '

0

( , , , , , , −& )



'

2E

: 

! 1 ( , , & )  '

:

( − )2    4  

: 1 ( , ,  , ,  ),

;



8  ) cos[ ( − )] exp(−J 2 8   ),  ;  =0 =1  R   G     J 2 8 H 2 (J 8 C 2 ) 1 T 2 for  = 0, , =     8 = 2   2  1: for  ≠ 0, : H (J 8 C 1 ) − H (J 8 C : 2 ) G S ( J 8 ) = H (J 8 C 1 ) U (J 8 ) − U (J  8 C 1 ) H (J 8 ),  : : 8 are positive roots of the transcenwhere the H ( ) and U ( ) are the Bessel functions   and the J 1(

, , , , ) = ;

2



8 S



8

dental equation

(J C H

1 )U

(J C

8

(J

2)

−U

)S

(J

(J C

1) H



(J C

2)

= 0.

:

3.1.2-14. Domain: C

1

≤

≤C

2,

0≤ ;

≤ 2 , − D

< < D . Second boundary value problem.

An infinite hollow circular cylinder is considered. The following conditions are prescribed: :

 

' L  ' L

= ( , , ) ;

at

= ! 1 ( , ,  ) at ; = ! 2 ( , ,  ) at ;

: = 0

(initial condition),

: =C

1

=C

2

(boundary condition), (boundary condition).

© 2002 by Chapman & Hall/CRC Page 232


 

Solution: :  ( , , , ) = " ;

−

− C

2E

 " 

+ C

1

2

0

( , , ) ( , , ,  , , ,  ) $  $ $

F

;

1

F

: 

2

"

2E

:



! 1 ( , , & ) ( , , , C ; : 2 E   " % "  " ! 2 ( , , & ) ( , , , C ; 0 − 0 "

0

% "

 "

−

0

1, 

, , , −& )$ $ $ &

2,

, , −& )$ $ $ & .



Here, : 

( , , ,  , , ,  )=

:

( − )2    exp  − 4      1( , ; ,  , ,  ), 2    : 1

         J 2 8 S (J 8 ) S (J 8  ) cos[ ( − )] exp(−J 2 8   ) 1  1  + ,  ; 1( , ,  , ,  ) =  J 2 8 C 2 −  2) S 2 (J 8 C 1) ;  ( C 22 − C  12) :   8  (J  2 8 C  22 −:  2) S  2 (J  8 C 2)−( : 1 =0 =1 G S (J 8 ) = H I (J 8 C 1) U (J 8 )− U I (J 8 C 1) H (J 8 ),    : :  where 0 = 1 and = 2 for  = 1, 2, KKK ; the H ( ) and U ( ) are the Bessel functions (the prime denotes the derivative with respect to the argument); and the J 8 are positive roots of the G G transcendental equation     ;

:

H I (J C

1 )U

I (J C

≤C

0≤

2)

− U I (J C

I (J C

1) H

= 0.

2)

:

3.1.2-15. Domain: C

≤

1

2,

;

≤ 2 , − D

< < D . Third boundary value problem.

An infinite hollow circular cylinder is considered. The following conditions are prescribed: :  

= ( , , )

: = 0

at

;

− 1 1

= ! 1 ( , ,  ) at ; + 1 2  = ! 2 ( , ,  ) at

' L  ' L :

;

(initial condition),

: =C

1

=C

2

(boundary condition), (boundary condition).

The solution  ( , , ,  ) is given by relations in Paragraph 3.1.2-14 in which :

( − )2  ( , , ,  , , ,  ) =  exp  −     ; 4   2    :

 

: 1(

1 

, , , , )= ;

;



1





S



=0

8

(J



VZJ H I (J C

G

1)

− 1 1H

8 U I (J

− VZJ



1 )X

VWJ U I (J C

1)

,



( ) are the Bessel functions; and the J

8





(J C

, ,  , ,  ),

 ;    2 8 ) S (J 8  ) cos[ ( − )] exp(−J 8    ) ; 2) S 2 ( J  8 C  ) − ( 1 2 C  2 + J  2 8 C 2 −  2) S 2 ( J 8 C 2 1 1 : 1 ) 1) − 1 1 H  ( J  8 C 1)XYU  ( J  8 : 8 C 1) − 1 1 U (J 8 C 1)X H (J 8 ).   : :

Here, 0 = 1 and = 2 for  = 1, 2, KKK ; H ( ) and U are positive roots of the transcendental equation 

1(

 J 2 8 S (J ( 1 22 C 22G  + J 2 8 C  22 −  =1: 8 ) = VWJ  8 H I (J  8 C



G

:

2)

+ 1  2U

= VZJ U I (J C

(J C

2 )X

1)

− 1 1U



 

(J C

1 )X

VZJ H I (J C

2)

+ 1 2H

(J C

2 )X

.

© 2002 by Chapman & Hall/CRC Page 233

:

3.1.2-16. Domain: C

≤

1

≤C

0≤

2,

≤ 2 , 0 ≤ [ < D . First boundary value problem. ;

A semiinfinite hollow circular : cylinder is considered. The following conditions are prescribed: 

= ( , , [ ) at ; = ! 1 ( , [ , \ ) at 

;



= ! 2 (: , [ , \ ) at ; = ! 3 ( , , \ ) at 

;

Solution:

:  ( , ,[ ,\ ) = " ;

0

+a C

Here,

2E

 "

0

1

=C

2

(boundary condition), (boundary condition).

[ =0

(] , ^ , _ ) ` ( , , [ , ] , ^ , _ , \ )] $ ] $ ^ $ _

F

;

1

F

: =C

(initial condition), (boundary condition),

:

2

"

:\ = 0

2E

:

` ( , , [ , ] , ^ , _ , \ − & )c $ ^ $ _ $ & ; ( = 1 : 2E F −a C 2" % " d " ! 2 (^ , _ , & ) b ' ` ( , , [ , ] , ^ , _ , \ − & )c $ ^ $ _ $ & ; ] ( = 2 0 0 0 ' : 2E F 2 +a " % " " F ! 3 (] , ^ , & ) b ' ` ( , , [ , ] , ^ , _ , \ − & )c ] $ ] $ ^ $ & . ; _ * =0 0 0 1 ' F 1

"

0

% "

0

 "

! 1 (^ , _ , & ) b '

0

'

]

: 1 ( [ − _ )2 ( [ + _ )2 c − exp b − c h ` 1 (i , j , ] , ^ , \ ), ` ( , ,[ ,] ,^ ,_ ,\ ) = exp b − ; 4a \ 2e f a \ l g o l l 4l a \ l l l k f kl m p (q m i ) p (q m ] ) cos[r ( j − ^ )] exp(−q 2 m s t ), d d ` 1 (i , j , ] , ^ , \ ) = l l l 2 m =0 =1 n l o l l l l l q 2 m v 2 (q m w 2 ) 1 u 2 for r = 0, m , l l l = 2 l l l l l l= l v (q m w 1 ) − v 2 (q m w 2 ) for r ≠ 0, g 1 n l p (l q m i ) = v (q m w 1 ) U (q m i ) − U (q l m w 1 ) v (q m i ),

where the v (i ) and U dental equation

(i ) arel the Bessel functions l l and thel q (q w v

3.1.2-17. Domain: w

1

≤i ≤w

(q w

1 )U

2)

−U

(q w

1) v

m

(q w

are positive roots of the transcen2)

= 0.

0 ≤ j ≤ 2x , 0 ≤ y < z . Second boundary value problem.

2,

A semiinfinite hollow circular cylinder is considered. The following conditions are prescribed: { } ~ { } ~ {

= | (i , j , y ) at =  1 ( j , y , t ) at

=  2 ( j , y , t ) at } € { =  3 (i , j , t ) at Solution: { (i , j , y , t ) = 

0

2ƒ

‚  1



+s w

2



0

−s w

„

0

%  % 

0

2ƒ

„ 0

0

‚  ‚ 

2ƒ 0

0 2



„

1

i =w

2

y =0

(initial condition), (boundary condition), (boundary condition), (boundary condition).

| (] , ^ , _ ) … (i , j , y , ] , ^ , _ , t )] † ] † ^ † _

1

0

−s  %  0

2



t =0 i =w

2ƒ

 1 (^ , _ , ‡ ) … (i , j , y , w

1, ^

,_ ,t −‡ )† ^ † _ † ‡

 2 (^ , _ , ‡ ) … (i , j , y , w

2, ^

,_ ,t −‡ )† ^ † _ † ‡

 3 (] , ^ , ‡ ) … (i , j , y , ] , ^ , 0, t − ‡ )] † ] † ^ † ‡ .

1

„ © 2002 by Chapman & Hall/CRC Page 234

ˆŠ

Here, … (i , j , y , ] , ^ , _ , t ) = 1( i , j , ] , ^ , t ) =

…

1 2Ž x s t 

1 1 k l ‚ + 2 2 x ( w 2l − w l 1) x l =0 p

k ml

=1

( y − _ )2 ( y + _ )2 h … +exp  − l 4s t l ‘ l 4l s t ‘

m p 2 n (q l m w l

m w

(q l m i )− U ’ (q l m w

1) U

l

where 0 = 1 and = 2 for r = 1, 2, “““ ; the v (i ) and U m q are n positive roots n of the transcendental equation l l l l v ’ (q w

3.1.2-18. Domain: w

≤i ≤w

1

2,

’ (q w

1 )U

2)

1( i

, j , ] , ^ , t ), l

l l l l l (q m i ) p (q m ] ) cos[r ( j − ^ )] exp(−q 2 m s t ) , 2 2 p l 2 (q l m w l q 2l m w 2 − r 2) p 2 (q m w 1) 2)−( 2−r ) 1

l2

q

‚

m i ) = v ’ (q

(q l

exp l  − l

− U ’ (q w

m i ),

(q

(i ) are the Bessel functions; and the

’ (q w

1) v

1) v

2)

= 0.

0 ≤ j ≤ 2x , 0 ≤ y < z . Third boundary value problem.

A semiinfinite hollow circular cylinder is considered. The following conditions are prescribed: { } ~ { } ~ {

− ” 1{

} € {

− ” 3{

= | (i , j , y )

at

=  1 ( j , y , t ) at { + ” 2 =  2 ( j , y , t ) at =  3 (i , j , t )

t =0 i =w

1

i =w

2

(initial condition), (boundary condition), (boundary condition),

y =0

at

(boundary condition).

The solution { (i , j , y , t ) is determined by the formula in Paragraph 3.1.2-17 where … (i , j , y , ] , ^ , _ , t ) = …

1( y

, _ , t ) … 2(i , j , ] , ^ , t ), 1 (y − _ ) ( y + _ )2 ( y + _ + • )2 ‚ … 1( y , _ , t ) = exp  − +exp  − −2 ”  exp  − 3 l 4s t l ‘ l l l − ” 3• ‘ † • h , 4s t l ‘ l 4s t 2Ž x s t  0 2

…

2( i , j , ] , ^ , t ) =

1 kl l x

k ‚ =0

p

ml

(q

l l l l l (q m i ) p (q m ] ) cos[r ( j − ^ )] exp(−q 2 m s t ) ( ” 22 w 22n l + q 2l m w l 22 − r 2) p 2 (q l m w l 2)−( ” 12w l 12 + q l 2 m w 12 − r 2) p 2 (q m w q

‚ =1

l2m p

m i ) = –—q l m v l’ (q l m w

− –Wq l

m U ’ (q

m w

l (q l m w

1)− ” 1 v 1)− l ” 1U

(q

l

l

Here, 0 = 1 and = 2 for r = 1, 2, “““ ; the v (i ) and U q m are n positive roots n equation l of the transcendental l l l –Zq v ’ (q w

1)

(q w

− ” 1v

1 )˜

–Wq U ’ (q w

2)

+ ” l 2U

(q w

2 )˜

1)

− ” 1U

= –Zq U ’ (q w 3.1.2-19. Domain: w

1

≤i ≤w

2,

m w

,

l (q l m i )

1)˜YU 1) ˜

1)

v

m i ).

(q

(i ) are the Bessel functions; and the

l

l l

(q w

1 )˜

–Zq v ’ (q w

2)

+ ” 2v

(q w

2 )˜

.

0 ≤ j ≤ 2x , 0 ≤ y < z . Mixed boundary value problems.

1 ™ . A semiinfinite hollow circular cylinder is considered. The following conditions are prescribed: { { { } € {

= | (i , j , y ) at =  1 ( j , y , t ) at

t =0 i =w

1

=  2 ( j , y , t ) at =  3 (i , j , t ) at

i =w

2

y =0

(initial condition), (boundary condition), (boundary condition), (boundary condition).

© 2002 by Chapman & Hall/CRC Page 235

Solution: { (i , j , y , t ) = 

2ƒ

‚ 

0

+s w

0



1

−s w

2



1

0



2

0

„

%  % 

0

 2 (^ , _ , ‡ )  }

0

… (i , j , y , ] , ^ , _ , t − ‡ ) ]

‘ š

† ^ † _ † ‡

=

1

„ ‘ š

† ^ † _ † ‡

=

2

„  3 (] , ^ , ‡ ) … (i , j , y , ] , ^ , 0, t − ‡ )] † ] † ^ † ‡ .

2

0

] }

2ƒ

„ 

… (i , j , y , ] , ^ , _ , t − ‡ )

 1 (^ , _ , ‡ )  }

0

‚ 

0

}

2ƒ

‚ 

0

2ƒ

−s  % 

| (] , ^ , _ ) … (i , j , y , ] , ^ , _ , t )] † ] † ^ † _

„

1

„

Here,

1

… (i , j , y , ] , ^ , _ , t ) = …

1 (i

,j ,] ,^ ,t ) =

2Ž x s t l  o l k

‚

2

l

l

kl x

( y − _ )2 ( y + _ )2 h … + exp  − ‘ l 4s t ‘ l 4l s t l

exp  − m p

‚

m

=0

(q m i ) p o l

=1 n

(q l

m

(i ) arel the Bessel functions l l and thel q (q w v

(q w

1 )U

2)

(q w

−U

1) v

, j , ] , ^ , t ), l

m ] ) cos[r ( j − ^ )] exp(−q l l

l l l q 2 m v 2 (q m w 1 u 2 for r = 0, l l l ml = 2 l l l l= l v (q m w 1 ) − v 2 (q for r ≠ 0,  1 n p (l q m i ) = v (q m w 1 ) U (q m i ) − U (q l m w 1 ) v (q

where the v (i ) and U dental equation

1 (i

(q w

l

2)

l

m w 2) m i ),

2m

s t ),

,

are positive roots of the transcen2)

= 0.

2 ™ . A semiinfinite hollow circular cylinder is considered. The following conditions are prescribed: { } ~ { } ~ { {

= | (i , j , y )

at

=  1 ( j , y , t ) at =  2 ( j , y , t ) at =  3 (i , j , t )

t =0 i =w

1

i =w

2

(initial condition),

y =0

at

(boundary condition), (boundary condition), (boundary condition).

Solution: { (i , j , y , t ) = 

0

2ƒ

‚  1



+s w

2



0

−s w

0

%  % 

0

2ƒ

„ 0

0

2ƒ

‚ 

0

‚ 

0 2

1, ^

,_ ,t −‡ )† ^ † _ † ‡

 2 (^ , _ , ‡ ) … (i , j , y , w

2, ^

,_ ,t −‡ )† ^ † _ † ‡

}  3 (] , ^ , ‡ )  }

„ 

2ƒ

 1 (^ , _ , ‡ ) … (i , j , y , w

1

_

… (i , j , y , ] , ^ , _ , t − ‡ )

„

Here, … (i , j , y , ] , ^ , _ , t ) = …

| (] , ^ , _ ) … (i , j , y , ] , ^ , _ , t )] † ] † ^ † _

„ 1

0

+s  %  0

2



1( i , j , ] , ^ , t ) =

1 2Ž x s t 

1 1 k l ‚ + 2 2 l l x ( w 2 − w 1) x l =0 p

(q

m i ) = v ’ (q

k ml

exp l  − l

=1

m w

=0

1( i

] † ] † ^ † ‡ .

, j , ] , ^ , t ), l

l l l l l (q m i ) p (q m ] ) cos[r ( j − ^ )] exp(−q 2 m s t ) , l q 2l m w 2 − r 2) p 2 (q m w 1) n (q l 2 m w l 2 − r 2) p l 2 (q l m w 2)−( 2 1 q

‚

( y − _ )2 ( y + _ )2 h … −exp  − l 4s t l ‘ l 4l s t ‘

‘ ›

1) U

l2 m p

(q

m i )− U ’ (q

m w

1) v

(q

m i ),

© 2002 by Chapman & Hall/CRC Page 236

l



ˆŠ l

l l

where 0 = 1 and = 2 for r = 1, 2, “ ““ ; the v (i ) and U (i ) are the Bessel functions (the n n prime denotes the derivative with respect to the argument); and the q m are positive roots of the l l l l transcendental equation v ’ (q w

3.1.2-20. Domain: w

≤i ≤w

1

’ (q w

1 )U

2,

2)

− U ’ (q w

1) v

’ (q w

2)

= 0.

0 ≤ j ≤ 2x , 0 ≤ y ≤ œ . First boundary value problem.

A circular cylinder of finite length is considered. The following conditions are prescribed: {

= | (i , j , y ) {

at

=  1 ( j , y , t ) at =  2 ( j , y , t ) at { {

=  3 (i , j , t ) { =  (i , j , t ) 4 Solution: { (i , j , y , t ) = 

0

2ƒ



+s w

0

0



2

0

„

%  % 

0

0

2ƒ

+s  %  0

2

−s  %  0

„

0



2

„

}

… (i , j , y , ] , ^ , _ , t − ‡ ) _

}  4 (] , ^ , ‡ )  }

… (i , j , y , ] , ^ , _ , t − ‡ ) _

1

… (i , j , y , ] , ^ , _ , t − ‡ ) ]

 3 (] , ^ , ‡ )  }

1

2ƒ

… (i , j , y , ] , ^ , _ , t − ‡ ) ]

}  2 (^ , _ , ‡ )  }

0

„

0

y =œ

}

2ƒ



(boundary condition), (boundary condition).

 1 (^ , _ , ‡ )  }

0



2

(boundary condition), (boundary condition),

y =0

at at

2ƒ



1

i =w

(initial condition),

| (] , ^ , _ ) … (i , j , y , ] , ^ , _ , t )] † ] † ^ † _

1



1

−s w

2

„ 

t =0 i =w

‘ š

… (i , j , y , ] , ^ , _ , t ) = … …

1 ( 

,¡ ,] ,^ ,t ) =

2

l

l

kl x

1 (i

k ‚ =0

m

‘ š

,j ,] ,^ ,t )  l o l m p ‚ =1 n

2 kl œ

‚

sin ž =1l

l

r x y l

(q m   ) p o l

œ

Ÿ

sin ž

l

(q l

‘ ›

=0

‘ ›

=

3.1.2-21. Domain: w

1

≤  ≤w

(q w 2,

† ^ † _ † ‡ 2

„ ] † ] † ^ † ‡

] † ] † ^ † ‡ .

r x _ œ

1 )U

(q w

2)

−U

(q w

1) v

exp ž − Ÿ

(q w

m

l

2)

l

m w

2)

2 2t

s r

m ] ) cos[r ( ¡ − ^ )] exp(−q l l

(  ) arel the Bessel functions l l and thel q v

=



l l l q 2 m v 2 (q m w 1 u 2 for r = 0, l l l ml = 2 l l l l= l v (q m w 1 ) − v 2 (q for r ≠ 0,  1 n p (l q m   ) = v (q m w 1 ) U (q m   ) − U (q l m w 1 ) v (q

where the v (  ) and U dental equation

1

„

„

Here,

† ^ † _ † ‡

=

œ

x

2

l 2m

Ÿ ‘

,

s t ),

,

m   ),

are positive roots of the transcen2)

= 0.

0 ≤ ¡ ≤ 2x , 0 ≤ y ≤ œ . Second boundary value problem.

A circular cylinder of finite length is considered. The following conditions are prescribed: {

} ~ { } ~ { } € { } € {

= | (  , ¡ , y )

at

=  1 ( ¡ , y , t ) at =  2 ( ¡ , y , t ) at =  3 (  , ¡ , t ) =  4 (  , ¡ , t )

at at

t =0   =w

1

  =w

2

y =0 y =œ

(initial condition), (boundary condition), (boundary condition), (boundary condition), (boundary condition).

© 2002 by Chapman & Hall/CRC Page 237

Solution: { (  , ¡ , y , t ) = 

0

2ƒ



2



0

−s w

1



+s w

2



„

0

0

−s  %  0

% 

0

„

% 

0

+s  % 

2ƒ

2ƒ

0

0



0

0

2ƒ

0 2

„ 

 1 (^ , _ , ‡ ) … (  , ¡ , y , w

1, ^

,_ ,t −‡ )† ^ † _ † ‡

 2 (^ , _ , ‡ ) … (  , ¡ , y , w

2, ^

,_ ,t −‡ )† ^ † _ † ‡

 3 (] , ^ , ‡ ) … (  , ¡ , y , ] , ^ , 0, t − ‡ )] † ] † ^ † ‡

1

„

2



„

 4 (] , ^ , ‡ ) … (  , ¡ , y , ] , ^ , œ , t − ‡ )] † ] † ^ † ‡ .

1

„

… (  , ¡ , y , ] , ^ , _ , t ) = … 1(  

2ƒ



Here,

…

| (] , ^ , _ ) … (  , ¡ , y , ] , ^ , _ , t )] † ] † ^ † _

1

, ¡ , ] , ^ , t )=

1(  

1 1 k l ‚ + x ( w 22l − w l 12) x l =0

k ml

m   ) = v ’ (q

p l (q

1 2 k l + ‚

,¡ ,] ,^ ,t ) 

œ l œ l

=1

l

l

=1

cos ž

lŸ œ

r x _

l

œ

Ÿ

exp ž −

2 2t

s r

x

œ

2

lŸ ‘

,

l l l l (q m   l ) p (q m ] ) cos[ r ( ¡ − ^ )] exp(−q 2 m s t ) , l q 2l m w 2 − r 2) p 2 (q m w 1) n (q l 2 m w l 2 − r 2) p l 2 (q l m w 2)−( 2 1 l2 m p

q

‚

r x y

cos ž

m w

1) U

m   l )− U ’ (q

(q

m wl

1) v

(q

m   ),

l

where 0 = 1 and = 2 for r = 1, 2, “ ““ ; the v (  ) and U (  ) are the Bessel functions (the n n prime denotes the derivative with respect to the argument); and the q m are positive roots of the transcendental equation l l l l v ’ (q w

3.1.2-22. Domain: w

≤  ≤w

1

2,

’ (q w

1 )U

2)

− U ’ (q w

’ (q w

1) v

2)

= 0.

0 ≤ ¡ ≤ 2x , 0 ≤ y ≤ œ . Third boundary value problem.

A circular cylinder of finite length is considered. The following conditions are prescribed: {

= | (  , ¡ , y ) at =  1 ( ¡ , y , t ) at

} ~ { } ~ {

− ” 1{

} € {

+ ” 4 { =  4 (  , ¡ , t )

+ ” 2 { =  2 ( ¡ , y , t ) at } € { − ” 3 { =  3 (  , ¡ , t ) at at

t =0   =w

1

  =w

2

(initial condition), (boundary condition),

y =0

(boundary condition), (boundary condition),

y =œ

(boundary condition).

The solution { (  , ¡ , y , t ) is determined by the formula in Paragraph 3.1.2-21 where … (  , ¡ , y , ] , ^ , _ , t ) = …

Here, the first factor has the form …

1(  

,¡ ,] ,^ ,t )=

1 kl x p

‚ l

k =0

(q

ml

l

1 ( 

, ¡ , ] , ^ , t )…

2(y

, _ , t ).

l l l l l l l l q m p (q m   ) p (q m ] ) cos[r ( ¡ − ^ )] exp(−q 2 m s t ) ( ” 22 w 22n l + q 2l m w l 22 − r 2) p 2 (q l m w l 2) − ( ” 12w l 12 + q l 2 m w 12 − r 2) p 2 (q m w l

l

l

l2

‚ =1

m   ) = –Wq l m v l’ (q l m w

− –Zq

m U ’ (q

m w

1) − ” 1 v 1) − ” 1 U

l (q l m w

(q

m w

,

l (q l m   )

1)˜YU 1) ˜

1)

v

(q

m   ),

© 2002 by Chapman & Hall/CRC Page 238

l

l ˆŠ

l

l

where 0 = 1 and = 2 for r = 1, 2, “““ ; the v (  ) and U n positive n q m are roots equation l of the transcendental l l l –Zq v ’ (q w

1)

(q w

− ” 1v

–Wq U ’ (q w

1 )˜

2)

+ ” l 2U

The second factor is given by

2 )˜

1)

− ” 1U

¢

¢ ¢

£

where the ¥

¢

3

¤

¤

£ ¢

=1

£

¤

sin( ¥ y ),

¥

¢

,_ ,t ) = ‚

¢

”

( y ) = cos( ¥ y ) +

¢

(y ) ( _ ) ¤ 2£

2

(q w

≤  ≤w

2,

(q w

+ ” 2v

2 )˜

.

exp(− s ¥ 2 t ),

2¥

4 2

¥

2¢

+” 2 ¥ +”

£

1

2)

¢

2 3 2 4

+

¥

¢

”

2¥

tan( ¥ œ )

are positive roots of the transcendental equation

3.1.2-23. Domain: w

–Zq v ’ (q w

1 )˜

¢

¢

”

=

l l

¢

kO¢

2(y

…

l

(q w

= –Zq U ’ (q w

(  ) are the Bessel functions; and the

3 2

=

œ

+

2

”

” ¥

2¢ 3 2

, Ÿ

+” 4 . − ” 3” 4

3 2

¥

ž 1+

0 ≤ ¡ ≤ 2x , 0 ≤ y ≤ œ . Mixed boundary value problems.

1 ™ . A circular cylinder of finite length is considered. The following conditions are prescribed: { { { } € { } € {

= | (  , ¡ , y ) at =  1 ( ¡ , y , t ) at

t =0   =w

1

(initial condition), (boundary condition),

=  2 ( ¡ , y , t ) at

  =w

2

(boundary condition),

=  3 (  , ¡ , t ) =  4 (  , ¡ , t )

y =0

at at

(boundary condition), (boundary condition).

y =œ

Solution: { (  , ¡ , y , t ) = 

0

2ƒ



+s w

0

2

| (¦ , § , ¨ ) … (  , ¡ , y , ¦ , § , ¨ , t )¦ † ¦ † § † ¨

1



1

−s w

2

„ 

0



0

„

%  % 

0

0

2ƒ

−s  %  0

0

+s  %  0

2ƒ



0 2ƒ



0 2



„

} … (  , ¡ , y , ¦ , § , ¨ , t − ‡ )

 1 (§ , ¨ , ‡ )  } ¦ }  2 (§ , ¨ , ‡ )  } ¦

… (  , ¡ , y , ¦ , § , ¨ , t − ‡ )

‘ š

† § † ¨ † ‡

=

1

„ ‘ š

† § † ¨ † ‡

=

 3 (¦ , § , ‡ ) … (  , ¡ , y , ¦ , § , 0, t − ‡ )¦ † ¦ † § † ‡

2

„

1

2ƒ

„

2

„

0



 4 (¦ , § , ‡ ) … (  , ¡ , y , ¦ , § , œ , t − ‡ )¦ † ¦ † § † ‡ .

1

„

Here, … (  , ¡ , y , ¦ , § , ¨ , t ) = … …

1 ( 

,¡ ,¦ ,§ ,t ) = l

1 ( 

kl x

2

,¡ ,¦ ,§ ,t )  k

‚ =0

m

l o l

1 œ m p

‚ =1 n

+ l

2 kl œ l

‚

r x y

cos ž =1

(q m   ) p o l

l l

(q

œ l

Ÿ

cos ž

r x ¨ œ

exp ž − Ÿ

l

2)

l

x

l

m ¦ ) cos[r ( ¡ − § )] exp(−q l l

l l l q 2 m v 2 (q m w 1 u 2 for r = 0, l l l ml = 2 l l l l= l v (q m w 1 ) − v 2 (q for r ≠ 0,  1 n p (q m   ) = v (q m w 1 ) U (q m   ) − U (q m w 1 ) v (q

2 2t

s r

m w

2)

œ

2m

2

Ÿ ‘

,

s t ),

,

m   ),

© 2002 by Chapman & Hall/CRC Page 239

l

l

l

where the v (  ) and U dental equation

m

(  ) arel the Bessel functions l l and thel q

are positive roots of the transcen-

(q w 1 ) U (q w 2 ) − U (q w 1 ) v (q w 2 ) = 0. 2 ™ . A circular cylinder of finite length is considered. The following conditions are prescribed: { = | (  , ¡ , y ) at t = 0 (initial condition), v

} ~ { } ~ { { {

=  1 ( ¡ , y , t ) at

  =w

1

=  2 ( ¡ , y , t ) at =  3 (  , ¡ , t ) at

  =w

2

y =0

(boundary condition), (boundary condition),

=  4 (  , ¡ , t )

y =œ

(boundary condition).

Solution: { (  , ¡ , y , t ) = 

0

2ƒ



2



0

−s w

1



+s w

2



„

0

0

+s  %  0

„ 0

% 

0

2ƒ



2ƒ

0

 1 (§ , ¨ , ‡ ) … (  , ¡ , y , w

1, §

,¨ ,t −‡ )† § † ¨ † ‡

 2 (§ , ¨ , ‡ ) … (  , ¡ , y , w

2, §

,¨ ,t −‡ )† § † ¨ † ‡

}

1

„

¨ }

2

 4 (¦ , § , ‡ )  }

„



0

… (  , ¡ , y , ¦ , § , ¨ , t − ‡ )

 3 (¦ , § , ‡ )  }

„

… (  , ¡ , y , ¦ , § , ¨ , t − ‡ ) ¨

1

‘ ›

=0

‘ ›

=

„

Here, … (  , ¡ , y , ¦ , § , ¨ , t ) = … 1(  

0

2

2ƒ

0

2ƒ





0

−s  % 

…

| (¦ , § , ¨ ) … (  , ¡ , y , ¦ , § , ¨ , t )¦ † ¦ † § † ¨

1

% 

at

(boundary condition),

, ¡ , ¦ , § , t )=

1(  

¦ † ¦ † § † ‡ . 

,¡ ,¦ ,§ ,t ) 

1 1 kl ‚ + x ( w 22l − w l 12) x l =0 p l (q

¦ † ¦ † § † ‡

m   ) = v ’ (q

k ml

2 kl lœ l

=1

sin ž l

=1

m w

1) U

3.1.2-24. Domain: 0 ≤   < z , 0 ≤ ¡ ≤ ¡

0,

r x y l œ

Ÿ l

sin ž l

r x ¨ œ

Ÿ

exp ž −

2 2t

s r

x

œ

2

Ÿ ‘l

,

l l l l l (q m   ) p (q m ¦ ) cos[r ( ¡ − § )] exp(−q 2 m s t ) , 2 2 p l 2 (q l m w 2l m w 2 − r 2) p 2 ( q l m w 1) 2)−( q 2−r ) 1

l2

m p n (q l 2 m w l q

‚

‚

l

l

(q m   )− U ’ (q m w 1) v (q m   ), l where 0 = 1 and = 2 for r = 1, 2, “““ ; the v (  ) and U (  ) are the Bessel functions (the n n prime denotes the derivative with respect to the argument); and the q m are positive roots of the l l l l transcendental equation v ’ (q w 1 ) U ’ (q w 2 ) − U ’ (q w 1 ) v ’ (q w 2 ) = 0. −z

< y < z . First boundary value problem.

A dihedral angle is considered. The following conditions are prescribed: { = | (  , ¡ , y ) at t = 0 (initial condition), {

=  1 (  , y , t ) at =  2 (  , y , t ) at {

Solution: { (  , ¡ , y , t ) = 

0

−

‚ 

0

© 

0

=0 ¡ =¡ 0

| (¦ , § , ¨ ) … (  , ¡ , y , ¦ , § , ¨ , t )¦ † ¦ † § † ¨ ‚

‚

+ s  %  ‚  ‚  1 (¦ , ¨ , ‡ ) 0

−s 

0

−

% 

1 ¦

0

‚

‚ 

−

‚

0

(boundary condition), (boundary condition).

¡

‚

 2 (¦ , ¨ , ‡ )

1 ¦

} … (  , ¡ , y , ¦ , § , ¨ , t − ‡ )

 } }

§

 } §

… (  , ¡ , y , ¦ , § , ¨ , t − ‡ )

‘ ª

=0

‘ ª

=

† ¦ † ¨ † ‡ † ¦ † ¨ † ‡ . 0

©

© 2002 by Chapman & Hall/CRC Page 240




Here,  

1 (

where the

, , , , ) =

1 

0

exp  −



2

+ 4 

( − )2      4  

exp −

2   

2





1

( , , ,  , , ,  ) =







=1

0





 

2  

1 (

, ,  , ,  ),

sin  



0

sin   



0

, 

( ) are the modified Bessel functions. !

3.1.2-25. Domain: 0 ≤  < " , 0 ≤ ≤

0,

−"

< < " . Second boundary value problem.

A dihedral angle is considered. The following conditions are prescribed:  #

 =0

 #

= & 1 ( , ,  ) at

=0

−1 %

#

= & 2 (  , ,  ) at

=



= $ ( , , ) at

−1 %



(initial condition), (boundary condition), (boundary condition).

0

Solution: '

−

'

0

−  ' ) '

0

'

−

0

+ ' ) '  ' −

0

Here,

1(  , ,  , ,  ) =

where the

0

 & 2 ( , , * ) ( , , ,  ,

0

( , , ,  , , ,  ) =

1 

 & 1 ( , , * ) ( , , ,  , 0, ,  − * ) (  ( ( *

 



 $ ( , , ) ( , , ,  , , ,  ) (  ( (

0

# ( , , ,  ) = '

0

exp  −

 2+



2

4  



1

exp −

2    1 2

0

  

2  



+

 

( − )2  4     

=1



0



0,

, −* )(  ( ( * .

1( 

, ,  , ,  ),

 

2  

cos  



0



cos  



0

 

,

( ) are the modified Bessel functions. !

3.1.2-26. Domain: 0 ≤  < " , 0 ≤ ≤

0,

0 ≤ < " . First boundary value problem.

The upper half of a dihedral angle is considered. The following conditions are prescribed: # # # #

= $ ( , , )

at

 =0

= & 1 ( , ,  ) = & 2 ( , ,  )

at at

= & 3 ( , ,  ) at

(initial condition),

=0 = 0

=0

(boundary condition), (boundary condition), (boundary condition).

Solution: # ( , , ,  ) = '

0

'

0

+ ' ) ' 0

'

0

0

'

0

0

− ' ) '  ' 0

+ ' ) ' 0

0

0

0

0

'

0

 $ ( , , ) ( , , ,  , , ,  ) (  ( (

1

 %

 ( , , ,  , , ,  − * ) (  ( ( * %   + =0   1 %  & 2 ( , , * ) (  , ,

,  , ,

,  − * ) (  ( ( * %   + = 0  %  & 3 ( , , * ) % ( , , ,  , , ,  − * )  (  ( ( * .  , =0

& 1 ( , , * )

© 2002 by Chapman & Hall/CRC Page 241

Here,  

1 (

where the



0



 ( − )2 ( + )2 − exp − 4       4   .

exp −

2    -

1

, , , , ) =



1

( , , ,  , , ,  ) =

exp  −

2



2

+ 4 







=1

0





 

2  

sin  

1 (



sin   

0

, ,  , ,  ), 

0

, 

( ) are the modified Bessel functions. !

3.1.2-27. Domain: 0 ≤  < " , 0 ≤ ≤

0,

0 ≤ < " . Second boundary value problem.

The upper half of a dihedral angle is considered. The following conditions are prescribed:  #  

−1 %

 #

−1 %

# % / #

= $ ( , , )

at

 =0

= & 1 ( , ,  )

at

=0

= & 2 ( , ,  ) at = & 3 ( , ,  ) at

=

(initial condition), (boundary condition), (boundary condition), (boundary condition).

0

=0



Solution: # ( , , ,  ) = '

 $ ( , , ) ( , , ,  , , ,  ) (  ( (

0

'

0

− ' ) ' 0

'

0

0

'

0

0

+ ' ) '  ' 0

− ' ) ' 0

 & 1 ( , , * ) ( , , ,  , 0, ,  − * ) (  ( ( *

0

 & 2 ( , , * ) ( , , ,  ,

0

0

, −* )(  ( ( *

 & 3 ( , , * ) ( , , ,  , , 0,  − * ) (  ( ( * .

0

'

0,

0

Here, 



( − )2 (

( , , ,  , , ,  ) = exp − +exp − 4   2         2 2      + 1 1  exp  − + 1(  , ,  , ,  ) = 0 0  0 4   2 2    1





where the

=1



+ )2  4   .  

2  

1( 

cos  

, ,  , ,  ), 

0



cos  



0

 

,

( ) are the modified Bessel functions. !

3.1.2-28. Domain: 0 ≤  < " , 0 ≤ ≤

0,

0 ≤ < " . Mixed boundary value problems.

1 0 . The upper half of a dihedral angle is considered. The following conditions are prescribed: # # # % / #

= $ ( , , ) = & 1 ( , ,  )

at at

= & 2 ( , ,  ) at = & 3 ( , ,  ) at

 =0

=0 = 0

=0

(boundary condition), (boundary condition).





(initial condition), (boundary condition),

© 2002 by Chapman & Hall/CRC Page 242


 



Solution:

 $ ( , , ) ( , , ,  , , ,  ) (  ( (

0

# ( , , ,  ) = '

'

0

'

0

+ ' ) ' 0

0

'

0

0

0

− ' ) ' 0

0

 & 3 ( , , * ) ( , , ,  , , 0,  − * ) (  ( ( * .

0

'

0

 %

 ( , , ,  , , ,  − * ) (  ( ( * %   + =0   1 %  & 2 ( , , * ) ( , , ,  , , ,  − * ) (  ( ( * %   + = 0 & 1 ( , , * )

0

− ' ) '  '

1

0

Here, 



 ( − )2 ( + )2 ( , , ,  , , ,  ) = exp − + exp − 1 (  , ,  , ,  ), 4       4   . 2          2 +  2  1 exp  −  sin   sin   , 1 ( , ,  , ,  ) = 0  0 4    2   0  0 

1



=1

where the



( ) are the modified Bessel functions. !

2 0 . The upper half of a dihedral angle is considered. The following conditions are prescribed:  #  

−1 %

 #

−1 %

# #

= $ ( , , )

at

 =0

= & 1 ( , ,  )

at

=0

= & 2 ( , ,  )

at

=

= & 3 ( , ,  ) at

(initial condition), (boundary condition), (boundary condition),

0

=0

(boundary condition).



Solution:

 $ ( , , ) ( , , ,  , , ,  ) (  ( (

0

# ( , , ,  ) = '

'

0

'

0

− ' ) ' 0

0

'

0

+ ' ) '  ' 0

+ ' ) ' 0

 & 1 ( , , * ) ( , , ,  , 0, ,  − * ) (  ( ( *

0

0

 & 2 ( , , * ) ( , , ,  ,

0

 %

0

0

'

& 3 ( , , * ) %

0



0,

, −* )(  ( ( *

( , , ,  , , ,  − * )

 ,

=0

 (  ( ( * .

Here,  

1( 

( , , ,  , , ,  ) =

, ,  , ,  )=

where the

1 

0

exp  −



1 2     2+

2

4  





exp − 1 2 

0



( − )2 ( + )2  −exp − 4       4    .  

2  

+





=1



0



 

2  

1( 

cos  

, ,  , ,  ),



 0



cos  

 0

 

,

( ) are the modified Bessel functions. !

© 2002 by Chapman & Hall/CRC Page 243

3.1.2-29. Domain: 0 ≤  < " , 0 ≤ ≤

0,

0 ≤ ≤ 1 . First boundary value problem.

A wedge domain of finite thickness is considered. The following conditions are prescribed: # # # # #

= $ ( , , )

at

 =0

= & 1 ( , ,  )

at

=0

= & 2 ( , ,  )

at

=

(initial condition), (boundary condition), (boundary condition),

0

= & 3 ( , ,  ) at

=0

(boundary condition),

= & 4 ( , ,  ) at

=1

(boundary condition).

Solution: 

 $ ( , , ) ( , , ,  , , ,  ) (  ( (

0

# ( , , ,  ) = '

0

2'

'

0

0

+ ' ) ' 2 ' 0

0

0

−  ' ) ' 2 ' 0

0

0

0

0

0

− ' ) ' 0

0

 %

1

0

( , , ,  , , ,  − * ) 

%







& 4 ( , , * ) %

( , , ,  , , ,  − * ) ( , , ,  , , ,  − * )

& 3 ( , , * ) %

 %

0

'

& 2 ( , , * )



%



 %

0

+ ' ) '  '

& 1 ( , , * )

 %

1

( , , ,  , , ,  − * )

 +

=0

(  ( ( *

  +

=

 ,

=0

 ,

=

(  ( ( * 0

 (  ( ( *

 (  ( ( * . 2

Here, 



( , , ,  , , ,  ) =



1

 1 (

2 

, , , , ) 2

1

where the



=1

  



2

+ exp  − 1 ( , ,  , ,  ) =  0 4  

sin  

 

=1



1



0





sin    

2  

 1

sin  

2 2





exp  −  

0





1

sin  

2





,

 

0



,

( ) are the modified Bessel functions. !

3.1.2-30. Domain: 0 ≤  < " , 0 ≤ ≤

0,

0 ≤ ≤ 1 . Second boundary value problem.

A wedge domain of finite thickness is considered. The following conditions are prescribed:  #  

= $ ( , , )

at

 =0

−1 %

 #

= & 1 ( , ,  )

at

=0

−1 %

#

= & 2 ( , ,  )

at

=

% / # % / #

(initial condition), (boundary condition), 0

(boundary condition),

= & 3 ( , ,  ) at

=0

(boundary condition),

= & 4 ( , ,  ) at

=1

(boundary condition).

© 2002 by Chapman & Hall/CRC Page 244


 



Solution:

 $ ( , , ) ( , , ,  , , ,  ) (  ( (

0

# ( , , ,  ) = '

0

2'

'

0

0

− ' ) ' 2 ' 0

0

0

+  ' ) ' 2 ' 0

 & 1 ( , , * ) ( , , ,  , 0, ,  − * ) (  ( ( *  & 2 ( , , * ) ( , , ,  ,

0

0

0

0

0

 & 4 ( , , * ) ( , , ,  , , 1 ,  − * ) (  ( ( * .

0

+ ' ) ' 0

'

0

, −* )(  ( ( *

 & 3 ( , , * ) ( , , ,  , , 0,  − * ) (  ( ( *

0

− ' ) '  '

0,

0

Here,  

1( 

1

, , , , )= 



where the

2(

2

 +

exp  −

0



( , , ,  , , ,  ) =

, , )=



2

4  

1 1

+

2  1



1 2

0

1( 

  



cos  

1

=1



=1

2(

0



cos   





+

2  



, ,  ,  ,  )





, ,  ),  

2  

2 2



1





cos  

exp  −  1







0



2

cos  



0

 

,

, 

( ) are the modified Bessel functions. !

3.1.2-31. Domain: 0 ≤  < " , 0 ≤ ≤

0,

0 ≤ ≤ 1 . Mixed boundary value problems.

1 0 . A wedge domain of finite thickness is considered. The following conditions are prescribed: # # # % / #

= $ ( , , )

at

 =0

(initial condition),

= & 1 ( , ,  ) = & 2 ( , ,  )

at at

=0 = 0

(boundary condition), (boundary condition),

=0

(boundary condition), (boundary condition).

= & 3 ( , ,  ) at = & 4 ( , ,  ) at

% / #

=1



Solution: # ( , , ,  ) = '

 $ ( , , ) ( , , ,  , , ,  ) (  ( (

0

0

2'

'

0

0

+ ' ) ' 2 ' 0

0

0

−  ' ) ' 2 ' 0

0

0

0

− ' ) '  ' 0

+ ' ) ' 0

0

0

0

0

'

0

& 1 ( , , * ) & 2 ( , , * )

1 

1 

 % 

( , , ,  , , ,  − * )

%

 %

%



( , , ,  , , ,  − * )

 +

=0

(  ( ( *



 +

=

(  ( ( * 0

 & 3 ( , , * ) ( , , ,  , , 0,  − * ) (  ( ( *  & 4 ( , , * ) ( , , ,  , , 1 ,  − * ) (  ( ( * .

Here, 

( , , ,  , , ,  ) =



1 (

, , , , )



2(

, ,  ),

© 2002 by Chapman & Hall/CRC Page 245



1

2



where the

1

, , ) =

2(

2 

+

1

1



  



2

+ exp  − 1 ( , ,  , ,  ) =  0 4  





=1





cos  

1

=1

  

0

2   

cos   



sin  

2 2



1



exp  −  



0

1



sin   

2



0

, 

, 

( ) are the modified Bessel functions. !

2 0 . A wedge domain of finite thickness is considered. The following conditions are prescribed:  #  

−1 %

 #

−1 %

# #

= $ ( , , )

at

 =0

= & 1 ( , ,  )

at

=0

= & 2 ( , ,  )

at

=

= & 3 ( , ,  ) at # = & ( , ,  ) at 4

(initial condition), (boundary condition), (boundary condition),

0

=0

(boundary condition), (boundary condition).

=1



Solution: # ( , , ,  ) = '

 $ ( , , ) ( , , ,  , , ,  ) (  ( (

0

0

2'

'

0

0

− ' ) ' 2 ' 0

0

0

+  ' ) ' 2 ' 0

 & 1 ( , , * ) ( , , ,  , 0, ,  − * ) (  ( ( *  & 2 ( , , * ) ( , , ,  ,

0

0

0

− ' ) ' 0

0

0

0

0

'

, −* )(  ( ( *

 %  & 3 ( , , * ) % ( , , ,  , , ,  − * )  ,

 %  & 4 ( , , * ) % ( , , ,  , , ,  − * )  ,

0

+ ' ) '  '

0,

0

=0

2



1( 

 (  ( ( * .

=

Here,



 (  ( ( *

( , , ,  , , ,  ) =

, ,  , ,  )=

where the

1 



 1(  , ,  , ,  )

0

exp  −

 2+

2

4  



1 2

2  1 0



sin   =1

  

2  





+



1

  

 

=1

sin  



0



 1

 

2  

2 2





exp  −  cos  



1



 0



2



,

 

cos  

 0

 

,

( ) are the modified Bessel functions. !

3.1.2-32. Domain: 0 ≤  ≤ 3 , 0 ≤ ≤

−"

0,

< < " . First boundary value problem.

An infinite cylindrical sector is considered. The following conditions are prescribed: # # # #

= $ ( , , ) at = & 1 ( , ,  ) at

 =0

= & 2 ( , ,  ) = & 3 ( , ,  )



at at

 =3

(initial condition), (boundary condition),

=0 = 0

(boundary condition), (boundary condition).

© 2002 by Chapman & Hall/CRC Page 246


 



Solution: 0

# ( , , ,  ) = '

'

−

'

0

0

− 3  ' ) '

'

−

0

'

−

0

+ ' ) '

0

 4 $  ( , , ) ( , , ,  , , ,  ) (  ( (

0

0

 %



& 1 ( , , * ) %   %

1

4 & 2 ( , , * )



%



( , , ,  , , ,  − * )

 5

( , , ,  , , ,  − * )

( ( ( *

=

4  +

  1 %  −  ' ) '  ' 4 & 3 ( , , * ) ( , , ,  , , ,  − * ) %   + = 0 − 0

Here, 

  

where the equation 7

=1

1

7

7

=1

( ) are the Bessel functions and the 8 7 9 0 (8 3 ) = 0. 0

7

3.1.2-33. Domain: 0 ≤  ≤ 3 , 0 ≤ ≤

(  ( ( * . 0



( − )2  ( , , ,  , ,  ,  ) =   exp   −  4    1( , ,  , ,  2         (8  6   ) (8 6  )   4 0 0 ) = 2 sin   sin  1(  , ,  ,  ,   6 3 )]2  3 0  6 [ ( 8 0  0 

(  ( ( *

=0

6

), 

 



0



exp(−8

26

  ),

are positive roots of the transcendental

0 ≤ < " . First boundary value problem.

0,

A semiinfinite cylindrical sector is considered. The following conditions are prescribed: # = $ ( , , ) at  = 0 (initial condition), # = & ( , ,  ) at  = 3 (boundary condition), 1 # # #

= & 2 ( , ,  ) = & 3 ( , ,  )

at at

=0 = 0

= & 4 ( , ,  )

at

=0



(boundary condition), (boundary condition), (boundary condition).

Solution: # ( , , ,  ) = '

0

'

0

'

0

0

− 3 ' ) '

0

0

+ ' ) ' 0

− ' ) ' 0

+ ' ) ' 0

0

'

0

'

0

 '

0

0

0

0

 4 $  ( , , ) ( , , ,  , , ,  ) (  ( (

'

0

 %

& 1 ( , , * ) %   %

4 & 2 ( , , * ) 4 & 3 ( , , * )

1 

1 

 

( , , ,  , , ,  − * )

( , , ,  , , ,  − * )

%

 %

%



( , , ,  , , ,  − * )

 %  4 & 4 ( , , * ) % ( , , ,  , , ,  − * )  ,

0

Here, 

 5

( ( ( *

=

4  +

=0

(  ( ( *



 +

=0

=

(  ( ( * 0

 (  ( ( * .



( − )2 ( + )2  ( , , ,  , , ,  ) =    exp − −exp − 1(  , ,  , ,  ),   4      4   . 2      (8   6   ) (8 6  )    4   0 0 26 ) = 2 sin   sin   exp(−8   ), 1(  , ,  ,  ,   3 0  6 [ (8 6 3 )]2  0  0  0 7 =1 =1 7    where the ( ) are the Bessel functions and the 8 6 are positive roots of the transcendental 7 9 0 equation ( 8 3 ) = 0. 0 1



7

7 © 2002 by Chapman & Hall/CRC Page 247

3.1.2-34. Domain: 0 ≤  ≤ 3 , 0 ≤ ≤

0 ≤ < " . Mixed boundary value problem.

0,

A semiinfinite cylindrical sector is considered. The following conditions are prescribed: # # # # % / #

= $ ( , , ) at = & 1 ( , ,  ) at

 =0

= & 2 ( , ,  ) = & 3 ( , ,  )

at at

= & 4 ( , ,  )

at

Solution:

 =3

(initial condition), (boundary condition),



=0 = 0

(boundary condition), (boundary condition),

=0

(boundary condition).

 0

# ( , , ,  ) = '

'

0

'

0

− 3 ' ) ' 0

+ ' ) ' 0

0

 4 $  ( , , ) ( , , ,  , , ,  ) (  ( ( 0

0

'

0

'

0

0

 %

& 1 ( , , * ) %   %

4 & 2 ( , , * )

−  ' ) '  ' 4 & 3 ( , , * ) 0

0

− ' ) ' 0

0

0

0

'

0

1 

1

( , , ,  , , ,  − * ) 

%

 5

( , , ,  , , ,  − * )

%

 %







( , , ,  , , ,  − * )

( ( ( *

=

4  +

=0

(  ( ( *

  +

=

(  ( ( * 0

 4 & 4 ( , , * ) ( , , ,  , , 0,  − * ) (  ( ( * .

Here,



1( 





( − )2 ( + )2  ( , , ,  , , ,  ) =     exp − +exp −  4      4   . 2    -  1



, ,  , ,  )=

where the equation

     



3

4 2





0

 =1 6

=1

7

, ,  , ,  ),

 (8   6   ) (8 6  )   0 0 sin   sin   exp(−8 [ ( 8 6 3 )]2 0  0  0 7

7 9

 26

  ),



( ) are the Bessel functions and the 8 (8 3 ) = 0. 0 0

7

1( 

6

are positive roots of the transcendental

7

3.1.2-35. Domain: 0 ≤  ≤ 3 , 0 ≤ ≤

0 ≤ ≤ 1 . First boundary value problem.

0,

A cylindrical sector of finite thickness is considered. The following conditions are prescribed: #

= $ ( , , ) #

at

= & 1 ( , ,  ) at #

= & 2 ( , ,  ) # = & ( , ,  ) 3

at at

= & 4 ( , ,  ) = & 5 ( , ,  )

at at

# #

 =0

(initial condition),

 =3

(boundary condition),



=0 = 0

(boundary condition), (boundary condition),

=0

(boundary condition), (boundary condition).

=1

© 2002 by Chapman & Hall/CRC Page 248


 



Solution: # ( , , ,  ) = '

  4 $ ( , , ) ( , ,

0 0  % 0 ' ) ' 2' & 1 ( , , * ) %  0 0 0  % 1 ) ' 2 ' 4 & 2 ( , , * ) %  0 0 0  % 1 ) ' 2 ' 4 & 3 ( , , * ) %  0 0 0  %  0 ) '  ' 4 & 4 ( , , * ) %

0 0 0  % 0  ) ' ' 4 & 5 ( , , * ) %

0 0 0 0

0

− 3 + ' − ' + ' − '

2'

,  , , ,  ) (  ( (

'



( , , ,  , , ,  − * ) 

 5



( , , ,  , , ,  − * ) ( , , ,  , , ,  − * ) ( , , ,  , , ,  − * ) ( , , ,  , , ,  − * )

( ( ( *

=

4  +

=0

 +

=

(  ( ( *



 ,

=0

 ,

=

(  ( ( * 0

 (  ( ( *

 (  ( ( * .

2           ( , , ,  , , ,  ) = 1( , ,  , ,  ) 2( , ,  ),      (8   6   ) (8 6  )    4 0 0 26 sin   sin   exp(−8   ), 1(  , ,  , ,  ) = 3 2 0  6  [ (8 6 3 )]2 0  0  0 7 =1 =1 7  

  2 2 2  7 9 (

,

,  ) = sin   sin   exp  − ,  2 2  1      1  1  1  =1    where the ( ) are the Bessel functions and the 8 6 are positive roots of the transcendental 0 equation (8 3 ) = 0. 0 7 7 3.1.2-36. Domain: 0 ≤  ≤ 3 , 0 ≤ ≤ 0 , 0 ≤ ≤ 1 . Mixed boundary value problem.

Here,



A cylindrical sector of finite thickness is considered. The following conditions are prescribed: #

=$ =& =& =& =& = &

# # # % / # % / #

Solution:

( , , ) 1( , ,  ) 2 ( , ,  ) 3 ( , ,  ) 4 ( , ,  ) 5 ( , ,  )

at at at at at at

 =0  =3

=0 = 0

=0

=1

  4 $ ( , , ) ( , ,

0 0 0  % 0 − 3 ' ) ' 2 ' & 1 ( , , * ) %  0 0 0  % 1 +  ' ) ' 2 ' 4 & 2 ( , , * ) %  0 0 0  % 1 −  ' ) ' 2 ' 4 & 3 ( , , * ) %  0 0 0

# ( , , ,  ) = '

(initial condition), (boundary condition), (boundary condition), (boundary condition), (boundary condition), (boundary condition).

0

2'

− ' ) ' 0

+ ' ) ' 0

,  , , ,  ) (  ( (

'



0

'

0

0

0

0

'

0

 

( , , ,  , , ,  − * )

 5



( , , ,  , , ,  − * ) ( , , ,  , , ,  − * )

( ( ( *

=

4  +

=0

 +

=

(  ( ( *



(  ( ( * 0

 4 & 4 ( , , * ) ( , , ,  , , 0,  − * ) (  ( ( *  4 & 5 ( , , * ) ( , , ,  , , 1 ,  − * ) (  ( ( * .

© 2002 by Chapman & Hall/CRC Page 249

Here,  

1( 

     

where the equation

2

3

1 2  +

1(  , ,  , ,  )

1

  



4

, ,  , ,  )=

 

( , , ,  , , ,  ) =





=1



2 2



1

exp  −  



1



2

0

 =1 6

=1

7

7 9 ( ) are the Bessel functions and the 8 ( 8 3 ) = 0. 0 6

0

7

1



cos  

 (8   6   ) (8 6  )   0 sin   sin   exp(−8 [ ( 8 6 3 )]2  0  0  0 7



0

   

1



cos  

   26

,   ),

are positive roots of the transcendental

7

3.1.3. Problems in Spherical Coordinates The heat equation in the spherical coordinate system has the form 

% #

1 %

=

% 



2

% #

2

%   

+

%  

% % # 1  sin : % + % sin : :  : 

2



2

% 2# 1 , 2 sin : % 2 

 = ;


6

=1

=0

sin (  ( (  5

sin ( ( ( * ,

=

4 > 

6 >  @

1 for D = 0, - 2 for D ≠ 0,

2

( , : , ,  , , ,  − * )

> A 

×> C

A 

@ 



& ( , , * ) %  

>

=

(initial condition), (boundary condition).

 

Solution:

 =0

6 

 

+1 2 ( B



> 6  ) 

 

6  )

+1 2 ( B

7

7

(cos : ) C (cos ) cos[ D ( − )] exp(− B (2 + 1)( −  D )! =   . 2 ( > + D )! E +1 2 ( B 6 3 )F

 26

  ),



9 Here, the +1 2 ( ) are the Bessel functions, the C (8 ) are 7 the associated Legendre functions >  expressed in terms> of the Legendre >G polynomials C (8 ) as follows:  7

C 

and the B

HJI



6

2

(8 ) = (1 − 8 )

2

( ( 8



>

C



(8 ), C



1 ( (8 ) =  ! 2 ( 8 

are positive roots of the transcendental equation

+1 2 ( B



(8

2

− 1) ;

3 ) = 0.

References: B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980), H. S. Carslaw and J. C. Jaeger (1984).

7

© 2002 by Chapman & Hall/CRC Page 250


 

3.1.3-2. Domain: 0 ≤  ≤ 3 , 0 ≤ : ≤  , 0 ≤ ≤ 2 . Second boundary value problem. A spherical domain is considered. The following conditions are prescribed: #

= $ ( , : , ) at = & (: , ,  ) at

% K #

# ( , : , ,  ) = '

2

'

0

'

0

2

+ 3

'

0

0

) '

 4  $ (  , , ) (  , : , ,  , , ,  )  2

'

0

2

sin (  ( (

 & ( , , * ) ( , : , , 3 , , ,  − * ) sin ( ( ( * ,

0



where 

(initial condition), (boundary condition).

 =3

 

Solution:

 =0

( , : , ,  , , ,  ) =

3 4 3

+

3



1 2   

 

6

> A 

?> 

> 

6

>

 

 6  )

> +1 2 ( B



 

6  )

+1 2 ( B

 7 7  26 × C (cos : ) C (cos ) cos[ D ( − )] exp(− B   ), > A  >  26    B (2 + 1)( − D )! 1 for D = 0, 6   = = . 2 2 for D ≠ 0, 2 2 ( + D )!> E 3 B 6 − ( + 1)F E +1 2 ( B 6 3 )F @  -      Here, the +1 2 ( ) are the Bessel functions, the C (8 ) are the associated7 Legendre functions (see Paragraph 3.1.3-1), and the B 6 are positive roots of the transcendental equation   7

2B 3

=0

=1

@

=0

3 )−

+1 2 ( B

7 9

3 ) = 0.

+1 2 ( B

7

3.1.3-3. Domain: 0 ≤  ≤ 3 , 0 ≤ : ≤  , 0 ≤ ≤ 2 . Third boundary value problem. A spherical domain is considered. The following conditions are prescribed: # % K #

+D #

= $ ( , : , ) at

 =0

(initial condition),

= & (: , ,  )

 =3

(boundary condition).

at

The solution # ( , : , ,  ) is determined by the formula in Paragraph 3.1.3-2 where 

( , : , ,  , , ,  ) =

2    A 

L

= @



1

1 for N = 0, for N ≠ 0,

- 2





 =0

6

L A 

ML =1 =0

L 

@

6



=









6



B





6  )

+1 2L ( B

(cos : ) C 

( + N )! E 3



7

×C L

L

+1 2 ( B



6  ) 

7

26

(cos ) cos[ N ( − )] exp(− B

26

(2 + 1)( − N )!

2B 2 6 L





 + ( D 3 +  )( D 3 −  − 1)F E

 

+1 2 ( B

  ),

 6 3 )F

2

.

7 Here, the +1 2 ( ) are the Bessel functions, the C (8 ) are the associated Legendre functions (see 6 Paragraph 3.1.3-1), and the B are positive roots of the transcendental equation     7

+1 2 ( B

B 3 HJI

3 ) + OPD 3

− 12 Q

+1 2 ( B

3 ) = 0.

9

7 7 Reference: B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

© 2002 by Chapman & Hall/CRC Page 251

3.1.3-4. Domain: 3

≤ ≤3

1

0 ≤ : ≤  , 0 ≤ ≤ 2 . First boundary value problem.

2,

A spherical layer is considered. The following conditions are prescribed: #

= $ ( , : , ) #



Solution: # ( , : , ,  ) = '



2

'

0

2 2

− 3

'

1

0

0

4

) '

2

0

) '

'

0

2

0



8   

=0

6

=1



=0 @

Here,



 

 

> S 

> 6

HJI

+1 2 ( B

6 3 )T A  >1

 

6  )

+1 > 2 ( B



(cos : ) C





sin ( ( ( *

=

1

4  5

sin ( ( ( * ,

=

4

2



 6  )

+1 2 ( B



(cos ) cos[ D ( − )] exp(− B

 

S 

 5

+1 2 ( B

6  )−T

 

 6 3





26

  ).





6  ),   2 B (2  + 1)( − D )! +1 27 ( B 6 3 2 ) 1 for D 7 = 0,  6 = = , >  -  2 for D ≠ 0, ( + D )! E  2 +1 2 ( B 6 3 1 ) 7 − 2 +1 2 ( B 6 3 2 )F @   where the +1 2 ( ) are the Bessel functions, the C 7 (8 ) are the associated Legendre functions >  7 >G polynomials C (8 ) as follows:  expressed in terms > of the Legendre  >    7 ( 1 ( C (8 ), C (8 ) = (1 − 8 2 ) 2 C (8 ) = (8 2 − 1) ;  ( 8 !2 ( 8  S   and the B 6 are positive roots of the transcendental equation >

6  )=



sin (  ( (

( , : , ,  , , ,  − * )

> A 

R>

2

( , : , ,  , , ,  − * ) 

×C S 

(boundary condition), (boundary condition).



& 2 ( , , * ) %  

 

( , : , ,  , , ,  ) =

 =3

 & 1 ( , , * ) %  0  %

 '

where 

1

 %

2

'

(initial condition),

 =3

2  4  $ (  , , ) ( , : , ,  , , ,  )

'

0

2 1

+ 3

 =0

at

= & 1 (: , ,  ) at # = & (: , ,  ) at 2

+1 2 ( B 26

+1 2 ( B

3

2)

+1 2 ( B

1)

+1 2 ( B

= 0.

Reference: B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

3.1.3-5. Domain: 3

1

≤ ≤3

0 ≤ : ≤  , 0 ≤ ≤ 2 . Second boundary value problem.

2,

A spherical layer is considered. The following conditions are prescribed: # % K # % K # 

Solution: # ( , : , ,  ) = '

= $ ( , : , ) at = & 1 (: , ,  ) at

 =0  =3

1

(initial condition), (boundary condition),

= & 2 (: , ,  ) at

 =3

2

(boundary condition).



2

'

0 2 1

'

+ 3

2 2

'

0

− 3

0

0

2  4  $ (  , , ) ( , : , ,  , , ,  )

'

2

sin (  ( (

1

) ' ) '

4 0

0

2

 '

2

0

'

0

 & ( , , * )  ( , : , , 3 1

1,

, ,  − * ) sin ( ( ( *

 & 2 ( , , * ) ( , : , , 3

2,

, ,  − * ) sin ( ( ( * ,

© 2002 by Chapman & Hall/CRC Page 252


  

where 

( , : , ,  , , ,  ) =

3 3 2

4 ( 3

1 4   

+

3 1)

−3

 



=0

?>

>  6  =1

>

A 

= @



2(B  ) = B

+1





6  )

1)

−



9

1 7

23

1

+1 2 ( B

T

 

6  ) 

+1 2 ( B

(cos ) cos[ D ( − )] exp(− B

(cos : ) C

S 

2(B 3

+1

S 

+1 2 ( B

>

>



 7 9  − B T 



 

2 ( + D )! 6 =  ' 4  (2 + 1)(  − D )! 1      4 1 +1 2 ( B 3 1 ) T +1 2 ( B 3 1 ) − 23 1    

1 for D = 0, D ≠ 0, for 

- S 2

S  >

@ 6

=0

×C Here,

> A 

3 1> )  

  ).

 

2

26

6  )(  ,

+1 2 ( B



+1 2 ( B

 )

+1 2 ( B

 ),



where the +1 2 ( ) and T +1 2 ( ) are the Bessel functions, the C (8 7 ) are the associated Legendre 6 are positive functions (see Paragraph 3.1.3-4), S  and S   roots of the transcendental equation  the B 7

1 A  +1> 2 ( B 3 2 ) = 0. 23 2 6 The integrals that determine 9 the coefficients can be expressed in terms of the Bessel functions and their derivatives; see Budak, Samarskii, and Tikhonov (1980). +1 2 ( B

B

3.1.3-6. Domain: 3

1

≤ ≤3

2,

2)

3

−

0 ≤ : ≤  , 0 ≤ ≤ 2 . Third boundary value problem.

A spherical layer is considered. The following conditions are prescribed: #

% K # % K #

= $ ( , : , ) at − D 1 # = & 1 (: , ,  ) at

 =0

+ D 2 # = & 2 (: , ,  ) at

 =3

1

(initial condition), (boundary condition),

 =3

2

(boundary condition).

 The solution # ( , : , ,  ) is determined by theL formula in  Paragraph 3.1.3-5 where  S   S   A  L   ML  1 6  ) +1 2 ( B 6  ) ( , : , ,  , , ,  ) = L +1 2 (L B  4     =0 6  =1 =0 @ 6 

×C Here,

A 

L

(cos ) cos[ N ( − )] exp(− B

(cos : ) C

L

2 ( + N )! 1 for N = 0, 6 =  = ' 4  2 for N ≠ 0,  (2 + 1)( − N )!  1 @ S  -     4 1 +1 2 ( B  ) = B +1 2 ( B 3 +1 2 ( B 3 1 ) −  D 1 + 23 1   



 

 7 9 − B T



+1

2(B 3

−  D

1)

1



+

7

1

23

1



T

L 3

+1 2 ( B

S 

+1 2 ( B



1)

T 

1)



26

  ).

 

2





6  )(  , 

+1 2 ( B

 )

+1 2 ( B

 ),



where the +1 2 ( ) and T +1 2 ( 9 ) are the Bessel functions, the C (8 ) are7 the associated Legendre functions (see Paragraph 3.1.3-4), of the transcendental equation S  roots S   and the B 6 are positive  7

B

+1 2 ( B

3

2)

+  D

2

1 − A  L 23 2 

9 The integrals that determine the coefficients functions and their derivatives.

HJI

6

+1 2 ( B

3

2)

= 0.

can be expressed in terms of the Bessel

Reference: B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

© 2002 by Chapman & Hall/CRC Page 253

3.1.3-7. Domain: 0 ≤  < " , 0 ≤ : ≤ : 0 , 0 ≤ ≤ 2 . First boundary value problem. A cone is considered. The following conditions are prescribed: # #

= $ ( , : , ) at

 =0

= & ( , ,  )

: =:

at

(initial condition), 0

(boundary condition).



Solution: # ( , : , ,  ) = '

2 0

 $ ( , , ) ( , : , ,  , , ,  )

0

'

U '

0

− ' ) ' 0

2 0

0

'

2

sin (  ( (



%  & ( , , * ) sin % ( , : , ,  , , ,  − * )  +

0

(  ( ( * ,

=

0

U

where 

( , : , ,  , , ,  ) = −



1

A 6 (2 V + 1)  2+ exp  − @ 6 4  

4      6 

=0

A 6 −6

Here, C

=

@

1 for W for W

- 2

×C 

= 0, ≠ 0,

−6

(cos : ) C (

= (1 − 8 )2 6

−6

C

( 8

−6



2



+1 2



C Z

6

1−8 1 (8 ) = Y  (1 + W ) 1 + 8 

2Z

2  

(cos ) cos[W ( − )],

(8 )

( ( V

−6

C

(8 )

.

 X

=cos

0

U

 (8 ) is the modified Legendre function expressed as −6

  

O − V , V + 1, 1 + W ;

1 2

− 12 8 Q ,

Y

where (  , [ , \ ; 8 ) is the Gaussian hypergeometric function and ( ] ) is the gamma function. The 6 summation with respect to V is performed over all roots of the equation C − (cos : 0 ) = 0 that are greater than −1 ^ 2.

HJI

Reference: H. S. Carslaw and J. C. Jaeger (1984).

3.2. Heat Equation with Source _

= `

_

b

c

3`

a

+ d

(e , f , g , a )

3.2.1. Problems in Cartesian Coordinates In the Cartesian coordinate system, the three-dimensional heat equation with a volume source has h i the form h i h i h i h j

2

=k l h m

2

+ h n

2

2

+ h

m

2

2

]

n

j

+ p ( , , ] , ). o

It describes three-dimensional unsteady thermal phenomena in quiescent media or solids with constant thermal diffusivity. A similar equation is used to study the corresponding three-dimensional mass transfer processes with constant diffusivity. 3.2.1-1. Domain: − "



2

2

+"

+ # 2 ) + 8 3 & % − '39 *: + ( , , ,  ), 7

1 exp 8 4 & % .9 + 5 7 4 & % 7 leads to the three-dimensional heat equation , ; + = & ( , -.-/+ + , 0/0 + + , 121 + ) that is dealt with in Subsection 3.1.1.  = ! exp 8 2

3.

   

= " exp 8 2

% .9 , 7 &

=  

2

  

2

+ 

2

  

2

+ 

% .9 , 7 &

2

  

2



= # exp 8 2

+ $ – (

2



+

2



+

2



 =

% .9 , 7 &

)+

1



+

2



+

3



+ s) 

.

This is a special case of equation 3.3.2.3 with < = ( ) = '>= , ? ( ) = @ .

© 2002 by Chapman & Hall/CRC Page 261

4.

 

=  

 

2

  

2

2

+ 



2

 

2

+ 



+ 

2

 

+

 

1

2

+

 

 

3

 

+

 

. 

 

This equation governs the nonstationary temperature (concentration) field in a medium moving with a constant velocity, provided there is volume release (absorption) of heat proportional to temperature (concentration). The substitution (! , " , # ,  ) = exp 8A5

+5

1!

2"

+5

+ B .9 C (! , " , # ,  ),

3#

where

% 1 % 2 % 3 1 2 2 2 8*% 1 + % 2 + % 3 9 , , 5 2=− , 5 3=− , B =' − 2& 2& 2& 4& leads to the three-dimensional heat equation ,  C = & D 3 C that is considered in Subsection 3.1.1. 5 1 =−

5.

 

=  

 

2

  

2

2

+ 



2

 

2

+ 



– 

2

 

.

  

 

This equation is encountered in problems of convective heat and mass transfer in a simple shear flow. Fundamental solution: E J4K

6.

$H! −  − 12 %2 (" + )) (! , " , # ,  , , ,  ) = exp G − 1 2 2 1 2 2 1F 2 4 &  8 1 + 12 %  9 (4 &  )3 F 2 8 1 + 12 %  9

1

2

−

( " − )2 + ( # − ) 2 . I 4& 

Reference: E. A. Novikov (1958).

 

=  

 

2

  

2

2

+ 

  

2

+ 

2



+ 

2

 

1

  

+

2

 

+

  

 

3



+ L ( , , , ).

 



 







This equation is encountered in problems of convective heat and mass transfer in a linear shear flow. Domain: −  < ! <  , −  < " <  , −  < # <  . Cauchy problem. An initial condition is prescribed: = < (! , " , # ) at

 = 0.

Solution: 

 

(! , " , # ,  ) = 

+

−

 −

 





< ( , , )  (! , " , # ,  , , ,  )  !  "  #

−



−

0

 





−

−

( , , ,  )  (! , " , # ,  , , ,  −  )  !  "  #   ,

where  (! , " , # ,  , , ,  ) = M

(! ,  ,  ; % 1 ) M (" , ,  ; % 2) M ( # , ,  ; % 3),

2 & 2 O  M (! ,  ,  ; % ) = 6 (N − 1):

−1 F 2

%

7.

   

=  

2

  

2

+ 

2

  

2

+ 

2

  

2



+ (

This is a special case of equation 3.3.2.5.

1



 2 % 8P! N O −  9 exp 6 − : .  2 & ( N 2 O − 1)

+  1)  

+ (

 

+ (

3



2



+  2)  

+  3)    

 

+ (s1 + s2 + s3 + Q ) . 







© 2002 by Chapman & Hall/CRC Page 262

8.

T R S  

+

T 2 S

2

  

+ 

2



2

+ 



= 0.

2



Three-dimensional Schro¨ dinger equation, V Fundamental solution:

2

 

2U

2

 

E

2

 

 

3F 2

V

(! , " , # ,  ) = − W 

exp 6YV X W (! 2S 

X W

2 S  

S

J4K

= −1.

2

2

+"

+ # 2) − V

3 : . 4

Reference: V. S. Vladimirov, V. P. Mikhailov, A. A. Vasharin, et al. (1974).

9.

   

= 

 

 

 Z

 

+  

 

 

 

 

 [

+  

 

 

 

 

 \

. 

 

This equation describes unsteady heat and mass transfer processes in inhomogeneous (anisotropic) media. It admits separable solutions, as well as solutions with incomplete separation of variables (see Subsection 0.9.2-1). In addition, for ] ≠ 2, ≠ 2, ^ ≠ 2 there are particular solutions of the X form =

 2 =46

( ,  ),

! 2− _

2 & (2 − ] )

+

" 2− `

% (2 −

)2 X

+

= # 2−

' (2 − ^ )2

: ,

where the function ( ,  ) is determined by the one-dimensional nonstationary equation ,

=

, 

, 2

5 + ,  2 

,

,

, 

5

=2

1 1 + 2−] 2−

+ X

1 − 1. 2−^ 

For solutions of this equation, see Subsections 1.2.1, 1.2.3, and 1.2.5.

3.3.2. Equations Containing Arbitrary Functions 1.

 

=  

 

2



2

 

2

+ 



2

 

+ 

2



+a ( ) . 

2

 

 

This equation describes three-dimensional unsteady thermal phenomena in quiescent media or solids with constant thermal diffusivity, provided there is unsteady volume heat release proportional to temperature. The substitution (! , " , # ,  ) = exp $3b < ( )  c)3C (! , " , # ,  ) leads to the usual heat equation , d.C = & D 3 C that is dealt with in Subsection 3.1.1. 2.

 

=  

 

2

  

2

o = r + 2s h

  

The transformation (! , " , # ,  ) = exp G ! f

2

+ 

1 (g

f 1 (g ) k g ,

2

+ 

2

  

2



+ $ a 1 ( ) + a 2 ( ) + a 3 ( ) + e ( )) 



) + " f 2(g ) + # f 3 (g ) + & h i f p = t + 2s h

f 2 (g ) k g ,



2 1 (g



)+f

2 2 (g





)+f

2 3 (g

q = u + 2s h





.

)j k g + h l (g ) k g m n (o , p , q , g ),

v 3 (w ) k w ,

v x (w ) = h

y x (w ) k w ,

leads to the three-dimensional heat equation z d*n = s {Az |.|/n + z 0/0 n + z 1213n } that is dealt with in Subsection 3.1.1.

© 2002 by Chapman & Hall/CRC Page 263

3.

~ 

2

= ‚ ~

~ €

2

 ~ ƒ

+~  2 ~ „

2

2

+ ~  2 † + i –‡ ( ƒ

2

+

~ …

2

„

+

2

…

)+

1(

ƒ ˆ

)+ €

„ ˆ

2(

€

)+

3(

… ˆ

€

)+ ‰ ( )j €



.

1 Š . Case ‹ > 0. The transformation Œ s ‹

(r , t , u , w ) = n (o , p , q ,  ) exp Ž  o = r exp { 2

p = t exp { 2

s ‹ w.} , 

q = u exp { 2

s ‹ w.} , 

{Pr 2 + t 2 + u 2 (}  ,

2s

s ‹ w.} , 

 =

1

exp { 4 s ‹ w.} + 5 ,

4 s ‹





where 5 is an arbitrary constant, leads to an equation of the form 3.3.2.2: z n z w

z 2n

=s ‚

z o 2

1

z 2n

+

z p 2

ln( ’( )

v x ( ) = ( ’( )3 “ 2 y x ‚

+ io v

z q 2†

1 (

1

‘ ( ) =

, †

’

z 2n

+

)+p v

2 (

)+q v

3 (

+

3 , 4

’ =4

ln( ’( )

‚ ’( l

† ’

) + ‘ ( )j/n , 

” = 1, 2, 3.

s ‹ ,

2 Š . Case ‹ < 0. Œ The transformation −s ‹ (r 2s

(r , t , u , w ) = • (o 1 , p 1 , q 1 ,  1 ) exp Ž  r

o 1 =

p 1=

,

cos { 2 − s ‹ w.}

t

cos { 2 − s ‹ w.}



q 1=

,



2

2

+t

+ u 2 ) tan { 2 − s ‹ w.}( , 

u

cos { 2 − s ‹ w.}

,

 1 =



1 tan { 2 − s ‹ w.}  2 −s ‹ 

also leads to an equation of the form 3.3.2.2 for • = • (o 1 , p 1 , q 1 ,  1 ) (the equation for • is not written out here). 4.

~ 

2

=  1( ) ~ €

~ €

 ~ ƒ

2

2

+  2( ) ~ €

 ~ „

+  3( ) ~

2

€

2



Here, 0 < s x (w ) < — ; ” = 1, 2, 3. Domain: − — < r < — , − — < t < — , − — Œ An initial condition is prescribed:

~ …

2

+ – ( , , , ). ƒ

„

…

€

< u < — . Cauchy problem.

= y (r , t , u ) at

w = 0.

ΠSolution:

(r , t , u , w ) = h

d 0

h

−

h ˜

−

˜

h

−

+h ˜ h ˜ ˜ h ˜ ˜ −

˜

−

where

‘ (r , t , u , o , p , q , w ,  ) = d

›

1

(o , p , q ,  ) ‘ (r , t , u , o , p , q , w ,  ) k o k p k q k 

˜ y (o , p , q ) ‘ (r , t , u , o , p , q , w , 0) k o k p k q ,

−

˜

™ ˜

˜

1 8š 3 “

s 1( ) k  , =h œ

2

 › ›

2

1 2 3

›

› d

exp Ž −

( r − o )2 ( t − p )2 ( u − q )2 − −  , 4 1 4 2 4 3

= h s 2( ) k  , œ

›

›

3

=h œ

d › s 3 ( ) k  .

›

See also the more general equation 3.4.3.3, where other boundary value problems are considered.

© 2002 by Chapman & Hall/CRC Page 264

5.

~ 

2

=  1( ) ~ €

~ €



2

~ ƒ

2

+  2( ) ~



€

2

2

+  3( ) ~ €

~ „ + i ‡ 2( ) + ž 2( ) j ~  Œ € „ € ~ „



2

+ i ‡ 1( ) + ž 1( ) j ~ 

€ ƒ € ~ … ~ ƒ + i ‡ 3( ) + ž 3( )j ~  + i s1( ) +s2( ) +s3( ) + Ÿ ( )j . € … € € ƒ € „ € … €  ~ …

The transformation

(r , t , u , w ) = exp i y 1 (w )r + y 2 (w )t + y 3 (w ) u + l (w )j(n (o , p , q , w ), o = ¡ 1 (w )r + ¢ 1 (w ), p = ¡ 2 (w )t + ¢ 2 (w ), q = ¡ 3 (w ) u + ¢ 3 (w ), where ¡ x (w ) = 5 x exp Ž h

‹ x (w ) k wc ,

y x (w ) = ¡ x (w ) h ¢ x (w ) = h

i 2 s x ( w ) y x ( w ) + ’ x ( w )j ¡ x ( w ) k w + ¥ x , ¦ 3

l (w ) = h

( ” = 1, 2, 3; « ¬ , ¤ ¬ , ¥ ¬ , ª z ­ z ©

6.

2

³ ´

= ¶ 1( ) ³ ´ µ

³ µ

x (w ) k w + ¤ x ¡ x (w ), x (w )

¡£

x

=1

2 i s x ( w ) y x ( w ) + ’ x ( w ) y x ( w )j k w + §

¨ (© ) k © + ª ,

are arbitrary constants) leads to an equation of the form 3.3.2.4: z 2­

= ® 1 (© ) ¡ 21 (© ) 2

+ ¶ 2( ) ³ ´ 2 µ

+ ® 2 (© ) ° 22 (© )

z ¯ 2

2

+ ¶ 3( ) ³ ´ 2 µ

³ · ³ ¸ + º4» 3( ) + ¼ 3( )½ ³ ´ µ ¹ µ ³ ¹ The substitution À

³ ¹

+ º s 1( )

2

z 2­ z ± 2

+ ® 3 (© ) ° 23 (© )

z 2­

. z ² 2

+ º » 1( ) + ¼ 1( )½ ³ ´

+ º4» 2( ) + ¼ 2( )½ ³ ´

³ ·

³ ¸

µ ·

2

µ ·

µ

2

+s2( )

µ ¸

µ ¸

µ

+s3( ) 2 + ¾ 1( ) + ¾ 2( ) + ¾ 3( ) + ¿ ( )½ µ ¹

µ ·

µ ¸

µ ¹

µ

´

.

(Á ,  , à , © ) = exp º Ä 1 (© )Á 2 + Ä 2 (© ) 2 + Ä 3 (© ) à 2 ½/­ (Á ,  , à , © ), where the functions Ä ¬ = Ä ¬ (© ) are solutions of the respective Riccati equations Ä ¬ Å = 4 ® ¬ (© ) Ä ¬ 2 + 2 ÆǬ (© ) Ä ¬ + È2¬ (© ) ( É = 1, 2, 3), leads to an equation of the form 3.3.2.5 for ­ = ­ (Á ,  , à , © ). 7.

³ ´

ÊË 2

+

Ë

Ë

( )+Î

ºÍÌ

µ

=1

³ µ

( )

µ ·

Ë 3½

³ ´

–¶ ³ ´

³ ·

³ ·

=

3

Ë

Ë Ê 2 ,Ï

=1

Ð

Ï ( ) µ

Ë 2 ³ ³ ·

´ ³ ·

Ï

+ Ð

33 (

µ

)³

2

³ ·

´2 . 3

Equation of turbulent diffusion. It describes the diffusion of an admixture in a horizontal stream whose velocity components are linear functions of the height. Fundamental solution: d 3 Ñ 1 1 Ê × (Û ) Ü Û . ×Â × (© ) = § exp (Á 1 , Á 2 , Á 3 , ¯ 1 , ¯ 2 , ¯ 3 ) = Ö− Õ × −1 ( © )  , Õ Ø ( Ø Ù Ø Ø 4× (4Ò )3 Ó 2 Ô det |Õ | 0 Ú Here, the following notation is used (Ý , Þ

= 1, 2):

Â ß = Á ß − ¯3ß − à ß (© ) − Á 3 á ß (© ) − ® § Ú

ß ã (© ) = ä Ú

d

0

,Ø =1

( © − Û )â ß ( Û ) Ü Û ,

 3=Á 3−¯ 3+® © ,

ß 3 (© ) = 3 ß (© ) = − ä 33 (© ) á ß (© ), ß ã (© ) + ä 33 (© ) á ß (© ) á ã (© ), d d Ú Ú à ß (© ) = § Ä ß (© ) Ü © , á ß (© ) = § â ß (© ) Ü © , 33 ( © ) = ä 33 ( © ), 0

0

det |Õ | is the determinant of the matrix T with entries Õ × Ø (© ), Õ × −1 Ø (© ) are the entries of the inverse 2 of T. The inequalities Õ 11 (© ) > 0, Õ 11 (© )Õ 22 (© ) − Õ 12 ( © ) > 0, and det |Õ | > 0 are assumed to hold. å4æ Reference: E. A. Novikov (1958).

© 2002 by Chapman & Hall/CRC Page 265

3.3.3. Equations of the Form ç ( è , é , ê ) ë ì = div[ î ( è , é , ê )∇ì ] – ï ( è , é , ê )ì

+

(è , é , ê , í ) ð

ë í

Equations of this form are often encountered in the theory of heat and mass transfer. For brevity, the following notation is used: À

À

div º4® (r)∇ ½ = ñ

Öò® (r) ñ Á

ñ

À

Ù + ñ Â ñ Á

ñ

Öò® (r) ñ

À

Ù + ñ Ã ñ Â

ñ

Öò® (r) ñ

r = {Á , Â , Ã }.

Ù , Ã

ñ

The problems presented in this subsection are assumed to refer to a simply connected bounded domain ó with smooth boundary . It is also assumed that ô (r) > 0, ® (r) > 0, and õ (r) ≥ 0. Ú

3.3.3-1. First boundary value problem. The following conditions are prescribed: À

= Ä (r) À

ö =0

at

= â (r, ö ) for r ÷

(initial condition), (boundary condition).

Ú

Solution: À d

(r, ö ) = −

0

ø ù d

(û , Û ) ü (r, û , ý − Û ) þ ÿ ú



(û ) (û ) ü (r, û , ý ) þ ÿ




ø ù



(û , Û ) ® (û ) 

ø 0 ø

þ Û +



ü (r, û , ý − Û )Ù þ 



þ Û . Ú

(1)

Here, the modified Green’s function is given by





ü (r, û , ý ) =

(r)­ ­





=1

(û )



­

exp(−

2



ý ),







­

2

=

2

(r)­ 

ø ù

û = { 1 , 2 , 3 },

(r) þ ÿ ,







(2)

where the and ­ (r) are the eigenvalues and corresponding eigenfunctions of the Sturm–Liouville problem for the following elliptic second-order equation with a homogeneous boundary condition of the first kind:

− (r)­ + (r)­ = 0, ­ = 0 for r .

div 4® (r)∇­ 





(3) (4)





Ú

The integration in solution (1) is carried out with respect to 1 , 2 , 3 ; along the outward normal to the surface with respect to 1 , 2 , and 3 . Ú General properties of the Sturm–Liouville problem (3)–(4): 









denotes the derivative 









1 . There are countably many eigenvalues. All eigenvalues are real and can be ordered so that , with as ; consequently, there can exist only finitely many 1 ≤ 2 ≤ 3 ≤ negative eigenvalues.





















2 . For (r) > 0, ® (r) > 0, and (r) ≥ 0, all eigenvalues are positive: 





> 0.



3 . The eigenfunctions are defined up to a constant multiplier. Any two eigenfunctions ­ (r) and ­ (r) corresponding to different eigenvalues and are orthogonal with weight (r) in the domain ÿ :















ø ù

(r)­

(r)­ 

(r) þ ÿ

= 0 for 

≠ 

.

© 2002 by Chapman & Hall/CRC Page 266

4 . An arbitrary function (r) that is twice continuously differentiable and satisfies the boundary condition of the Sturm–Liouville problem ( = 0 for r ) can be expanded into an absolutely and Ú uniformly convergent series in the eigenvalues: 















1



(r) = 



(r), ­





= 



=1

2

­

(r) (r)­ 

(r) þ ÿ ,



ø ù

2 where the formula for ­ is given in (2). In a three-dimensional problem, to each eigenvalue there generally correspond finitely many linearly independent eigenfunctions ­ (1) , ­ (2) , , ­ ( ) . These function can always be replaced by their linear combinations ¬ ¬ ¬ ­ ¯ ( ) = « ¬ ,1 ­ (1) + + « ¬ , ¬ −1 ­ ( −1) + ­ ( ) , = 1, 2, , , (1) (2) ( ) such that ­ ¯ , ­ ¯ , , ­ ¯ are now pairwise orthogonal. Thus, without loss of generality, we assume that all eigenfunctions are orthogonal.



"!

#

$&%





'









(((

















)

(((





(((

3.3.3-2. Second boundary value problem. The following conditions are prescribed: = (r) at

ý =0




*

(initial condition),

 *

= (r, ý ) for r

(boundary condition).





Solution: (r, ý ) = *

+



d

+

(û , ) ü (r, û , ý − ) þ ÿ ,

ø 0 ø ù d



ú

(û , ) ® (û ) ü (r, û , ý − ) þ 

,

ø 0 ø

þ

,

,

+ ,

þ +




(û ) (û ) ü (r, û , ý ) þ ÿ 

ø ù

. ,





(5)



Here, the modified Green’s function is given by (2), where the and ­ (r) are the eigenvalues and corresponding eigenfunctions of the Sturm–Liouville problem for the elliptic second-order equation (3) with a homogeneous boundary condition of the second kind,



­



= 0 for r





+

.

(6)

For (r) > 0 the general properties of the eigenvalue problem (3), (6) are the same as for the first boundary value problem (with all > 0).



3.3.3-3. Third boundary value problem. The following conditions are prescribed: = (r)

at




*

ý =0

(initial condition),

 *







+ (r)

= (r, ý ) for r *



)



+

(boundary condition).



The solution of the third boundary value problem is given by formulas (5) and (2), where the and ­ (r) are the eigenvalues and corresponding eigenfunctions of the Sturm–Liouville problem for the second-order elliptic equation (3) with a homogeneous boundary condition of the third kind,





­ 

+ (r)­ = 0 for r )



+

.

(7)

For (r) ≥ 0 and (r) > 0 the general properties of the eigenvalue problem (3), (7) are the same as for the first boundary value problem (see Paragraph 3.3.3-1). Let (r) = = const. Denote the Green’s functions of the second and third boundary value problems by 2 (r, û , ý ) and 3 (r, û , ý , ), respectively. If (r) > 0, then the following limiting relation holds: 2 (r, û , ý ) = lim 3 (r, û , ý , ). 

)

)

)

-

-

-

¬

/.

021

)

0

-



)

References for Subsection 3.3.3: V. S. Vladimirov (1988), A. D. Polyanin (2000a, 2000c).

© 2002 by Chapman & Hall/CRC Page 267

3.4. Equations with

Space Variables 3

3.4.1. Equations of the Form 4

= 5

4

7

8

+

9 5

( :

1, ;




cos(

?



=1

@





exp −

B

)





*

2



=1 

*

 )



+® ý

=

>

2 . Fundamental solution: 

G

M

(x, ) = H

B

3 . Suppose P

= ( 1, Q

Q

(((

|x|2 exp − , 4®

1

,

2 F

>

® I

B

|x|2 =

?

K

=1

B

JK

2

LNM

.

O

, ) is a solution of the homogeneous equation. Then the functions B

K O

1 Q

= C

( Q

R

1 S

+ A

O

M

1,

,

((( M

R

+ S

M

O

2 Q

=

S

+

K

T

S

=1 3

=

2

LNM K

V

B

S

( Q

X

Y

B

|

2

Z

X

+ B

1, A

+2

,

(((

V

S

+ B

)

+1 ), A

K

W

E Q

=1

B

K

M

[

, + A

K

O

2 K

Y

1 S

K

LNM

4 ( +

V

O

Y

V

K

+2

1

=1 O

exp −

C

| +

+1 ), A

K

D

Q

+ B

K

O LUM

exp C

2

, A

W

K

T

X

+ O

1 Y

, [

(((

, X

O

+

, K Y

[

\

X

+ + Y

S

[

[

, S

X

−

= 1, Y

\

O

where , 1 , , +1 , , 1 , , , and are arbitrary constants, are also solutions of the equation. The signs at in the formula for 1 can be taken independently of one another. ]

^

(((

^

S

K

S

(((

S

Y

X

K

S

Q

© 2002 by Chapman & Hall/CRC Page 268

M

3.4.1-2. Domain:

= {−

` K

< a


0.

¬ ³

The transformation ( is any number) ¾

Ä

( 1, ´

‰

,

(((

, ) = ( 1, ‰

¿

À

,

(((

1 , ) exp 2 À

—

·

À

1

=

1 ‰

exp 2 Å

š

,

Ã Ç ¿ È

,

(((

Á

š

= À

2

¨NÄ

exp 2 ‰

Å

+

‰ à µ Â

š

,

Ã Ç ¿

µ

=1

È

—

− Ã

ÅÆ

Ç

¶È

,

¿ÊÉ

µ

1

=

4

µ

exp 4 Å

Ã

+

Ã Ç ¿ È

¾

Ç µ

Ä š

= š

leads to the -dimensional heat equation Æ

¸

Ë

µ

¸

· š ° °

3.

¬

2

¨ °

=

«

®

2 ¬

=1

«

°

« ¯

­

¨ °

+ –

2

¼

¯

=1 ³

«

±

+

» œ

=1

·

+s Ä

=1

œ

½

² ¯

»

°

¨ °

±

. Ä

±

This is a special case of equation 3.4.3.2 with

¬

( )= Ì

that is dealt with in Subsection 3.4.1.

º Ã

¿

and ( ) = . ¶

¿

4. «

«

2

¨°

= ¬

®

¯

«

=1 ­

The substitution ´

( 1, ‰

,

(((

«

° °

¬

2

+

¨°

« ¯

=1

±

+ ¬

. ²

¬ ³ «

±

Ä

, ) = exp ‰

¿

¼

¾

¿

1 − 2

Ä ½

Ä Î

¨ Ä

( 1, ‰

‰

Ã

š

=1

leads to the -dimensional heat equation

Î

(((

,

, ), ‰

¿

Î

µ

š

Æ

Í

ž °

¸

¹

= Ã

where ¾

1 = − 4

2

¨ Ä

¶

Ã

š

=1

,

µ

that is dealt with in Subsection 3.4.1. š

Ï

š

© 2002 by Chapman & Hall/CRC Page 270

°

5.

=

«

¬

2

¨°

®

¯

¬

«

­

«

°

¨°

+

2

«

=1

°

°

¯

+ Å

=1

±

²

« È

2 Õ

+

Õ

Ó

Ô

«

¬

«

2

2

¨ °

Ô

¯

­

«

Ù

2 ¼

Ô

Ý2Þ

Ú

Û

±

ª

+

. Ò

¬ ±

exp Ô

= −1. ½

Ù

ˆ

2 ×

2

×

¿

s ¯

=1

±

½

(x, ) = −

¼

= 0.

2 ¬

«

+ ¬

±

This is the n-dimensional Schro¨ dinger equation, Fundamental solution: Ø

½

°

«

=1 Ö

¨Ñ°

³

This is a special case of equation 3.4.3.4. 6.

°

° °

¼

2 ª

×

¿

|x|2 − Ù

Ô

ˆ

4 ×

¿

|x|2 =

,

Æ

2 1 ‰

+

2

+

ÜÜÜ

.

Ú

‰

Reference: V. S. Vladimirov, V. P. Mikhailov, A. A. Vasharin, et al. (1974).

3.4.3. Equations Containing Arbitrary Functions å

1.

=

ß

à

2

äå

â

ã

à

á

ß

Á

å

( )+ ( ) ã

è

=1

æ

The transformation

ç

äå

+

2

ß

=1

ß

å

æ

á

. É

á à

é

( 1, ê

,

(((

, ) = exp Ú

ê

¿

Ä

Ä

= ò

Ä Ú

ë

+2 ê

ë

Ä

Á

() í

ë

î

2

Ã

()

¿

=1

ó

Ä

äNÄ

( )+ ê

Ä

ë Ú

äNÄ

=1 Ä

()

í

í

ì ì

î

+ () í

ï

í

ð

( 1, ñ

ò

,

(((

, ), Ú

ò

í

Ä ì

( )=

, í

í

î

, í

( )= ï

()

í

í

ô

ì

î

, í

õ

Ä

leads to the -dimensional heat equation Æ

ö

=

÷Êñ

2. ß

= à

ß

â

ã

ß

=1




á

å

à

ß

äå

+ –

2

þ

2

ÿ

=1

æ

ç

ö

úüûýúüû

ñ

å

äÑå

+

ã

ø

=1

å å

2

äå

that is discussed in Subsection 3.4.1.

ù

ó

( )+ ( ) ã

è

=1 æ

æ

á

. ð

á à

1 . Case > 0. The transformation é

( 1, ê

,

(((



, ) = ( 1, ê

í

ñ

ò

(((

, ò

ø

ò

1

=

1




exp 2 

,



ø

,

=




1 ,  ) exp 2


exp 2 

,



Ä



ø

ó

Å

í

(((

ò

ê

ó

í

=1

1

=



,

ê

ó




4 is an arbitrary constant, leads to an equation of the form 3.4.3.1: ê



2

NÄ




exp 4 

+ ,



ó

í

ó

ø

where 

ö

ë



1 (  )3 

( ) =


= ñ

ö



 ö

ö

ò

ln(  )

2

+

ñ

ø

=1 



2

ó

í



ø

2

,





ë



( ) + ( )

ø þ

ò

ï

=1



1

( ) = ï

ln(  )



, ñ

+



ô

ð



,

Æ

4





é

ê

,

(((

, ) =  ( 1, ê

í

(((

,  ,  ) exp

ø

 1

=

ê

1

cos 2  −


ó

í

,

(((

,

=  ø

− 

2 þ

ø



.

ó

õ

2 . Case < 0. The transformation ( 1,




= 4

cos 2  −


ó

tan 2  −


ó





í



, 

=

2 ø

í

ê

1 2 −

ð

,

=1

ó

ê

ø







tan 2  −


ó

 í

ó

also leads to an equation of the form 3.4.3.1 (this equation is not specified here).

© 2002 by Chapman & Hall/CRC Page 271



3.

=

 





=1 

 

2 

( ) 



+

2

( 

 

1,   



,

, ). 





The solutions of various problems for this equation can be constructed on the basis of incomplete separation of variables (see Paragraphs 0.6.1-2 and 0.9.2-1) taking into account the results of Subsections 1.1.1 and 1.1.2. Some examples of solving such problems are given below. It is assumed that 0 < ( ) <  ,  = 1, , . ó

í

(((

1 . Domain:  = {−  < :

}

(


≥ 0, satisfy the

x

− ® (Š Š



,  )ž

=0

(  = 1,

‹Œ‹Œ‹

, )

(2)

x x with nonhomogeneous initial conditions of ax special form,

= ¯ (Š ž

− ” ) at

= , 

x the first kind, x and homogeneous boundary conditions of ž

x

x

In determining the function ž delta function.

ž

= 0 at = 0 x at

, the quantities ”

x Š

= =

Š

(3)

x – —

x

, .

and  play the role of parameters; ¯ (Š ) is the Dirac

2 ˜ . The second and third boundary value problems. The following conditions are prescribed: ‰ x ‰ ‰ z z

¢

M

M ¢

‰

− +±

x ° ‰

x

= L (x)



= ™ (x,  ) at = (x,  ) at Š

x

at

¨

x

Š

x

=0 =– = —x

The second boundary value problem corresponds to °

x

(initial condition),

x

(boundary conditions), x (boundary conditions). =±

= 0.

© 2002 by Chapman & Hall/CRC Page 274

Solution: ‰

(x,  ) = Ÿ

0

w³x

−

š Ÿ ›

v

Ÿ

0

=1

+

wx

Ÿ v

0

=1

(y,  ) ž (x, y,  ,  )   y   œ

x

š Ÿ ¡

š Ÿ ¡

(¢ )

£

(¢ )

£

(–

, )

x

ƒ

(—

›

(y) ž (x, y,  , 0)x   y L

(y,  ) ž (x, y,  ,  )† ™

x

+Ÿ 

x

x

¥

¢ =¦ ¢

x

(y,  ) ž (x, y,  ,  )†

, )

¥

¢ =© ¢

ƒ´¨

  §

  §

¥( ) x ¥( )

  

.

  

The Green’s function can be represented as the product (1) of the corresponding one-dimensional Green’s functions satisfying the linear equations x x (2) with thex initialx conditions (3) and the homogenx z eous boundary conditions x x x x x −° ž = 0 at Š = – , z ¢ ž M +± ž = 0 at Š = — . ¢ ž

µV¶

7.

M Reference: A. D. Polyanin (2000a, 2000b).

{ | { }

= ·

w

{ ~

,¸ =1 { 

·

· ¸

(

‘A€

1,   



,

) 

{ |

~

{ 

–ˆ ( ¤

¸

1,   



,

)  ~

+ |

“

( 

1,   

,

, ).  ~

}

The problems considered below are assume to refer to a bounded domain • with smooth surface § . We introduce the brief notation x = {Š 1 , ‹Œ‹Œ‹ , Š } and assume that the condition v w

¹

¹

v ,º =1

£

¹ » º

(x) » º

≥

w ®

2 » ¹ ,

v ¹

®

> 0,

=1

is satisfied; this condition imposes the requirement that the differential operator on the right-hand side of the equation is elliptic. 1 ˜ . First boundary value ‰ problem. The following conditions are prescribed: = L (x) ‰

at 

=0

(initial condition),

¼

(boundary condition).

= ™ (x,  ) for x Solution:

§

‰

(x,  ) = Ÿ

−Ÿ

0

0

š Ÿ ›

(y,  ) ž (x, y,  −  )   œ

š Ÿ ¡ ™

z

z ½

(y,  )

¥ ž

•

¥

+Ÿ

  

(x, y,  −  )¤

  §

›

L ¥

(y) ž (x, y,  )     

•

¥

.

(1)

‘

Here, the Green’s function is given by ž

(x, y,  ) =

¿

(x) ¾ =1 À

v

v

Á

(y) Á

v2

À

exp(− »

Á

),

Á 2

=

2

Ÿ ›

(x)   Ã

,

y = {Ä 1 , ŌŌŠ, Ä

},

(2)

v  v v À v À v À (x) are the eigenvalues and corresponding eigenfunctions of the Sturm–Liouville

where the » and problem forv the following elliptic second-order equation with homogeneous boundary condition of v À the first kind: ¿ ¹

v ,º =1

Æ

¹ È £

¹ º

(x)

Æ À º

¤

− É (x) + »

= 0Æ Ç for x

Æ Ç

¼

À §

.

À

= 0,

(3) (4)

À © 2002 by Chapman & Hall/CRC Page 275

The integration in solution (1) is carried out with respect to operator defined as ½

≡ ¥

Æ Î

,Ä Ê ;

Ë Ë Ì Í

is the differential

Ê

¿ ¹

Ä 1 , ŌŌÅ

¹ º

(y) Ð º

,

Ä ¹ Æ Î

,º =1 Ï

(5)

Æ surface Ñ . In the special case where where N = { Ð 1 , ŌŌŠ, Ð Ê } is the unitÆ outward normal to the ¹´¹ (x) = 1 and ¹ (x) = 0 for Ò ≠ Ó , the operator of (5) coincides with the usual operator of differentiation º Ï Ï along the direction of the outward normal to the surface Ñ . General properties of the Sturm–Liouville problem (3)–(4): 1. There are countably many eigenvalues. All eigenvalues are real and can be ordered so that » 1 ≤ » 2 ≤ » 3 ≤ ÔŒÔŒÔ , with » Ê Õ  as Ö Õ  ; consequently, there can exist only finitely many negative eigenvalues. 2. For É (x) ≥ 0 all eigenvalues are positive: × Ê > 0. 3. The eigenfunctions are defined up to a constant multiplier. Any two eigenfunctions Ê (x) and (x) corresponding to different eigenvalues × Ê and × are orthogonal in the domain à :

Ü ÝSÞ

À

Ù

À Ø

› À

ß à?á â

To each eigenvalue (2) dent eigenfunctions (1) Ê , Ê , ŌŌŠ, combinations À

À

¯ (Ê ã ) =

(x) Ê

(1)

+

ä ã ,1 Ê

(x) Ú

= 0 for Ã

Ø

≠ Ö

Û

.

À Ø

there generally correspond finitely many linearly indepen) . These functions can always be replaced by their linear

× Ê (

Ê

Ø À

ԌԌÔ

( ã −1)

+ä

ã , ã −1 Ê

+

(ã )

Ê

,

= 1, 2, å

ŌŌÅ

, Û

,

(2) ( )À À such that ¯ (1) are now pairwiseÀ orthogonal. Thus, without loss of generality, we Ê , ¯ Ê À , ŌŌŠ, ¯ Ê assume that all eigenfunctions are orthogonal. Ø

À

À

À

2 ˜ . Second boundary value problem. The following conditions are prescribed: æ

= ç (x)

æ

at Â

= ™ (x, ) for x

Æ

=0

(initial condition),

ê

(boundary condition). Ñ

Â

Here, the left-hand side Æ ofè the é boundary condition is determined with the help of (5), where , Ä , y, and Ä ã must be replaced by æ , , x, and ã , respectively. Î Solution: æ

Ù

(x, ) = Â Ù

+

Ç

Ù 0

0

š

›

Ç

ë

(y, ì ) (x, y, − ì ) Ú

Ù î š ™

Ù

+

à í Ú ì

 Î

(y, ì ) (x, y, − ì ) Ú

Ñ í Ú ì

› ç

(y) (x, y, ) Ú Î

.

à í

Â

(6)

Â

Î

Here, the Green’s function is defined by (2), where the × Ê and Ê (x) are the eigenvalues and corresponding eigenfunctions of the Sturm–Liouville problem for the elliptic second-order equation (3) À with a homogeneous boundary condition of the second kind: = 0 for x ê

Ñ

.

(7)

Æ À

For É (x) > 0 the general properties Æ è ofé the eigenvalue problem (3), (7) are the same as for the first boundary value problem (see Item 1 ˜ ). For É (x) ≡ 0 the zero eigenvalue × 0 = 0 arises which corresponds to the eigenfunction 0 = const. It should be noted that the Green’s function of the second boundary value problem can be À expressed in terms of the Green’s function of the third boundary value problem (see Item 3 ˜ ).

© 2002 by Chapman & Hall/CRC Page 276

3 ˜ . Third boundary value problem. The following conditions are prescribed: æ æ

= ç (x)

at Â

+ å (x)æ = ™ (x, ) for x

Æ

=0

(initial condition),

ê

(boundary condition). Ñ

Â

The solution of Æ the è é third boundary value problem is given by relations (6) and (2), where the × Ê and Ê (x) are the eigenvalues and corresponding eigenfunctions of the Sturm–Liouville problem for the second-order elliptic equation (3) with a homogeneous boundary condition of the third kind: À

+ å (x) = 0 for x Æ À

ê

Ñ

.

(8)

À

For É (x) ≥ 0 and å (x) > 0, the Ægeneral properties of the eigenvalue problem (3), (8) are the same è é as for the first boundary value problem (see Item 1 ˜ ). Let å (x) = å = const. Denote the Green’s functions of the second and third boundary value problems by 2 (x, y, ) and 3 (x, y, , å ), respectively. Then the following relations hold:  Î

Î

)=

2 (x, y,

Î Ù

where à õVö

0

= ›

Ú Ã

Â

 lim 3 (x, y, ñPò ô ã ó 0  ð 1 Î Ã 0

+ lim

if É (x) > 0;

, å ),

, å ), if É (x) ≡ 0;

3 (x, y,

ãôó 0 Î

Â

is the volume of the domain in question.

References: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), A. D. Polyanin (2000a, 2000b).

© 2002 by Chapman & Hall/CRC Page 277

Chapter 6

Hyperbolic Equations with Three or More Space Variables 2m

6.1. Wave Equation l

=

2 l

2p o

m

3

n

6.1.1. Problems in Cartesian Coordinates The wave equation with three spaceK variables in the rectangular Cartesian coordinate system has the form P 2

P

=

N 2

2r q

2t

s

+

2

s u

2t

s

s v

2

+

2t

s

.

2 x

s w

This equation is of fundamental importance in sound propagation theory, the propagation of electromagnetic fields theory, and a number of other areas of physics and mechanics. 6.1.1-1. Particular solutions and their properties. 1 y . Particular solutions: t

(u , v , w , z ) = {

exp |

} 1u

t

(u , v , w , z ) = {

sin( }

1u

(u , v , w , z ) = t

sin( }

(u , v , w , z ) =

{

t

(u , v , w , z ) = t

where { ,

{

sinh( }

1, f 2, f 3, } 1, } 2, f

+f +f

1u

sinh( } {

+}

1u 1u

3

2 ) sin( } 3 w

+f

+}

2 3 

+f

3 ) sin |"q z

+f

, b

+f

b

+}

2 2

+}

2 3 

,

+}

2 2

+}

2 3 

,

2

+}

2 2

+}

2 3 

,

2

+}

2 2

+}

2 3 

,

2

} 1

3 ) sinh |"q z

+f

2 ) sinh( } 3 w

2

} 1

3 ) cos | q z

2 ) sinh( } 3 w

+f

1 ) sinh( } 2 v

2 2

+}

2 ) sin( } 3 w

+f

1 ) sinh( } 2 v

+f

} 1

+f

1 ) sin( } 2 v

2

q z

3w ~

1 ) sin( } 2 v

+f

and }

b

+}

2v

3 ) cosh |"q z

b

} 1 b

} 1

are arbitrary constants.

2 y . Fundamental solution: € (u , v , w , z ) =

1 2 q

‚

(q

2 2

− ƒ 2 ),

z

ƒ

= „

u

2

+v

+ w 2,

2

where (… ) is the Dirac delta function.

†k‡

‚

Reference: V. S. Vladimirov (1988).

‹ 3 y . Infinite series solutions containing arbitrary functions of space variables: ‹

(u , v , w , z ) = ˆ (u , v , w ) + t

Š+‹ ‰

=1 t

(u , v , w , z ) =

z

(u , v , w ) + z

( Œ z )2 (2 )!

Š+‹ ‰

=1

Ž

‹

ˆ

(u , v , w ),

( Œ z )2 (2 + 1)! Ž

‹



Ž

≡

s s u

2 2

+

s s v

2 2

+

s s w

2 2

,

(u , v , w ),

© 2002 by Chapman & Hall/CRC Page 393

where ˆ (u , v , w ) and  (u , v , w ) are any infinitely differentiable functions. The first solution satisfies the initial conditions t (u , v , w , 0) = ˆ (u , v , w ), s ‘ t (u , v , w , 0) = 0, and the second solution initial conditions t (u , v , w , 0) = 0, s ‘ t (u , v , w , 0) =  (u , v , w ). The sums are finite if ˆ (u , v , w ) and  (u , v , w ) are polynomials in u , v , w .

†k‡

Reference: A. V. Bitsadze and D. F. Kalinichenko (1985).

4 y . Suppose t = t (u , v , w , z ) is a solution of the wave equation. Then the functions t t

1

t ‹

=

2

=

3

=

t

(~

{

r t

{

+“

’ u

1− „

{ ƒ 2

−” u

−Œ

z

( ” • Œ )2 r

t

2z 2

+“

1, ~ ’ v

u

,

2z 2

−” z

,v ,w ,

−Œ

ƒ 2

+“

2, ~ ’ w

Œ

1 − (” „

ƒ 2

3, ~ ’ z −2

v

−Œ

2z 2

u

,

ƒ 2

4 ),

,

)2 x

• Œ

+“

w

−Œ

2z 2

,

z ƒ 2

−Œ

2z 2 x

,

where { , “ , ” , and ’ are arbitrary constants, are also solutions of the equation. The signs at ’ in the expression of t 1 can be taken independently of one another. The function t 2 is a consequence of the invariance of the wave equation under the Lorentz transformation.

†k‡

References: G. N. Polozhii (1964), W. Miller, Jr. (1977), A. V. Bitsadze and D. F. Kalinichenko (1985).

6.1.1-2. Domain: − –


0, where the ( ) are the Bessel functions (the prime denotes the derivative with respect to the argument) and the ç are positive roots of the transcendental equation ( ý ) = 0. 



=

í ç

2í ç



+



þ

ÿ 2

í

,

=







!





!



!



õ

6.1.2-2. Domain: 0 ≤

≤

,0≤ ý

≤ 2þ , 0 ≤ "

#

≤ ÿ . Second boundary value problem.

A circular cylinder of finite length õ is considered. The following conditions are prescribed: = 0 (õ , , ) at = 0 (initial condition), $

"

= = $




$



% $



' $



= =

#



1 ( , , ) at at 1 (õ , , ) 

"

"

#

2 (õ (

(

#

õ



, , ) 3( , , ) "



"



at at

=0 =

(initial condition), (boundary condition),

#

#

=0 =

(boundary condition), (boundary condition).

&

)

'

© 2002 by Chapman & Hall/CRC Page 399

Solution: õ 

( , , , )= $

"

#





2 

 *

õ +

0(



0

 



+



0

2

0 *

0



.

.

0 .

2



2 



2



"

#

1(

Ü



Ü 

-





-

Ü



Ü 

-



, ,Ý ) ( , , , , , , −Ý )Ü 

-



"

#

&



-



Ü 

2 ( , , Ý ) ( , , , , , 0, − Ý ) Ü 



"

#





Ü





Ü Ý -

Ü Ý 



õ +

0

-

õ *







õ

0 *

-

+

*

#



0



"





0

, ,Ý ) ( , , , , , , −Ý )Ü

3( (

0

-

, , ) ( , , , , , , )Ü

(









0

2



0

+



,

(



, , ) ( , , , , , , )Ü

õ





2 

,

&

0

−



1(

0



2

+

0 +









"

#





)

Ü





.

Ü Ý 



Here, õ

( , , , , , , )= 

"

#



-

&

1

+

 

/ ) !

æ

2

/

 !







( !

2− &

!



)

!

2ç 

2 /

2ç 

(

æ æ =0 ç =1 =0

2

+





2 &



æ õ =1

) ç

!

2)[ 

1

0



( !





/

cos

2



)

( !



ç !

/

cos 

1

) ç

!

)]2 &

3

)



"

= ç

!



5

!

6

,

2

7

cos

1

1 for 2 for

:

6

= !



where the ( ) are the Bessel functions and the ( ) = 0. ! ;

sin( /





 3

2

)

2 2

+

2ç 6

)

9

4



)

cos[ ( − )] cos



 1

3

/

/

!



 1



3

)

 3

ç !

!



) 4

,

ç





9

/

sin

 1

ç !

= 0, > 0, 8

8

are positive roots of the transcendental equation

&

6.1.2-3. Domain: 0 ≤

≤
0, 8
( ) are the Bessel functions (the prime denotes the derivative with respect to the argument) and  the : are positive roots of the transcendental equation > (  ) = 0.

© 2002 by Chapman & Hall/CRC Page 422

 

2 I . A circular cylinder of finite length is considered. The following conditions are prescribed:        

=  0 ( ,  ,  ) at

 =0

(initial condition),

=  1 ( ,  ,  ) at =  1 (  ,  ,  ) at

 =0

(initial condition), (boundary condition),

=  2 ( ,  ,  ) =  3 ( ,  ,  )

 =0

=

at at

(boundary condition), (boundary condition).

 =

Solution: 

 ( ,  ,  ,  ) = 





+

 

0 0 2 



0



+ $ 2 +$

2

−$

2

  





0

0

0

2

.0/

0



2





0



0



 1 ( , , ! ) " ( ,  ,  , , , ! ,  ) # # # !      2  1 ( , ! , % ) " ( ,  ,  ,  , , ! ,  − % ) #

0 2



0

 2 ( , , % ) J  " ( ,  ,  , , , ! ,  − % )K M !    



0



=0

# 

 3 ( , , % ) J  " ( ,  ,  , , , ! ,  − % )K M # # !   =  

 &

0

1

# ! # %



( , , ! , % ) " ( ,  ,  , , , ! ,  − % ) # 



Here, " ( ,  ,  , , , ! ,  ) =   

# !



0

2

 0 ( , , ! ) " ( ,  ,  , , , ! ,  ) # #    

0



0



0



0



+



0

 

2



#

#

# % # %

# ! # % .

/ /

6.3.2-5. Domain: 

1

-

≤ ≤

=1

2,

) 1

sin 2 3

  

5

sin 2 3

 !

sin '(76 1 , 8 < 8 5 8 < 8 8 < 8 < 8 8 / 8 2< 8 . .N/ : 2 .98 > ( : )> ( : ) + 2 :  2 − ? 2 )[ > ( :  )]2 ) @ :   - :( 8 /  =0 =1 =1 ;    ! × cos[? (  − )] sin 2 3 sin 2 3 sin '  6 @ : , ,  5  5 / < 8 8 8 / $ 2 2 2 $ 2 2 2 1 for ? = 0, 3 2 + + , @ : = $ 2 2: + 3 2 ++ , = B 1 = < 8 8 2 for ? > 0,   ; 8 < where the > ( ) are the Bessel functions and the : are positive roots of the transcendental equation > C (  ) = 0.  2

0 ≤  ≤ 2 , 0 ≤  ≤  . First boundary value problem.

A hollow circular cylinder of finite length is considered. The following conditions are prescribed:   

=  0 ( ,  ,  ) at

 =0

(initial condition),



=  1 ( ,  ,  ) at

 =0

(initial condition),

=  1( ,  ,  )  =  ( ,  ,  ) 2

at at

=

1

=

2

=  3 ( ,  ,  ) =  4 ( ,  ,  )

at at

 =0

 

 =

(boundary condition), (boundary condition), (boundary condition), (boundary condition).

© 2002 by Chapman & Hall/CRC Page 423

Solution:  ( ,  ,  ,  ) = 

  



+

 0



0

0

+ $ 2

1

2

−$

2



0

 



+



2





0

0



0

  

0

 



2

 0 ( , , ! ) " ( ,  ,  , , , ! ,  ) # #    

1

2

1

Here,

8 < 8 

" ( ,  ,  , , , ! ,  ) = 2 

.98

. -

=0

8 < 8

.0/

:-

=1

-

=1

>

8 < 8

2:

: 

2(

;

>

2

1)

−>

(

8

the > ( ) and P equation

8

2)

: 

2(

8 O

/

8

8


2,

8


(

$

2 2 2

3





2

++ ,

: );

are positive roots of the transcendental

8
+ 0 ? !

+1 @ 2 (

# = > A ) = ?

2

sin g ù g ù  ù

=

1

=

2

sin g ù g ù  ù

sin g ù ù g ù  ù ,

+1 @ 2 (

# = > )

× $ = ú (cos B ) $ = ú (cos g ) cos[ 4 ( C −  )] sin /.

+1 @ 2 (

+1 @ 2 (

# = > D




+0 1 .

# = > A )−E = # = > D 1) 5 = # = > A ), +1 @ 2 ( +1 @ 2 ( # = > (23 + 1)(3 − 4 )! 5 = 2 # = > D 2) 1 for 4 = 0, +1 @ 2 ( = > = 2 = , 2 for 4 ≠ 0, (3 + 4 )! [ 5 = 2 +1 @ 2 ( # = > D 1 ) − 5 = 2 +1 @ 2 ( # = > D 2 )] ú " ú ! where the 5 = +1 @ 2 (A ) are the Bessel functions, the $ = ú (' ) are the associated Legendre functions expressed in terms of the Legendre polynomials $ = (' ) as = = 1 2 @ 2 F ú F = = = = (' 2 − 1) , = $ (' ), $ ú (' ) = (1 − ' ) ú $ (' ) = 3 !2 F ' F ' ú and the # = > are positive roots of the transcendental equation = +1 @ 2 ( # D 2 ) = 0. ? =

?

# = > A )=5 =

sin g ù ù g ù 

( , g ,  , ) (ø , ÿ ,
, , g ,  ,  − )

 û

(ø , ÿ ,
, , g ,  ,  ) =

)



0

2

sin g ù ù g ù 

(ø , ÿ ,
, , g ,  ,  − )9 :

 1 (g ,  , ) 8 )

0



2

2

)



0



2

 0 ( , g ,  ) (ø , ÿ ,
, , g ,  ,  )

 1 ( , g ,  ) (ø , ÿ ,
, , g ,  ,  )

2



2

1

1

0



0

where

0*



* 



2 2

0

 

2 1

þ

− ÷ 2þ 

 

0

2

+



2

1)E

=

+1 @ 2 (

6.3.3-5. Domain: D 1 ≤ A ≤ D 2 , 0 ≤ B ≤ G , 0 ≤ C ≤ 2G . Second boundary value problem. A spherical layer is considered. The following conditions are prescribed: H = I (A , B , C ) at J = 0 (initial condition), 0 ) *H A = I 1 ( , B , C ) at J = 0 (initial condition), ) K H = L 1 (B , C , J ) at A = D 1 (boundary condition), ) K H = L 2 (B , C , J ) at A = D 2 (boundary condition). Solution: H (A , B , C , J ) = )

+ M

)

2N

J M 2N

2 1

+ T 2D

2 2

M

0U

M

0

M

O1 2

M 0U M

M

0U

M

0

M 0

N

2N

O

N

I 0 (P , Q , R ) S (A , B , C , P , Q , R , J )P

O I 1 (P , Q , R ) S (A , B , C , P , Q , R , J )P

O1

M

2N

2

0

N

0

− T 2D

+

M

0

M

0

N

0

M

0

2N

M

0

N

2

M

O1 Y

2

2

sin Q F P F Q F R

sin Q F P F Q F R

L 1 (Q , R , V ) S (W , X , C , D

1, Q

, R , J − V ) sin Q F Q F R F V

L 2 (Q , R , V ) S (W , X , C , D

2, Q

, R , J − V ) sin Q F Q F R F V

(P , Q , R , V ) S (W , X , Z , P , Q , R , J − V )P

2

sin Q F P F Q F R F V ,

O © 2002 by Chapman & Hall/CRC Page 432

6.3. EQUATIONS OF THE FORM [

2

[]

\

=^

2

2_

n

where 3 sin fJg h i

S (W , X , Z , P , Q , R , J ) =

3 2

4j ( k

3 1)g

−k

+ h

mon

1 4j g W q P n

×x Here,

n

q

=| n

us

l

m

q n

mrq

p l =1q n

=0

t s

=0

(cos Q ) cos[ y ( Z − R )]

(cos X ) x

+1 v 2 ( w

n

=

n

+1 v 2 ( w

n

sin fz/{ T 2n w {

T

2w 2 p

2p

p P )

+h i +h

.

n n

u

2

n n

u

p W )

+1 v 2 ( w

p

(} + y )! W (2} + 1)(} −n y )! M n O 1 1 € +1 v 2 ( w O p k p W ) = w p €  +1 v 2 ( w p k 1 ) − 2 k 1 n n n n n 1 ƒ +1 v 2 ( w q p k − „ w p ƒ  +1 v 2 ( w p k 1 ) − n 2k 1 n p

n n

q u

q

t 1 for y = 0, 2n for y ≠n 0, n

433

− a ` + b (c , d ,  , e )

3`

2

p W )~ W , n

+1 v 2 ( w

n

1 )‚

ƒ n €

1 )‚

+1 v 2 ( w +1 v 2 ( w

n

p W ) p W ),

n

where the € +1 v 2 (W ) and ƒ +1 v 2 (W ) are the Bessel functions, the x (… ) are the associated Legendre functions (see Paragraph 6.3.3-4), and the w p are positive roots of the transcendental equation n n

u



w

6.3.3-6. Domain: k

1

≤W ≤k

2,

+1 v 2 ( w

2)

k

−

u

1

2k

+1 v 2 ( w

2

k

= 0.

2)

0 ≤ X ≤ j , 0 ≤ Z ≤ 2j . Third boundary value problem.

A spherical layer is considered. The following conditions are prescribed: † ˆ † ˆ ‰ † ˆ ‰ †

= ‡ 0 (W , X , Z ) at = ‡ 1 (W , X , Z ) at

†

− y 1U = Š 1 (X , Z , z ) + y 2 † = Š 2 (X , Z , z )

z =0

(initial condition), (initial condition),

z =0 W =k

at at

(boundary condition), (boundary condition).

1

W =k 2 n † (W , X , Z , z ) is determined byŒ the formula in Paragraph 6.3.3-5 where The solution n n n n n Œ u u m‹n m m,Œ 1 p P ) S (W , X , Z , P , Q , R , z ) = t n +1 v 2 ( w Œ p W ) +1 v 2 ( w Œ 4j g W P l =0 p l =1 =0 s p n n sin fz/{ T 2n w 2 p + h i × x (cos X ) x (cos Q ) cos[  ( Z − R )] . { T 2w 2 p + h

Here,

n

Œ

u s

Œ

u

2 (} +  )! p = W (2} + 1)(} −  )!n M O 1 1 € +1 v O 2 ( w k +1 v 2 ( w W ) = „w €  +1 v 2 ( w k 1 ) − Ž y 1 + 2k 1  n

1 for  = 0, =| 2 for  ≠ 0,n n

t

n

−n w ƒ  +1 v 2 ( w k n

− Ž y

1)

1

+

n

1

2k



1

ƒ

+1 v 2 (Πw

n

n n 2

+1 v 2 ( w

n

1 )‚

k

ƒ

1 )‚

€

n

p W )~ W ,

+1 v 2 ( w

W )

+1 v 2 ( w

W ),

where the € +1 v 2 (W ) and ƒ +1 v 2 (W ) are the Bessel functions, the x (… ) are the associated Legendre functions (see Paragraph 6.3.3-4), and the w p are positive roots of the transcendental equation u w

n



+1 v 2 ( w

k

2)

+ Ž y

2

−

u

1 2k

2



n

+1 v 2 ( w

k

2)

= 0.

© 2002 by Chapman & Hall/CRC Page 433

434

HYPERBOLIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES

6.4. Telegraph Equation 



2‘ ’



+“

2

‘ 

= ’

2• ”

3

– ‘

+

– ‘

’

(˜ , ™ , š , ) —

6.4.1. Problems in Cartesian Coordinates A three-dimensional nonhomogeneous telegraph equation in the rectangular Cartesian system of coordinates has the form ˆ

ˆ †

2†

ˆ ›

2

ˆ

2

+œ ˆ › =

ˆ

2†

Ž ˆ ž

2

ˆ

2†

+ ˆ Ÿ

2†

+ ˆ  

2

2



ž Ÿ   › − ¡ † + ¢ ( , , , ).

6.4.1-1. Reduction to the three-dimensional Klein–Gordon equation. ž Ÿ   › ›¥¤/¦ ž Ÿ   › The substitution † ( , , , ) = exp £ − 12 œ ( , , , ) leads to the equation ˆ

2

ˆ ›

¦ 2

=

ˆ

2

2

Ž ˆ ž

¦

ˆ

2

¦

ˆ

+ ˆ Ÿ

2

2

+ ˆ  

2

¦ 2

− £7¡ − 14 œ 

2 ¤/¦

+ exp £

1 2œ

›¥¤

ž Ÿ   › ¢ ( , , , ),

which is discussed in Subsection 6.3.1. 6.4.1-2. Domain: 0 ≤

ž

Ÿ

≤ § 1, 0 ≤

 

≤ § 2, 0 ≤

≤ § 3 . First boundary value problem.

A rectangular parallelepiped is considered. The following conditions are prescribed: ¨

=© =© =¬ =¬ =¬ =¬ =¬ =¬

ª « ¨ ¨ ¨ ¨ ¨ ¨ ¨

0( 1( Ÿ 1( Ÿ 2( ž 3( ž 4( ž 5( ž 6(

Ÿ

ž

ž

 

, Ÿ ,   ,   ,   ,   , Ÿ , Ÿ ,

, )   , ) › , ) › , ) › , ) › , ) › , ) › , )

at at at at at at at at

› › ž ž Ÿ Ÿ  

 

=0 =0 =0 =§1 =0 =§2 =0 =§3

(initial condition), (initial condition), (boundary condition), (boundary condition), (boundary condition), (boundary condition), (boundary condition), (boundary condition).

Solution: ª ¨ (ž , Ÿ ,   , › ) = ª › ­

3

0

® ­

3

2

0

2

® ­

+­ ® ­ ® ­ ® «0

0

+  2­

3

0«

0

0«

­

0

−  2­

­

0«

0

® ­

­

0

® ­

0

0

ª

0

0

0

ž Ÿ   › ² ( , , , ¯ , ° , ± , − ¶ )¸ ¹

® ¬ 1 (° , ± , ¶ ) · ª ¯ ª

2

3

® ­

ž Ÿ   › © 1 (¯ , ° , ± ) + œ © 0 (¯ , ° , ± )µ/² ( , , , ¯ , ° , ± , ) ³ ¯ ³ ° ³ ± 2

3

2

+ ­

® ­

3

2

− ­

´

0

­

0

1

1 ž Ÿ   › ® © 0 (¯ , ° , ± ) ² ( , , , ¯ , ° , ± , ) ³ ¯ ³ ° ³ ±

® ¬ 2 (° , ± , ¶ ) · ª 1

ª

1

ª

¯

³ ° ³ ± ³ ¶ ³ ° ³ ± ³ ¶

=

1

® ž Ÿ   › ² ( , , , ¯ , ° , ± , − ¶ )¸ º ³ ¯ ³ ± ³ ¶

® ¬ 3 (¯ , ± , ¶ ) · ª ® ¬ 4 (¯ , ± , ¶ ) · ª

ž Ÿ   › ² ( , , , ¯ , ° , ± , − ¶ )¸ ¹

=0

° °

=0

ž Ÿ   › ² ( , , , ¯ , ° , ± , − ¶ )¸ º

³ ¯ ³ ± ³ ¶

=

®

2

© 2002 by Chapman & Hall/CRC Page 434

¼ = ¿ 2À » ½ »½ ª 2 1 ž ® ­ ® ¬ 5 (¯ , ° , ¶ ) · ª ² ( ± 0 0 ª 2 1 ž ® ­ ® ¬ 6 (¯ , ° , ¶ ) · ª ² ( ± 0 0

«

+  2­ −  2­ +­ ž

Ÿ

 

² ( , , ,¯ ,° ,± , )=

3

­

8

›

­

0

«

0

Here,

­

0«

2

0

® ­

0

® ­

0

Ð

+¾ »

ÊÉ =1

Ë

where

2

Ñ Ì

1

Ÿ

 

›

, , , ¯ , ° , ± , − ¶ )¸ È Ÿ

 

=0

³ ¯ ³ ° ³ ¶

›

, , , ¯ , ° , ± , − ¶ )¸ È

Ì

Ë

Ê,Í É

Î Ï

=

³ ¯ ³ ° ³ ¶ 3

= Ø × Ö , 2

×

Ì

Ï

×

Ë

Í

Ì

3

sin( Ð

× sin( Ð Ë

= ÙÚ× Ö , Ò

Ë Í

1

=1 =1

Í

= Õ×Ö ,

435

− Â Á + Ã (Ä , Å , Æ , Ç )

3Á

® 1 ž Ÿ   › ® ¢ (¯ , ° , ± , ¶ ) ² ( , , , ¯ , ° , ± , − ¶ ) ³ ¯ ³ ° ³ ± ³ ¶ .

› ¤ ‹Ê Ë É

exp £ − 12 œ

§ 1§ 2§ 3

¼

2

6.4. TELEGRAPH EQUATION »

= Û 2(Ð

Í ž

) sin(Ñ Ì

Ÿ

) sin(Ò

¯ ) sin(Ñ Ì ° ) sin(Ò Ë Í 2

 

Í

)

± ) sin £

Ë ›/Ó

Ï Ì

Í Ô

,

+ Ñ Ì2 + Ò 2 ) + Ü − 14 Ý 2 .

×

6.4.1-3. Domain: 0 ≤ Þ ≤ 1 , 0 ≤ ß ≤ 2 , 0 ≤ à ≤ 3 . Second boundary value problem. A rectangular parallelepiped is considered. The following conditions are prescribed: á = â (Þ , ß , à ) at ã = 0 (initial condition), 0 ä åá = â 1 (Þ , ß , à ) at ã = 0 (initial condition), ä æ á = ç 1 (ß , à , ã ) at Þ = 0 (boundary condition), ä æ á × = ç 2 (ß , à , ã ) at Þ = 1 (boundary condition), ä è á = ç 3 (Þ , à , ã ) at ß = 0 (boundary condition), ä è á × = ç 4 (Þ , à , ã ) at ß = 2 (boundary condition), ä é á = ç 5 (Þ , ß , ã ) at à = 0 (boundary condition), ä é á

Solution: á (Þ , ß , à , ã ) = ä

= ç 6 (Þ , ß , ã ) ä

3

ã ê 3

+

0ë

ê

−Û

2

+Û

2

−Û

2

+Û

2

−Û

2

+Û

2

+

0ë

ê

2

0ë

2

ê å0 ë ê ê

0ë

ñ

ê

0ë

0ë ê

0å ê

2

0ë

3

ê

ê

0ë

0å

ê

1

0ë

3

ê

0å ê

0ë ê

1

0ë

2

ê

0ë

ê

0å

1

0ë

ê 2

êå

0

ê

0ë

ê

3

ê

0ë

1

0ë 2

ê

0ë

ê

×

3

(boundary condition).

â 1 (ì , í , î ) + Ý â 0 (ì , í , î )òÚï (Þ , ß , à , ì , í , î , ã ) ð ì ð í ð î 2

3

à =

â 0 (ì , í , î ) ï (Þ , ß , à , ì , í , î , ã ) ð ì ð í ð î

0ë

1

0ë

0å

ê

0

ê

3

ê

ê

1

at

ç 1 (í , î , ó ) ï (Þ , ß , à , 0, í , î , ã − ó ) ð í ð î ð ó × ç 2 (í , î , ó ) ï (Þ , ß , à , 1, í , î , ã − ó ) ð í ð î ð ó ç 3 (ì , î , ó ) ï (Þ , ß , à , ì , 0, î , ã − ó ) ð ì ð î ð ó × ç 4 (ì , î , ó ) ï (Þ , ß , à , ì , 2, î , ã − ó ) ð ì ð î ð ó ç 5 (ì , í , ó ) ï (Þ , ß , à , ì , í , 0, ã − ó ) ð ì ð í ð ó × ç 6 (ì , í , ó ) ï (Þ , ß , à , ì , í , 3, ã − ó ) ð ì ð í ð ó 1

0ë

ô

(ì , í , î , ó ) ï (Þ , ß , à , ì , í , î , ã − ó ) ð ì ð í ð î ð ó ,

© 2002 by Chapman & Hall/CRC Page 435

436

HYPERBOLIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES

where Ô

å÷

−ö

2

ï (Þ , ß , à , ì , í , î , ã ) = õ × × ×

ý

sin ùãú û

+

ú û ø

1 2 3

ü‹ý É

üÉ

ý

ü,ÿ É

þ

=0

1 for 2 for

1 4

û =Ü −



2

Ý

,



=

ý

ÿ

ý

ÿ Þ ) cos(Ñ þ ÿ

cos( Ð þ

ì ) cos(Ñ þ í ) cos(Ò ÿ

Ñ þ

ý ,

1 2

= Û 2 (Ð þ







× cos( Ð

= 0, Ð > 0,ý ÿ 

ú

=0 =0

ý

=

ÿ ýþ

=ÿ





,

2

= Ò

)

ý

ÿ



 

þ


,

,

3

+ Ñ þ2 + Ò 2 ) + Ü − 14 Ý 2 .

The summation is performed over the indices satisfying the condition corresponding to  = = = 0 is singled out. 

î ) sin ù







) cos(Ò 

+ 

+ 

> 0; the term 





6.4.1-4. Domain: 0 ≤ Þ ≤ 1 , 0 ≤  ≤ 2 , 0 ≤



≤ 3 . Third boundary value problem.

A rectangular parallelepiped is considered. The following conditions are prescribed: 

=  0 (Þ ,  ,  ) at =  1 (Þ ,  ,  ) at =  1 ( ,  ,  ) at

     

−

  

+  −

  

  

=  2 ( ,  ,  ) =  3 (Þ ,  ,  ) 

3

+  −  +

  



2 

  



1 



4

 

5 6 

=  4 (Þ , ,  ) =  5 (Þ ,  ,  ) =  6 (Þ ,  ,  ) 

=0  = 0 Þ =0

(initial condition), (initial condition), (boundary condition),

at at

Þ = 

=0

(boundary condition), (boundary condition),

at at at 

= 2 =0

= 3

(boundary condition), (boundary condition), (boundary condition).



1

 

The solution  (Þ ,  ,  ,  ) is determined by the formula in Paragraph 6.4.1-3 where 

ï (Þ ,  , ì , í ,  ) = 8 exp ù − 12 Ý

ü‹ý



ü



ü



ý



  



=1

=1

=1

ý

× sin( Ð Here, ý

ý

= arctan 

ý

 



where the Ð

Ð

1

 

, 

= arctan ý ý

Ñ

ý

,Ñ

, and Ò

ý

sin( Ð ý




ì +



Ò

ý

,



3  2

Þ + 

) sin(Ñ



= Û 2 (Ð

+Ñ

 

+ ý



î +



2



+  ) sin(Ò 



ý








) sin(Ò

 

( 3 4 + Ñ )( 3 +  4 )   2 +  2 ( 3 + Ñ 2 )( 42 + Ñ 2 ) ø 



í +

) sin(Ñ 

= arctan

2 

( 1 2 + Ð 2 )( 1 + 2 ) = 1 +  2   ( 1 + Ð 2 )( 22 + Ð 2 ) ø ý ø    

ý

 



,

ý

1

) sin ù!

)

  




.



2

+ Ò 2 ) + Ü − 14 Ý 2 , 

( 5 6 + Ò  2 )( 5 +  6 )   , 3 +  2 ( 5 + Ò 2 )( 62 + Ò 2 ) 



are positive roots of the transcendental equations Ð Ñ Ò

2 2 2

− −



− 



1 2

=(

3 4

=(

5 6

=(













1

+

3

+

5

+







2 )Ð

cot( 1 Ð ),

4 )Ñ

cot( 2 Ñ ),

6 )Ò

cot( 3 Ò ).



© 2002 by Chapman & Hall/CRC Page 436

6.4. TELEGRAPH EQUATION "

2#

+& "

2

"%$

2(

='

"%$

437

− * ) + + (, , - , . , / )

3)



6.4.1-5. Domain: 0 ≤ Þ ≤ 1 , 0 ≤  ≤ 2 , 0 ≤

#

≤ 3 . Mixed boundary value problems.

1 0 . A rectangular parallelepiped is considered. The following conditions are prescribed: 

=

  

= 

,  ,  ) at

 0 (Þ  1 (Þ

=  1 ( , ,  ) =  2 ( , ,  ) 

=  3 (Þ , ,  )

  



=  4 (Þ , ,  )

  



=0

(initial condition),

Þ =0

(boundary condition),

at

Þ =

1

(boundary condition),

at 

=0

(boundary condition),

=

2

(boundary condition),

=0

(boundary condition),

=

(boundary condition).

at

=  5 (Þ ,  ,  )

  

(initial condition),

at



  

=0

,  , ) at 







=  6 (Þ ,  ,  )



at



at







3

Solution:  

(Þ

,



, , ) =

3



3

+ ê

0ë

+Û

2

−Û

2

+Û

2

−Û

2

+Û

2

ë 0

ê

ê

0ë

ñ

ê

0ë

3

0ë

ê

0

2

0ë

ê 3

ê

0ë

0

1

ê

0ë

3

ê

1

0ë ê

0

ê

0ë

2

ê

0ë

ê

0

1

0ë

ê 2

ê

0

1

0ë ê

ê

0ë

3

ê

0

0ë

ê

2

0ë

ê

,í ,î )+ Ý

 1 (ì 2

, í , î ) ï (Þ ,  ,  , ì , í , î ,  ) ð ì ð í ð î

 0 (ì

0ë

ê 1

0ë

0

ê

ê

1

3

ê 2

0ë

ê 2

ê

ê

−Û

+

0ë

ê 

2

, í , î )ò ï ( Þ ,  ,  , ì , í , î ,  ) ð ì ð í ð î

 0 (ì

 1 

 1 (í

,î ,ó )

ï (Þ ,  ,  , ì , í , î ,  − ó )



ì

ø

  2 (í

,î ,ó ) ø



ï (Þ ,  ,  , ì , í , î ,  − ó ) ì

 4 (ì

, î , ó ) ï ( Þ ,  ,  , ì , 2, î ,  − ó ) ð ì ð î ð ó

 5 (ì

, í , ó ) ï (Þ ,  ,  , ì , í , 0,  − ó ) ð ì ð í ð ó

 6 (ì

, í , ó ) ï ( Þ ,  ,  , ì , í , 3,  − ó ) ð ì ð í ð ó

0ë



(ì , í , î , ó ) ï (Þ ,  ,  , ì , í , î ,  − ó ) ð ì ð í ð î ð ó , ô



ï (Þ ,  ,  , ì , í , î ,  ) =



1 2 3

exp ù − 12 Ý

ü‹ý



ü



=1



þ

ü,ÿ

ÿ 

=0 =0

þ ý ú

=

1 for = 0,  2 forý ÿ > 0,


þ



= Û 2(2

ý2 2

ý ÿ



1

,ÿ

Þ ) cos(3 þ

ý

3

þ

) cos(4 

ÿ

ì ) cos(3 þ í ) cos(4 ÿ

× sin( 2 =

ÿ

sin( 2 þ




ý þ

1

, î , ó ) ï (Þ ,  ,  , ì , 0, î ,  − ó ) ð ì ð î ð ó ë

1

ð í ð î ð ó ð í ð î ð ó

=

 3 (ì

where 2

 1 =0

=

+ 3 þ2 + 4 2 ) + Ü −





2 1 2 4Ý .

, 4

=





3



)

î ) sin ù

ý 


þ

ÿ 

,

,

© 2002 by Chapman & Hall/CRC Page 437

438

HYPERBOLIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES

2 0 . A rectangular parallelepiped is considered. The following conditions are prescribed:  =  0 (Þ ,  ,  ) at  = 0 (initial condition),    =  1 (Þ ,  ,  ) at  = 0 (initial condition), 

= = = =

       

Solution:

at at at at

Þ =0

=  5 (Þ ,  ) =  6 (Þ ,  ,  )

at at



 

(Þ ,  , ,  ) = 

+

3



2

+@

2

+@

2

+@

2

+@

2

+@

2

1

1

5 0 6 5 0 6 5 0 6

2

where

2

=

3

2

5 0 5 06 5 06

5 0 5 06 5 06  1

5 0 5 06 5 06 2

5 0 5 06 5 06  1

5 0 5 0 6 5 0 6 3





1 2 3

2

exp E − 12 >

S

(2T + 1) , 2U 1

M I

= S

(2V

(; ,  ,  , 7 , 8 , 9 ,  − A )

 3 (7

,9 ,A )

 4 (7

, 9 , A ) : (; ,  ,  , 7 , 2, 9 ,  − A )
0,  

are positive roots of the‹ transcendental equation ‰

, 0 ≤ ¨ ≤ 2’ , 0 ≤ §

^

( , ,  , , v , w , € ) y x

p 2(

ˆ



, v , w )|

^

where the Œ (¥ ) are the Bessel functions and the Ž Œ  ( Ž § ) = 0. 6.4.2-3. Domain: 0 ≤

n 0(

y w

u

˜

(Ž Œ

y v

u

, w , ~ ) x ( , ,  , m , v , w , € − ~ ) y

`

q 0t

u

p 1 (v

p 3( u

2 −  2)[ Œ §

2 2 2 ‘

ˆ

ˆ

ˆ

‡Š

† † =0 ‰ =1 =0Š ‹

=™ −  

2s

^

, v , w ) x ( , ,  , , v , w , € ) y

,v ,w )+ {

˜

u

“

u z

q 0 q 0r q 0

( , ,  , , v , w , € ) = exp ” − 12 { 1

n 1(

q 0t

2s

qo 0 q 0

^

+

u

qo 0 q 0 r q 0 2s

Here, x

n 0(

q 0t

“

≤ . Third boundary value problem. 

A circular cylinder of finite length is considered. The following conditions are prescribed: ›

=® =® =± =± =±

œ ° › œ ž ›

+‘ −‘ +‘

œ Ÿ › œ Ÿ ›

1

›

2

›

3

›

0 (

,¨ , ) ,¨ , ) (¨ ,  , ¯ ) 2 ( , ¨ , ¯ ) 3 ( , ¨ , ¯ )

at at at at at

1 (

=0 =0 =§ =0 “ =

¯ ¯   

(initial condition), (initial condition), (boundary condition), (boundary condition), (boundary condition).

›

¶ º in ¶ º ¶ ¶ º ¶ where ¶ Paragraph The solution ( , ¨ ,  , ¯ ) is determined by the formula 6.4.2-2 ˜

²

(

1

,¨ , ,¥ ,© ,³ ,¯ ) = ’

´

´

µ¶

exp ” − 21 ¯ «

´

µ

º ¶

µ¸

=0 · =1

=1

(

2·

2· ¹

§

+

2

( »

¶

)» ( 2 − ¾ 2 )[ » ( ¸ ¸

2 ½ 1§

·

¼

¸

× cos[¾ ( ¨ − © )] ¿ ¶

Here, the

= ¸

¸ ¹

¿

Á

¿ Á

·

) ¥

·

( À ) = cos(Æ À

Å

1 for 2 for

)+ Æ

¸ ½ 2

¾ ¾

sin(Æ

= 0, > 0, ¸

¶

À

),

¸

= ·

Ã

º ¶

¸ Á ¿

Á

2

=

¬

2 2·

)]2 §

( ³ ) sin ¯ 2 ¦

¿

(¥ ) are the¶ Bessel functions, »

(À )

º ¶

¶ ¶

¦ Ã

¸ ¸·

à ·

Ä

.

¸ ¸

¸ Æ ¸2 ½ 3 Æ 2 Æ 2 2

+

2Æ 2 ¬

+½ +½

2 2 2 3

+

+Ç − ¸ ½ 2 Æ 2

2

1 2 4« ,

+

È

2 É

1+

2¸ Ê ½ 2 Æ 2

;

© 2002 by Chapman & Hall/CRC Page 440




2

6.4. TELEGRAPH EQUATION 

2




2

+   = 

3

441

−   + ( ,  , , )

the    and   are positive roots of the transcendental equations tan(  )

    (  ) +  1   (  ) = 0,





= 

2+ 3 2−  2 3

.

6.4.2-4. Domain: 0 ≤  ≤  , 0 ≤  ≤ 2 , 0 ≤  ≤  . Mixed boundary value problems. 1  . A circular cylinder of finite length is considered. The following conditions are prescribed: 

=  0 ( ,  ,  ) at

 =0

(initial condition),

=  1 ( ,  ,  ) at = " 1 (  ,  ,  ) at #  = " 2 ( ,  ,  ) at

 =0  =0

(initial condition), (boundary condition), (boundary condition),

= " 3 ( ,  ,  )

 =

(boundary condition).

! 



# 

 =

at

Solution: $

 ( ,  ,  ,  ) =  $

+

%$ 0 0 2& $

%$

$

2&

'

'

 0 ( , ) , * ) + ( ,  ,  , , ) , * ,  ) , , ) , * ( ( ( (

0

 1 ( , ) , * ) + .  0 ( , ) , * )/0+ ( ,  ,  , , ) , * ,  ) , , ) , * ( ( ( ( $ $ % $ 2& − 1 2 " 1 () , * , 2 ) 3 + ( ,  ,  , , ) , * ,  − 2 )4 5 , ) , * , 2 ( =' ! 0 0 0 ( $ $ 2& $ ' 2 " 2 ( , ) , 2 ) + ( ,  ,  , , ) , 0,  − 2 ) , , ) , 2 −1 0! 0 0 ( ( ( ( $ $ 2& $ ' 0

!

0

( -

0

2

+1

! $

+

0

$

0

%$ 0

0

2&

$

0

0

'

(

" 3 ( , ) , 2 ) + ( ,  ,  , , ) ,  ,  − 2 ) , , ) , 2 ( ( (

( 6

0

( , ) , * , 2 ) + ( ,  ,  , , ) , * ,  − 2 ) , (

(

Here,

(

, ) , * , 2 .


 > )]2 ? @  

× cos[A (  − ) )] cos B
0,

>

where the   ( ) are the Bessel functions (the prime denotes the derivative with respect to the argument) and ( the    are positive roots of the transcendental equation   (  ) = 0. 2  . A circular cylinder of finite length is considered. The following conditions are prescribed: 

= = I  ="  ="  =" !

0 ( 1 ( 1 ( 2 ( 3 (

, , , , ,

, ) , ) , ) , ) , )

at at at at at

 =0  =0  =  =0  =

(initial condition), (initial condition), (boundary condition), (boundary condition), (boundary condition).

© 2002 by Chapman & Hall/CRC Page 441

442

HYPERBOLIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES Solution: %$ $

 ( ,  ,  ,  ) =  $

0

%$

+

0

0

−1

$

$

+

$

%$

! $

'

0

$

(

" 2( , ) , 2 ) 3 ( *

+ ( ,  ,  , , ) , * ,  − 2 )4 J (

(

" 3( , ) , 2 ) 3 ( *

+ ( ,  ,  , , ) , * ,  − 2 )4 J , , ) , 2 ( =% (

0

$

2&

'

0

0

%$

' $

2&

0

 1 ( , ) , * ) + .  0 ( , ) , * )/ + (  ,  ,  , , ) , * ,  ) , , ) , * ( ( ( ( 2& " 1 () , * , 2 ) + ( ,  ,  ,  , ) , * ,  − 2 ) , ) , * , 2

( -

0

0

$

0

$

2&

0

!

0

0

 0 ( , ) , * ) + ( ,  ,  , , ) , * ,  ) , , ) , * ( ( ( (

0

'

0

! $

2

2

!

$

+ 1 2 +1

$

2&

0

0

' $

2&

( , ) , * , 2 ) + ( ,  ,  , , ) , * ,  − 2 ) ,

( 6

0

(

(

(

=0

+ ( ,  ,  , , ) , * ,  ) =   (

2

exp 7 − 21 . 98 :

 



< 

=

1

2 2 2

  

2

@  




1)

where the   ( ) and L  ( ) are the Bessel functions, the    are positive roots of the transcendental equation -

    ( 

1)

−  1   ( 

1 )/

-

 L   ( 

2)

+  2 L  ( 

2 )/

=  L   (  -

1)

−  1 L  ( 

and the @  are positive roots of the transcendental equation 6.4.2-8. Domain: 

≤ ≤

1

tan( @  ) @

1 )/

= @



-

    ( 

3+ 4 2−  3 4

2)

+  2   ( 

2 )/

,

.

0 ≤  ≤ 2 , 0 ≤  ≤  . Mixed boundary value problems.

2,

1  . A hollow circular cylinder of finite length is considered. The following conditions are prescribed:  !   #  # 

Solution:

= = =" =" =" ="

1 ( 2 ( 3 ( 4 (

%$

2&

$

 ( ,  ,  ,  ) =  $

+

0

%$

0

0

+ 1 2

1

2

+1

$ !

$

+

0!

0

$

0

$ %$

0

$

2&

2& 0

2

 =0  =0  =

1

 =

2

 =0  =

(initial condition), (initial condition), (boundary condition), (boundary condition), (boundary condition), (boundary condition).

 0 ( , ) , * ) + ( ,  ,  , , ) , * ,  ) , , ) , * ( ( ( (

1

 1 ( , ) , * ) + .  0 ( , ) , * )/0+ ( ,  ,  , , ) , * ,  ) , , ) , * ( ( ( ( ( % $ 2& " 1 () , * , 2 ) 3 + ( ,  ,  , , ) , * ,  − 2 )4 5 , ) , * , 2 0 0 ( =' 1 ( % $ 2& " 2 () , * , 2 ) 3 + ( ,  ,  , , ) , * ,  − 2 )4 5 , ) , * , 2 0 0 ( =' 2 ( $ ' $

0

'

-

1

2& 0

$

at at at at at at

2

0

$

, ) , ) , ) , ) , ) , )

'

0!

!

2

$

, , , , , ,

'

0

$

2

$

1 (

!'

$

− 1 2 −1

2&

0 (

2

' '

$ $

'

1 2

'

1

'

" 3 ( , ) , 2 ) + ( ,  ,  , , ) , 0,  − 2 ) , , ) , 2 ( ( ( ( " 4 ( , ) , 2 ) + ( ,  ,  , , ) ,  ,  − 2 ) , , ) , 2 ( ( ( ( 2

1

6

( , ) , * , 2 ) + ( ,  ,  , , ) , * ,  − 2 ) , (

(

(

(

, ) , * , 2 .

© 2002 by Chapman & Hall/CRC Page 445

446

HYPERBOLIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES
< sin 7  ? @   8 C  C * × cos[A (  − ) )] cos B cos B , ? @    D  D < 

:

2 2 2 1 for A = 0, @   = 1 2  2  + 1   + G − 14 . 2 , 2 for A ≠ 0, 2   ( ) =   (    1 ) L  (    ) − L  (    1 )   (    ),

 = H > K

where the   ( ) and L  ( ) are the Bessel functions, and the    are positive roots of the transcendental equation   (  1 ) L  (  2 ) − L  (  1 )   (  2 ) = 0. 2  . A hollow circular cylinder of finite length is considered. The following conditions are prescribed: 

=  0 ( ,  ,  ) at =  1 ( ,  ,  ) at

!  I  I 

=" ="  ="  ="

Solution:  $

+

%$

2

−1 

1

2

!

$

2

2

Here,

$ !

$

+

0!

0

$

0

0

$

$

0 2&

1

 =

2

(boundary condition), (boundary condition), (boundary condition), (boundary condition).

 =0  =

 0 ( , ) , * ) + ( ,  ,  , , ) , * ,  ) , , ) , * ( ( ( (

1

 1 ( , ) , * ) + .  0 ( , ) , * )/ + (  ,  ,  , , ) , * ,  ) , , ) , * ( ( ( ( ( % $ 2& " 1 () , * , 2 ) + ( ,  ,  ,  1 , ) , * ,  − 2 ) , ) , * , 2 0

%$ ' '

$

2

'

$ '

2&

" 2 () , * , 2 ) + ( ,  ,  , 

0

1 2

1

'

0

2

 =

-

0

2&

at at at at

(initial condition), (initial condition),

 =0

2

0

$

2&

%$ 0

$

0

$

'

1

0!

$

+ 1 2

−1

$

'

'

!'

0

, ) , ) , ) , )

$

2&

0 0 2& $

0

+1

, 2 ( ,  3 ( ,  4 ( ,  %$

$

 ( ,  ,  ,  ) =

1 (

 =0

'

" 3( , ) , 2 ) 3 ( *

+ ( ,  ,  , , ) , * ,  − 2 )4 J (

" 4( , ) , 2 ) 3 ( *

+ ( ,  ,  , , ) , * ,  − 2 )4 J , , ) , 2 ( ( =% (

2

( , ) , * , 2 ) + ( ,  ,  , , ) , * ,  − 2 ) , 6

1

(

(

(

=0 (

, (

, ) , 2

, ) , * , 2 . (




= 

2 2 2

1

  

  ( ) =    (    K

2 1 )L







+   = 

2

2

< @  

+ G − 14 . 2 ,

447

−   + ( ,  , , )

3

= 1 2  2  +

 (    ) − L   (   

1)

2 2 2

1

 

+ G − 14 . 2 ,

2



 (    );

the   ( ) and L  ( ) are the Bessel functions, and the    are positive roots of the transcendental equation    (  1 ) L   (  2 ) − L   (  1 )    (  2 ) = 0. 6.4.2-9. Domain: 0 ≤  ≤  , 0 ≤  ≤ 

0 ≤  ≤  . First boundary value problem.

0,

A cylindrical sector of finite thickness is considered. The following conditions are prescribed: 

= = =" =" =" =" ="

!     

Solution:

0 (

 $

+

%$

0

+1

2

−1

2

+1

2

−1

2

$

+

$ $ $ $

Here, + ( ,  ,  , , ) , * ,  ) = M 2  E  (

O

$

$

0

$

0!

0

$

0

%$ 0

0!

0

8 −N

$

%$ 0

!EO

2 0

=0  = 0  =0  = 

' 0

'

U 0$

0

' 0

3

(

, ) , * , 2 + ( ,  ,  , , ) , * ,  − 2 )4 5 ( ='

)

+ ( ,  ,  , , ) , * ,  − 2 )4 V ( )

+ ( ,  ,  , , ) , * ,  − 2 )4 V ( =U

3

(

1

" 3( , * , 2 ) (

0

(

1

" 2( , * , 2 ) ( '

0

;

 =

" 1 () , * , 2 ) 3

0

'

U 0$

0

 =0

(initial condition), (initial condition), (boundary condition), (boundary condition), (boundary condition), (boundary condition), (boundary condition).

 1 ( , ) , * ) + .  0 ( , ) , * )/0+ ( ,  ,  , , ) , * ,  ) , , ) , * ( ( ( ( ( %$ U 0

0

U 0$

 =0

-

0

%$

at at at at at at at

 0 ( , ) , * ) + ( ,  ,  , , ) , * ,  ) , , ) , * ( ( ( (

0

'

0

$

0

0!

! $

!

!

'

0

U 0$

0

− 1 2

U 0$

1 ( 2 ( 3 ( 4 ( 5 (

%$ 0

, ) , ) , ) , ) , ) , ) , )

1 (

$

 ( ,  ,  ,  ) =

, , , , , , ,

=0

,

, * , 2 (

, (

0

, * , 2

" 4( , ) , 2 ) 3 ( *

+ ( ,  ,  , , ) , * ,  − 2 )4 J (

" 5( , ) , 2 ) 3 ( *

+ ( ,  ,  , , ) , * ,  − 2 )4 J , , ) , 2 ( ( =% (

6

( , ) , * , 2 ) + ( ,  ,  , , ) , * ,  − 2 ) , (

(

(

(

=0 (

, (

, ) , 2

, ) , * , 2 .

O U 0 (    )   & U 0 (   ) A   A  ) : : : sin B O ( sin B 2 [    & U 0 (    )]  0 D  0 D  =1  =1 =1 sin 7E F 1 2  2  + 1 2  2  2  −2 + G − . 2T 4 8 C  C * × sin B sin B ,  D  D F 1 2  2  + 1 2  2  2  −2 + G − . 2T 4 ;

;
: 6

= arctan

 6

9

,

6

=  +

6.6.3-5. Domain: $ = {0 ≤
≤ ;  = 1,  ,  }. Mixed boundary value problem. The following conditions are prescribed: %

= & 0 (x) = & 1 (x)

' ( % %

at at

= + (x,  ) at ' ) * % = , (x,  ) at Solution: (x,  ) = -

'0

+ '

Here, (x, y,  ) =

(initial condition), (initial condition),

 =0
=0

(boundary conditions), (boundary conditions).


=

( %



 =0

2



1 2  





+

2

+

2

 

 - . / (y, 0 ) (x, y,  − 0 )  y  0  - . & 0 (y) (x, y,  )  (   - 1 * + (y, 0 ( ) @ 0 =1 (   - 1 * , (y, 0 ( ) @ =1 0



 1 =1  2 =1



 



y + - . & 1 (y) (x, y,  )  y '

)'

 

(x, y,  − 0 )A *



) (x, y,  − 0 )A *

1 + 1) , 2 1

# (2 "  1 =

)

 0

( ) *    0 .



sin(   1
1 ) sin(   2
2 )  sin(  
 ) =1

× sin(   1  1 ) sin(   2  2 )  sin(     ) where

=4

=0

  (

2 + 1) , 2 2

# (2 "  2 =

 ,

sin     2 (  2 1 +  +  2  ) + ! 

 2 (  2 1 +  +  2  ) +

,

# (2  + 1) "   = . 2 

© 2002 by Chapman & Hall/CRC Page 462

6.6. EQUATIONS WITH

463

SPACE VARIABLES

6.6.4. Equations Containing the First Time Derivative 1. B

2C 2

+E B

C

2G

=F

H C

–I C

+ J (K

1,

L L L , K H , ). D

B D B D telegraph equation with  space variables. Nonhomogeneous

1 M . The substitution % = exp  − 21 N  !O leads to the equation '

2

'

O

=  2 P  O −  − 14 N

2



2

!O + exp 

1 2N

 ! / (


1,

 ,
 ,  ),

which is considered in Subsection 6.6.3. 2 M . Domain: $ = {0 ≤
≤ ;  = 1,  ,  }. First boundary value problem. The following conditions are prescribed: %

= & 0 (x)

at

 =0

(initial condition),

= & 1 (x) at = + (x,  ) at % = , (x,  ) at

 =0

(initial condition), (boundary conditions), (boundary conditions).

' (% %

Solution:


=0
=

( % (x,  ) = -

'0

 - . / (y, 0 ) (x, y,  − 0 )  y  0

  - . & 0 (y) (x, y,  )  y + - . & 1 (y) + N & 0 (y)3 (x, y,  )  y  2 ( '    + 2 - 1 * + (y, 0 ) ' (x, y,  − 0 )A *   ( )  0 

=0 ( ) @ =1 0 (  '  − 2 - 1 * , (y, 0 ) ' (x, y,  − 0 )A * *   ( )  0 , 

=4 ( ) @ 0 =1

+ '

where x = {
1 ,  ,
 }, y = { 1 ,  ,   },  y =   ( )



Green’s function: (S 

(x, y,  ) =

2

 Q −R

2

1 2  



= {0 ≤  ≤ 

 



 1 =1  2 =1





for 

1

 

2

   ,   (

)

= 

1

 

−1

 

+1

   ,

= 1,  ,  − 1,  + 1,  ,  }.





sin(   1
1 ) sin(   2
2 )  sin(   
 ) =1

× sin(   1  1 ) sin(   2  2 )  sin(      )

sin     2 (  2 1 +  +  2  ) + − N 2T 4 ! ,   2 (  2 1 +  +  2 ) + − N 2T 4

where

 # 1# 2#  1 = " ,  2 = " ,  ,   = " .

1

2

 3 M . Domain: $ = {0 ≤
≤ ;  = 1,  ,  }. Second boundary value problem.

The following conditions are prescribed: %

=& =& =+ ' ) * % =, ' ( % ' ) * %

0 (x)

at at 1 (x) (x,  ) at (x,  ) at

 =0  =0
=0
=

(initial condition), (initial condition), (boundary conditions), (boundary conditions).

© 2002 by Chapman & Hall/CRC Page 463

464

HYPERBOLIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES Solution: ( %

(x,  ) = -

 - . / (y, 0 ) (x, y,  − 0 )  y  0

0



+ - . & 0 (y) (x, y,  )  y + - .  

2

−



2

+

(





0

=1



- 1 * (

0

=1



& 1 (y) + N & 0 (y)3

 * + (y, 0 ) (x, y,  − 0 )3

2

( )

2

=0

(x, y,  )  y

  ( )  0

 * * , (y, 0 ) (x, y,  − 0 )3 =4   ( )  0 . 2

- 1 *

( )

Here, 

(x, y,  ) =

Q −R

(S



2

1 2  



 



 1 =0  2 =0

 



 1  2    cos(   1
=0 5 5 5

× cos(   1  1 ) cos(   2  2 )  cos(      )

 2


1 ) cos( 

2)

 cos(   
 )

sin     2 (  2 1 +  +  2  ) + − N 2T 4 ! ,   2 (  2 1 +  +  2 ) + − N 2T 4

where 1#  1 = " ,

1

2#  2 = " ,

2

  = "

 ,

 # ;



6 5

= 7

1 for = 0, " 2 for ≠ 0, 

= 1, 2,  ,  .

"

4 M . Domain: $ = {0 ≤
≤ ;  = 1,  ,  }. Third boundary value problem. The following conditions are prescribed: %

=& =& ' ) * % % − =+ ' ) * % +8 % =, ' (%

at (x) at 1 (x,  ) at (x,  ) at

 =0

0 (x)

 =0
=0
=

(initial condition), (initial condition), (boundary conditions), (boundary conditions).

The solution % (x,  ) is given by the formula in Item 3 M with 

 Q (x, y,  ) = 2 −R

(S

2

 





 1 =1  2 =1





sin     2 (  2 1 +  2 2 +  +  2  ) + − N 2T 4 ! 9  =1  1  2     2 (  2 1 +  2 2 +  +  2 ) + − N 2T 4 × sin(   1
1 + :  1 ) sin(   2
2 + :  2 )  sin(  
 + :  ) × sin(   1  1 + :  1 ) sin(   2  2 + :  2 )  sin(     + :  ). 

9

9

Here, : 6

= arctan

 6



9

,

6

( ; 8 +  2 6 )( < + 8 ) , ( 2 +  26 )( 8 2 +  26 )

=  +



= 1, 2,  ,  ;

the  6 are positive roots of the transcendental equation 1 ; + 8 =

 −

; 8 

>

= cot( ?  ), 

= 1, 2,  ,  .

© 2002 by Chapman & Hall/CRC Page 464

6.6. EQUATIONS WITH

465

SPACE VARIABLES

5 M . Domain: $ = {0 ≤
≤ ;  = 1,  ,  }. Mixed boundary value problem. The following conditions are prescribed: %

= & 0 (x)

at

 =0

(initial condition),

= & 1 (x) % = + (x,  )

at at

 =0
=0

(initial condition), (boundary conditions),

= , (x,  ) at


=

(boundary conditions).

' ( % ' ) * %

Solution: ( %

(x,  ) = -

'0

+ '

where 

(x, y,  ) =

2

 Q −R

(S



2

1 2  

 - . / (y, 0 ) (x, y,  − 0 )  y  0   - . & 0 (y) (x, y,  )  y + - . & 1 (y) + N & 0 (y)3 (x, y,  )  y 2 ( '    - 1 * + (y, 0 ) ' (x, y,  − 0 )V *   ( )  0  =0 ( ) U 0 =1 (   - 1 * , (y, 0 ) (x, y,  − 0 )V * *   ( )  0 , =4 ( ) U =1 0



+

2

+

2

 



 1 =1  2 =1







sin(   1
1 ) sin(   2
2 )  sin(  
 )



=1

× sin(   1  1 ) sin(   2  2 )  sin(     ) 1 + 1) , 2 1

# (2 "  1 =

2.

2C

B

B D

2

+E B

C

2G

=F

H C

+

XW

2 + 1) , 2 2

# (2 "  2 = W H I

B

=1

B D

B

The transformation % (


C W K

sin     2 (  2 1 +  +  2  ) + − N 2T 4 ! 

 ,

(  2 1 +  +  2 ) + − N 2T 4 # (2  + 1) "    = . 2  2



,

+Y C .

 1 1  ,      ,
 ,  ) = (
,      ,
 , 0 ) exp −  − O 1 1 = 2N 2 2


=1

>

,

0 = 

leads to the equation ' '

2

0

O 2

=P  O + O ,

 = 

8 2

+ N 4

2 2

−

 1  4 4

2 =1

,

which is considered in Subsection 6.6.3.

Z\[

3.

Reference: R. Courant and D. Hilbert (1989).

B

2C 2

+ ]

–1 B

C

=G H C .

B D D B D Darboux equation. Cauchy problem. Initial conditions are prescribed: % ' ( %

= & (x) at =0 at

 = 0,  = 0.

© 2002 by Chapman & Hall/CRC Page 465

466

HYPERBOLIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES Solution:

1_

% (x,  ) = ^ _ 

where ^ _ e

_

_ S

−1

-

|x−y|=

(

& (y) ` a b ,

2 ^ _ = c 2# , (d T 2)

is the area of the surface of an d -dimensional sphere of radius e , and ` a b is the area element of this surface (i.e., the solution % is the average of the function & over the sphere for radius e with center at x).

Z\[

−1

Reference: R. Courant and D. Hilbert (1989).

© 2002 by Chapman & Hall/CRC Page 466

Chapter 7

Elliptic Equations with Two Space Variables =0 2g The Laplace equation is often encountered in heat and mass transfer theory, fluid mechanics, elasticity, electrostatics, and other areas of mechanics and physics. For example, in heat and mass transfer theory, this equation describes steady-state temperature distribution in the absence of heat sources and sinks in the domain under study. A regular solution of the Laplace equation is called a harmonic function. The first boundary value problem for the Laplace equation is often referred to as the Dirichlet problem, and the second boundary value problem as the Neumann problem. Extremum principle: Given a domain h , a harmonic function i in h that is not identically constant in h cannot attain its maximum or minimum value at any interior point of h . 7.1. Laplace Equation

f

7.1.1. Problems in Cartesian Coordinate System The Laplace equation with two space variables in the rectangular Cartesian system of coordinates is written as j j 2 j ki

2

2 i + j l 2 = 0.

7.1.1-1. Particular solutions and a method for their construction. 1 M . Particular solutions:

k l k l i ( , )=m +n +o , k l k l k l 2 2 i ( , )= m ( − )+n , k l k k l k l l 3 2 i ( , )= m ( −3 ) + n (3 2 − 3 ), k l k l m k + nl i ( , )= +o , 2+ 2 k l k l l i ( , ) = exp(p q )( m cos q + n sin q ), k l k k l i ( , ) = ( m cos q + n sin q ) exp(p q ), k l k k l l i ( , ) = ( m sinh q + n cosh q )( o cos q + h sin q ), k l k k l l i ( , ) = ( m cos q + n sin q )( o sinh q + h cosh q ), k l k k l l i ( , ) = m ln r ( − 0 )2 + ( − 0 )2 s + n , k l where m , n , o , h , 0 , 0 , and q are arbitrary constants.

2 M . Fundamental solution:

t k

l

1 1v ( , )= ln , 2u

v

=w

k

2

+

l

2.

© 2002 by Chapman & Hall/CRC Page 467

468

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES k

l

3 x . If i ( , ) is a solution of the Laplace equation, then the functions k

l

i

1

i

2

= m i (p y + o 1 , p y + o 2 ), k l k l = m i ( cos z + sin z , − sin z + cos z ),

i

3

=m i { k

k

2

l

+

l

2

,k

2

+

l

2

| ,

are also solutions everywhere they are defined; m , o 1 , o signs at y in i 1 are taken independently of each other.

2,

z , and y are arbitrary constants. The

4 x k. A method for constructing particular solutions involves the following. Let } ( ~ ) = l fairly general k l k l  ( , ) + €‚ ( , ) be any analytic function of the complex variable ~ = + € ( and  are real k l functions of the real variables and ; € 2 = −1). Then the real and imaginary parts of } both satisfy the two-dimensional Laplace equation,ƒ ƒ 2

 = 0,

Recall that the Cauchy–Riemann conditions j



j k

j

 = j l ,

j

2

 j l

= 0. j  j =− k

are necessary and sufficient conditions for the function } to be analytic. Thus, by specifying analytic functions } ( ~ ) and taking their real and imaginary parts, one obtains various solutions of the two-dimensional Laplace equation.

„\[

References: M. A. Lavrent’ev and B. V. Shabat (1973), A. G. Sveshnikov and A. N. Tikhonov (1974), A. V. Bitsadze and D. F. Kalinichenko (1985).

7.1.1-2. Specific features of stating boundary value problems for the Laplace equation. 1 x . For outer boundary value problems on the plane, it is (usually) required to set the additional condition that the solution of the Laplace equation must be bounded at infinity. 2 x . The solution of the second boundary value problem is determined up to an arbitrary additive term. 3 x . Let the second boundary value problem in a closed bounded domain h boundary … be characterized by the boundary condition*

with piecewise smooth

j

j †i

= } (r) for r ‡ … ,

where ˆ ‰ is the derivative along the (outward) normal to … . The necessary and sufficient condition ˆ Š of solvability of the problem has the form ‹ Œ } (r)  …

= 0.

Ž ‘ ’ “” •

The same solvability condition occurs for the outer second boundary value problem if the domain is infinite but has a finite boundary.

„\[

Reference: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964).

* More rigorously, – must satisfy the Lyapunov condition [see Babich, Kapilevich, Mikhlin, et al. (1964) and Tikhonov and Samarskii (1990)].

© 2002 by Chapman & Hall/CRC Page 468

7.1. LAPLACE EQUATION —

7.1.1-3. Domain: − ™

469

=0

2˜

< š < ™ , 0 ≤ › < ™ . First boundary value problem.

A half-plane is considered. A boundary condition is prescribed: ‹

Solution: „\[

œ ž

œ (š , › ) = 1 

= } (š ) at › } (Ÿ )  Ÿ

ž

( š − Ÿ )2 + ›

−

2

› = 0. ‹   ¡

= 

1

  ¡ −

2 2

} (š + › tan ¢ )  ¢ .

References: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), H. S. Carslaw and J. C. Jaeger (1984).

7.1.1-4. Domain: − ™

< š < ™ , 0 ≤ › < ™ . Second boundary value problem.

A half-plane is considered. A boundary condition is prescribed: £ ¤ œ ‹ ž = } (š )

Solution: œ

(š , › ) = 

1

where ¥ is an arbitrary constant.

„\[

ž

( š − Ÿ )2 + ›

} (Ÿ ) ln w

−

› = 0.

at

2

 Ÿ +¥ ,

Reference: V. S. Vladimirov (1988).

7.1.1-5. Domain: 0 ≤ š < ™ , 0 ≤ › < ™ . First boundary value problem. A quadrant of the plane is considered. Boundary conditions are prescribed: œ

Solution:

„\[

‹

œ (š , › ) = 4 š › 

= } 1 (› ) at

ž

0

[š

2

œ

š = 0,

= } 2 (š ) at ‹

} 1 (¦ )¦  ¦ 4 + š › + (› − ¦ )2 ][š 2 + (› + ¦ )2 ] 

› = 0.

ž

} 2 (Ÿ )Ÿ  Ÿ . [(š − Ÿ )2 + › 2 ][(š + Ÿ )2 + › 2 ]

0

Reference: V. S. Vladimirov, V. P. Mikhailov, A. A. Vasharin, et al. (1974).

7.1.1-6. Domain: − ™

< š < ™ , 0 ≤ › ≤ § . First boundary value problem.

An infinite strip is considered. Boundary conditions are prescribed: œ

Solution:

„\[

= } 1 (š ) at

› = 0, ‹ ž

 œ (š , › ) = 1 sin { › 2§ § |  1 › + sin { 2§ § | ‹

ž ž

−

ž −

œ

= } 2 (š ) at

} 1 (Ÿ cosh[ (š − Ÿ ) ¨ } 2 (Ÿ  cosh[ (š − Ÿ ) ¨

› =§ .

) Ÿ

§ ] − cos( › ¨ § )

) Ÿ . § ] + cos( › ¨ § )

Reference: H. S. Carslaw and J. C. Jaeger (1984).

7.1.1-7. Domain: − ™

< š < ™ , 0 ≤ › ≤ § . Second boundary value problem.

An infinite strip is considered. Boundary conditions are prescribed: £ ¤ œ

Solution:

= } 1 (š ) at ‹

ž

œ (š , › ) = 1 ‹ ž ž 2 −

› = 0,

£ ¤ œ

= } 2 (š ) at

› =§ .

} 1 (Ÿ ) ln © cosh[ (š − Ÿ ) ¨ § ] − cos( › ¨ § ) ª  Ÿ

ž 1 −  } 2 (Ÿ ) ln © cosh[ (š − Ÿ ) ¨ § ] + cos( › ¨ § ) ª  Ÿ + ¥ , 2 − where ¥ is an arbitrary constant.

© 2002 by Chapman & Hall/CRC Page 469

470

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES

7.1.1-8. Domain: 0 ≤ š < ™ , 0 ≤ › ≤ § . First boundary value problem. A semiinfinite strip is considered. Boundary conditions are prescribed: œ

= } 1 (› ) at

œ

š = 0,

= } 2 (š ) at

Solution: ž œ

(š , › ) =

‹

2 «X¬ §

 š

exp‹ { −ž ­

=1

 ›

| sin { ­ §

§

œ

› = 0,

= } 3 (š ) at

› =§ .

®

|

 ¦

} 1(¦ ) sin { ­

0

 ¦ |

§

 › 1 1 1 − sin { ‹ ž 2( Ÿ )  Ÿ    | 2§ § cosh[ (š + Ÿ ) ¨ § ] − cos( › ¨ § ) ° ± 0 ¯ cosh[ ( š − Ÿ ) ¨ § ] − cos( › ¨ § )  › 1 1 1 + sin ² − 3( Ÿ )  Ÿ .    2§ § ³ 0 ¯ cosh[ (š − Ÿ ) ¨ § ] + cos( › ¨ § ) cosh[ (š + Ÿ ) ¨ § ] + cos( › ¨ § ) ° ±

+

Example. Consider the first boundary value problem for the Laplace equation in a semiinfinite strip with ´ 1 ( µ ) = 1 and

´ 2 (¶ ) = ´ 3 (¶ ) = 0.

Using the general formula and carrying out transformations, we obtain the solution 2 sin( · µ ‘ ¹ º ) . ˜ (¶ , µ ) = · arctan ¸ sinh( · ¶ ‘ ¹ º )»

¼\[

Reference: H. S. Carslaw and J. C. Jaeger (1984).

7.1.1-9. Domain: 0 ≤ š ≤ § , 0 ≤ › ≤ ½ . First boundary value problem. A rectangle is considered. Boundary conditions are prescribed: œ

= 1 (› ) at ± œ = 3 (š ) at ž

œ (š , › ) =

ž

+

sinh ¿ ­

«X¬

sin ² ­ ¥ ¬

¬

where the coefficients ¬

= Ã ¾ É

¼\[

= Ë

¬

2 2

, ¾ Ä Ä

É

Å 0

0

( § − š )À sin ² ­

½

¾ ¬

=1

=1

Ê

  §

,¥

¬

š ³



sinh ¿ ­

, and

›

+ ž ³

1 (Æ

±

3 (Æ

Ã

) sin ² ­ Ç ) sin ² ­ ÌÇ

Æ ½

³

½

³ Ì

= Ã

Æ , È

Á É

Æ

= ½ sinh ² ­ Ç É

Á ¬

=1

«X¬

sin ² ­ Â

=1

 ±



sinh ² ­ 

½

§

š š



sin ² ­ ³

sinh ² ­ ³

›

½  §

› ³

³

,

are expressed as

Á

®

› =½ .

¬

«X¬

( ½ − › )À +

§

¬

½



š =§ ,

±

ž ¬

«X¬

= 2 (› ) at ± = 4 (š ) at œ

› = 0,

±

Solution:

œ

š = 0,

È

Æ ,

³

,

= Ë

2 2É

Ä Ä

Å 0

® ±

±

Æ

2 (Æ

) sin ² ­ Ç

4 (Æ

) sin ² ­ ÌÇ

0 Â É É ½ Ë = Ì sinh ² ­ Ì Ç . ³ É

½

³

Æ , È

Æ ³

È

Æ ,

References: M. M. Smirnov (1975), H. S. Carslaw and J. C. Jaeger (1984).

© 2002 by Chapman & Hall/CRC Page 470

7.1. LAPLACE EQUATION Í

471

=0

2Î

7.1.1-10. Domain: 0 ≤ Ï ≤ Ì , 0 ≤ Ð ≤ ½ . Second boundary value problem. A rectangle is considered. Boundary conditions are prescribed: £ Ñ Ò

= 1 (Ð ) at ± £ ¤ Ò = 3 (Ï ) at (Ï , Ð ) = −

( Ï − Ì )2 + Ì Ï ¾4 Ì Á4 0

0

Õ

−½ Ô × Ö

=1

ÉÕ

−Ì

−

4½

0

( Ï − ½ )2 +

 4½

0

Ð

× Ô

ÕÉ

Ù

É

Ä

Å 0

Ä

2 = Ì

É

Ù Õ

Ì Ë É cos Û ­ Ì Ç Ï Ü cosh Ø ­ Ì Ç ( − Ð )Ú +

2 = Ù ÊÖ

+Ó

1 (Æ

® Þ

Æ

3 ( Æ ) cos Û ­ ÌÇ

Þ

0

Æ

) cos Û ­ Ç Ù

×

= É

Ü Ü Ì

­ Ç

sinh Û ­ Ç Ù

È

ÄÉ

2Ý = Ù

É Ý

Æ ,

É,Ê

É

Ö

É

Å É Ä

2 = Ì

É

,

0

®

cos Û ­ Ì Ç Ï Ü cosh Û ­ Ì Ç Ð Ü ,

ÂË É

=1

,

Æ , È

Ô

cosh Û ­ Ù Ç Ï Ü cos Û ­ Ù Ç Ð Ü É

Ý

=1

É is an arbitrary constant, and the coefficients

É

2

cosh Ø ­ Ù Ç ( Ì − Ï )Ú cos Û ­ Ù Ç Ð Ü + É

Ð =½ .

±

Ê É

Ô =1

where Ó

Ê

2

Ì Ï = ,

= 2 (Ð ) at ± = 4 (Ï ) at

£ ¤ Ò

Ð = 0,

±

Solution: Ò

£ Ñ Ò

Ï = 0,

Þ

, Â É

×

, and Ë É

2 (Æ

) cos Û ­ Ç Ù

4 (Æ

) cos Û ­ ÇÌ

0 Þ Â É Ù Ë = sinh Û ­ Ì Ç Ü . ­ Ç É

Ü ,

are expressed as

Æ

É

Æ

Ü

Æ , È

Ü È

Æ ,

The solvability condition for the problem in ® question has the form (see Paragraph 7.1.1-2, ® Item 3 ß ) Ä

Å

1 (Ð

Þ

0

) Ð + È

Ä

Å

0

Þ

2 (Ð

) Ð − È

Ä

0

3 (Ï

Þ

Ä

) Ï − È

4 (Ï

Þ

0

) Ï = 0. È

Ù 7.1.1-11. Domain: 0 ≤ Ï ≤ Ì , 0 ≤ Ð ≤ . Third boundary value problem.

A rectangle is considered. Boundary conditions are prescribed: £ Ñ Ò

Ò

− à 1 = 1 (Ð ) at £ ¤ Ò Ò Þ − à 3 = 3 (Ï ) at Þ

£ Ñ Ò

Ï = 0,

Ò

+ à 2 = 2 (Ð ) at £ ¤ Ò Ò Þ + à 4 = 4 (Ï ) at

Ð = 0,

Þ

For the solution, see Paragraph 7.2.2-14 with á ≡ 0.

Ì Ï = , Ù Ð = .

Ù 7.1.1-12. Domain: 0 ≤ Ï ≤ Ì , 0 ≤ Ð ≤ . Mixed boundary value problems.

1 ß . A rectangle is considered. Boundary conditions are prescribed: £ Ñ Ò

= (Ð ) at Ò Þ = ã (Ï ) at

Solution: Ò

Ù Õ

(Ï , Ð ) = − Ç

Ô =1

­

Þ× É

Ï = 0,

£ Ñ Ò Ò

Ð = 0,

cosh Ø Ç Ù ­ ( Ì − Ï )Ú sin Û Ç ­ Ù

Ð Ü

Ï =Ì , Ù

= â (Ð ) at = ä (Ï ) at Ù Õ

+ Ç

â Ô =1

­

× É

Ð = .

cosh Û Ç ­ Ù

Ï

Ü sin Û Ç ­ Ù

Ð Ü

É É Éã É Õ Ù Ï ä Ï Ð cos Û Ç ­ Ü sinh Ø Ç ­ ( − Ð ) Ú + Ô cos Û Ç ­ Ë É ® Ì Ë É Ì Ü sinh Û Ç ­Ì Ü ® Ì =1 =1 Ù Ä Ä É É −Ð É É Ð + Ì Ù ã (Ï ) Ï + Ì Ù ä (Ï ) Ï , È È 0 0 Õ

+ Ô

© 2002 by Chapman & Hall/CRC Page 471

472

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES

where Ä

2

= Ù

Þ É É

0

Ä

2 = Ì

ã

Å ®

Æ

(Æ ) sin Û Ç ­ Ù Þ

Æ

ã (Æ ) cos Û Ç Ì­

0

×

Ü

â É

Æ ,

Ü È Ì

= sinh Û Ç ­ Ù

É Ë

Ü ,

Å ® â (Æ ) sin Û Ç ­ Ù

0

Ä

2 = Ì

ä

É

å\[

= Ù

Æ , È

Ä

2

ä (Æ ) cos Û Ç Ì­ Ù

0

= sinh Û Ç Ì ­ É

Æ Ü Æ

Æ , È

Ü

Æ , È

Ü ,

Reference: M. M. Smirnov (1975).

2 ß . A rectangle is considered. Boundary conditions are prescribed: Ò

= (Ð ) Ò

at

Ï = 0,

= ã (Ï ) at

Ð = 0,

Þ

£ Ñ Ò

Ì Ï = , Ù

= â (Ð ) at

£ ¤ Ò

= ä (Ï )

Ð = ,

at

where (0) = ã (0). Þ Solution: Ò

Õ

(Ï , Ð ) = Ô =0

Þ É cosh

ÕÉ

+ Ô =0

where

Þ É

É

2í

Þ

0

æ

0

×

Ü sin Û

Ð É

Ù

Ù Ü cosh Û Ë

(2ê + 1)

(è ) sin Ø é

Ù

î ã (è ) sin Ø é

(2ê í + 1)

−Ð Ù

Ü +

Ô

â ×

É cosh

=0

Ù ÕÉ Ô

É

2

è Ú ë è ,

â ì = Ù æ

è Ú ë è , í

äì =

2í

ï ì = é

É

É

0

Ï É Ï

sin Û Ë

É Ë cosh

É

×

sinh Û ×

ä

Ë

=0

(2ê + 1) Ù , 2

ì = é

Õ

Ì Ì Ü +

É

× å\[

Ì

É

2 = Ù æ ç

ã ì =

Ï

sin Û Ë

coshÉ Ë

É

−Ï

É É

ã

Ì ×

cosh Û ×

Ì Ü sin Û

ç â (è ) sin Ø é

î ä (è ) sin Ø é 0 Ù (2ê í + 1) æ

Ð É Ð

Ù Ü sinh Û Ë

É É

×

(2ê + 1) Ù

Ì Ü

Ù Ü ,

É

è Ú ë è ,

(2ê í + 1)

è Ú ë è ,

.

2

Reference: M. M. Smirnov (1975).

7.1.2. Problems in Polar Coordinate System The two-dimensional Laplace equation in the polar coordinate system is written as £

1ð ð £

Û

ð £ Ò ð £ Ü

7.1.2-1. Particular solutions: ð

Ò

( )= ð

Ò

where õ

= 1, 2, ööö ; Ö

ln + ð ó

ñ

Ö

, ô , and Ý

ð1 2

£

2

£ ñ

Ò ð 2

= 0,

=ò Ï

2

+ Ð 2.

ð

( , )Ö = Û

,

+

, Ý + ð ó Ý

Ü (ô

cos õ

ñ

+

sin õ

ñ

),

Â

are arbitrary constants.

Â

© 2002 by Chapman & Hall/CRC Page 472

7.1. LAPLACE EQUATION ÷

ð

473

=0

2ø

ð

7.1.2-2. Domain: 0 ≤

≤ù

≤

or ù

< ú . First boundary value problem.

The condition Ò

ð ñ

= ( ) at

=ù

Þ

ñ

is set at the boundary of the circle; ð ( ) is a given function. Þ

1 ß . Solution of the inner problem ( ≤ ù ): ð Ò

2û

1 æ 2

ñ

( , )=

Þ

0

é

− 2 ñ cos( − ü ) + ù

ð ù ð

(ü )

− 2ù

2

ð

2

2

ë ü .

This formula is conventionally referred to as the Poisson integral. Solution of the outer problem in series form: í

ð Ò

ñ

0

( , )=

í é Ù

1

ì = é

+ ý

2

æ æ

0

Þ

2û

Þ

0

Û

ì

2û

1

ì =

ð

Õ

ì í

ù

=1

ð ñ

( , )=

(ü ) sin(ê ü ) ë ü ,

ê = 1, 2, 3, ööö

2û

ð

ð

(ü ) Þ

0

é

),

ê = 0, 1, 2, ööö ,

ð

1 æ 2

ñ

Ù

+ ì sin ê

(ü ) cos(ê ü ) ë ü ,

2 ß . Bounded solution of the outer problem ( ≥ ù ): Ò

ñ

( ì cos ê Ü

2

−ù 2 ñ cos( − ü ) + ù

ð

− 2ù

2

2

ë ü .

Bounded solution of the outer problem in series form: Ò í 0,

ñ

( , )= í

where the coefficients

í

ð

Õ

0

+ ý

2

ì

ì í

ùð Û

( ì cos ê Ü

ñ

Ù

+ ì sin ê

ñ

),

=1

Ù ì , and ì are defined by the same relations as in the inner problem.

þ In hydrodynamics and other applications,ð outer problems are sometimes encountered in which one has to consider unbounded solutions for ú . Example. The potential flow of an ideal (inviscid) incompressible fluid about a circular cylinder of radius ÿ with a constant incident velocity at infinity is characterized by the following boundary conditions for the stream function:

ø = 0 at
= ÿ , ø

Solution:



ø (
,  ) = å\[


sin   ÿ


−

2

as




 .

sin  .




References: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), A. N. Tikhonov and A. A. Samarskii (1990).

ð

7.1.2-3. Domain: 0 ≤ The condition

ð

≤ù

or ù

≤

< ú . Second boundary value problem. £ Ò

ð ñ

= ( ) at Þ

ñ

=ù

is set at the boundary of the circle. The function ( ) must satisfy the solvability condition æ

Þ

2û 0

Þ

ñ

( )ë

ñ

= 0.

© 2002 by Chapman & Hall/CRC Page 473

474

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES ð

1 ß . Solution of the inner problem ( ≤ ù ): Ò

ð ñ

ù

( , )=

2

2û

æ

ð

2

(ü ) ln Þ

0

é

ð

ñ

− 2ù

2

cos( − ü ) + ù 2

ù

ë ü +ô ,

where ô is an arbitrary constant; this formula is known as the Dini integral. Series solution of the inner problem: ð Ò

ð



ñ

í

1

ì = é

æ

ì

2û

ê

=1

ù



 ( , )=−

ù

2

1

 ì =

2û

æ

0

ñ

)+ô ,

(ü ) sin(ê ü ) ë ü ,

ð

2 . Solution of the outer problem ( ≥ ù ): ñ

+  0)

Parabolic coordinates 9 ,:

= 7 9 : , < = 12 7 (: 2 − 9 2) −; = < 9 < = , 0 ≤ : < =

Elliptic coordinates A ,B

= 7 cosh cos B , < = 7 sinh sin B A ; 0 ≤ < = , 0 ≤ B < 2

A

Laplace operator, 8

2A

≤E ≤

1

2.

A

?

1

(cosh D − cos C )2 2

7

2

+ sin2 B ) > ?

7 2(sinh

Bipolar coordinates = , < = C ,D cosh D − cos C cosh D − cos C ; C  0 ≤ < 2 , −= < D < = 7.1.2-7. Domain: 

?

1

7 sin C



2 2 1 + 9 2 + : 2) > ? 2 ? : 2@

7 2(9

A

7 sinh D

2

? 2

2

? B ?

2

> ? C

+

2

+

?

@

2

2

? D

2

@

?

Mixed boundary value problem.

An annular domain is considered. Boundary conditions are prescribed: *

= F 1 ( G ) at

E =



1,

= F 2 ( G ) at

E =

2.

?

Solution:  (E , G ) =



1 (2) 1 (1) 2 0 + 2 0 

ln

1

H

E 

+ 

2

E



H



1

( "  cos  G + #  sin  G ) +  

=1

=1

( $  cos  G + %  sin  G ). E 

Here, the coefficients "  , #  , $  , and %  are expressed as "  = $  =

  

1



+  2  ( 2 +  +1



(2)

2







  

2

1



1

+1 (1)



 2

)

1 −1 (2)

 (

 2 2



#  =

, 

−  2 (1)   , +  12  )

 



(2)

+  2  ( 2 +  2

I 

%  =



1



1

+1



+1 (1) 2 1

I 

)

  

2

, 

1

−1 (2)

 (

I  2

2



−  2 I (1)  , +  12  )

where the constants ( & ) and I ( & ) (' = 1, 2) are defined by the same formulas as in the first boundary  value problem.

J)(

Reference: M. M. Smirnov (1975).

7.1.3. Other Coordinate Systems. Conformal Mappings Method 7.1.3-1. Parabolic, elliptic, and bipolar coordinate systems. In a number of applications, it is convenient to solve the Laplace equation in other orthogonal systems of coordinates. Some of those commonly encountered are displayed in Table 21. In all the coordinate systems presented, the Laplace equation 8 2  = 0 is reduced to the equation considered in Paragraph 7.1.1-1 in detail (particular solutions and solutions to boundary value problems are given there). The orthogonal transformations presented in Table 21 can be written in the language of complex variables as follows: + 'K< = − 21 'L7 (9 + 'K: )2 ; + 'K< = 7 cosh(A + K' B ) ; + 'K< = 'L7 cot M ;

1 C 2(

+ 'KD )N

(parabolic coordinates), (elliptic coordinates), (bipolar coordinates).

© 2002 by Chapman & Hall/CRC Page 476

7.1. LAPLACE EQUATION O

477

=0

2P

The real parts, as well as the imaginary parts, in both sides of these relations must be equated to each other (' 2 = −1). Example. Plane hydrodynamic problems of potential flows of ideal (inviscid) incompressible fluid are reduced to the Laplace equation for the stream function. In particular, the motion of an elliptic cylinder with semiaxes Q and R at a velocity in the direction parallel to the major semiaxis ( Q > R ) in ideal fluid is described by the stream function

Q +R

P (S , T ) = − R U

Q −R V

where S and T are the elliptic coordinates.

J)\

1 W 2 XZY

sin T ,

[

2

2

=Q

− R 2,

References: G. Lamb (1945), J. Happel and H. Brenner (1965), G. Korn and T. Korn (1968).

7.1.3-2. Domain of arbitrary shape. Method of conformal mappings. 1 . Let ] = ] ( ^ ) be an analytic function that defines a conformal mapping from the complex plane ^ = _ + `Ka into a complex plane ] = b + `Kc , where b = b (_ , a ) and c = c (_ , a ) are new independent variables. With reference to the fact that the real and imaginary parts of an analytic function satisfy the Cauchy–Riemann conditions, we have b = c and b = − c , and hence 2f

? _

2

2f

+

? a

? d 2

? e

= gh] i ( ^ ) g 2

2f

> ? b

? e

+

2

2f

? c

2

? d @ .

Therefore, the Laplace equation in the under a conformal mapping into the ? ? _ a -plane transforms ? ? Laplace equation in the b c -plane. 2 j . Any simply connected domain k in the _ a -plane with a piecewise smooth boundary can be mapped, with appropriate conformal mappings, onto the upper half-plane or into a unit circle in the b c -plane. Consequently, a first and a second boundary value problem for the Laplace equation in k can be reduced, respectively, to a first and a second boundary value problem for the upper half-space or a circle; such problems are considered in Subsections 7.1.1 and 7.1.2. Subsection 7.2.4 presents conformal mappings of some domains onto the upper half-plane or a unit circle. Moreover, examples of solving specific boundary value problems for the Poisson equation by the conformal mappings method are given there; the Green’s functions for a semicircle and a quadrant of a circle are obtained. A large number of conformal mappings of various domains can be found, for example, in the references cited below. J)\

References: V. I. Lavrik and V. N. Savenkov (1970), M. A. Lavrent’ev and B. V. Shabat (1973), V. I. Ivanov and M. K. Trubetskov (1994).

7.1.3-3. Reduction of the two-dimensional Neumann problem to the Dirichlet problem. Let the position of any point (_ l , a l ) located on the boundary m of a domain k be specified by a parameter n , so that _ l = _ l ( n ) and a l = a l ( n ). Then a function of two variables, F (_ , a ), is determined on m by the parameter n as well, F (_ , a ) gpo = F (_ l ( n ), a l ( n )) = F l ( n ). The solution of the two-dimensional Neumann problem for the Laplace equation q 2 f = 0 in k with the boundary condition of the second kind f

= F l ( n ) for r s m

?

can be expressed in terms of the solution ? r of the two-dimensional Dirichlet problem for the Laplace equation q 2 b = 0 in k with the boundary condition of the first kind b = t l ( n ) for r s m , where t l ( n ) = u

v l ( n ) w n , as follows: f (_ , a ) = u d 0

b x a

(y , a 0 ) w y − u e 0

b x _

(_ , y ) w y + z .

Here, (_ 0 , a 0 ) are the coordinates of danyx point in k , ande z x is an arbitrary constant.

J)\

Reference: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964).

© 2002 by Chapman & Hall/CRC Page 477

478

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES

7.2. Poisson Equation {

= – } (x)

2|

7.2.1. Preliminary Remarks. Solution Structure Just as the Laplace equation, the Poisson equation is often encountered in heat and mass transfer theory, fluid mechanics, elasticity, electrostatics, and other areas of mechanics and physics. For example, it describes steady-state temperature distribution in the presence of heat sources or sinks in the domain under study. The Laplace equation is a special case of the Poisson equation with ~ ≡ 0. In what follows, we consider a finite domain  with a sufficiently smooth boundary € . Let r s  and  s  , where r = {_ , a },  = {‚ , ƒ }, |r −  |2 = (_ − ‚ )2 + (a − ƒ )2 . 7.2.1-1. First boundary value problem. The solution of the first boundary value problem for the Poisson equation q

2

f

= − ~ (r)

(1)

in the domain  with the nonhomogeneous boundary condition f

= v (r) for r s €

can be represented as f (r) = u „ ~ (  ) … (r,  ) w  † − u ‡ v (  )

… x

†

w € † .

(2)

x r

Here, … (r,  ) is the Green’s function of the first boundary value problem, ˆ ‰ is the derivative ˆ Š ‹ of the Green’s function with respect to ‚ , ƒ along the outward normal N to the boundary € . The integration is performed with respect to ‚ , ƒ , with w  † = w ‚ w ƒ . The Green’s function … = … (r,  ) of the first boundary value problem is determined by the following conditions. 1 j . The function … satisfies the Laplace equation in _ , a in the domain  everywhere except for the point (‚ , ƒ ), at which … has a singularity of the form 21Œ ln |r−1 | . 2 j . With respect to _ , a , the function … satisfies the ‡ homogeneous boundary condition of the first kind at the domain boundary, i.e., the condition … | = 0. The Green’s function can be represented in the form … (r,  ) =

1 1 ln +b , 2Ž |r −  |

(3)

where the auxiliary function b = b (r,  ) is determined by solving ‡ the first boundary value problem for the Laplace equation q 2 b = 0 with the boundary condition b g = − 21Œ ln |r−1 | ; in this problem,  is treated as a two-dimensional free parameter. The Green’s function is symmetric with respect to its arguments: … (r,  ) = … (  , r).  /‘ ’ “3”

• –

When using the polar coordinate system, one should set

r = {— , ˜ },

 = {‚ , ƒ },

|r −  |2 = —

2

+‚

2

− 2— ‚ cos( ˜ − ƒ ),

w  † =‚ w ‚ w ƒ

in relations (2) and (3).

© 2002 by Chapman & Hall/CRC Page 478

7.2. POISSON EQUATION ™

2š

= − › (x)

479

7.2.1-2. Second boundary value problem. The second boundary value problem for the Poisson equation (1) is characterized by the boundary condition f = v (r) for r s € . x

The necessary solvability condition for x r this problem is u „ ~ (r) w  + u ‡ v (r) w €

= 0.

(4)

The solution of the second boundary value problem, provided that condition (4) is satisfied, can be represented as f (r) = u „ ~ (  ) … (r,  ) w  † + u ‡ v (  ) … (r,  ) w € † + z ,

(5)

where z is an arbitrary constant. The Green’s function … = … (r,  ) of the second boundary value problem is determined by the following conditions: 1 j . The function … satisfies the Laplace equation in _ , a in the domain  everywhere except for the point (‚ , ƒ ), at which … has a singularity of the form 21Œ ln |r−1 | .

2 j . With respect to _ , a , the function … satisfies the homogeneous boundary condition of the second kind at the domain boundary: œ … œœ 1 , œ‡ = x

€

0

where € 0 is the length of the boundary of  x . r The Green’s function is unique up to an additive constant.  /‘ ’ “3”  – The Green’s function cannot be determined by condition 1 j and the homogeneous œ‡ boundary condition ˆ ‰ œ = 0. The point is that the problem is unsolvable for … in this case, ˆ Š because, on representing … in the form (3), for ž we obtain a problem with a nonhomogeneous boundary condition of the second kind for which the solvability condition (4) now is not satisfied. 7.2.1-3. Third boundary value problem. The solution of the third boundary value problem for the Poisson equation (1) in the domain  with the nonhomogeneous boundary condition f x

+ Ÿ   = v (r) for r s €

r is given by formula (5) with z = 0,x where … = … (r,  ) is the Green’s function of the third boundary value problem and is determined by the following conditions:

1 ¡ . The function … satisfies the Laplace equation in ¢ , £ in the domain  everywhere except for the point (‚ , ƒ ), at which … has a singularity of the form 21Œ ln |r−1 | .

2 ¡ . With respect to ¢ , £ , the function … satisfies the homogeneous boundary condition of the third kind at the domain boundary, i.e., the condition ¤ ˆ ‰ + Ÿ … ¥ ‡ = 0. The Green’s function can be represented in theˆ Š form (3); the auxiliary function ž is identified by solving the corresponding third boundary value problem for the Laplace equation ¦ 2 ž = 0. The Green’s function is symmetric with respect to its arguments: … (r,  ) = … (  , r).

§)¨

References for Subsection 7.2.1: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), N. S. Koshlyakov, E. B. Gliner, and M. M. Smirnov (1970).

© 2002 by Chapman & Hall/CRC Page 479

480

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES

7.2.2. Problems in Cartesian Coordinate System The two-dimensional Poisson equation in the rectangular Cartesian coordinate system has the form 2

  2

x ¢

2

+

 

x £

x

+ ~ (¢ , £ ) = 0.

2

x

7.2.2-1. Particular solutions of the Poisson equation with a special right-hand side. ª ª 1 ¡ . If ~ (¢ , £ ) = « © ¬ © ®« ¬ exp( ¯ « ¢ + ° ¬ £ ), the equation has solutions of the form =1 =1 ­ ±   (¢ , £ ) = − © «

± ¬

=1

«®¬ © =1

¯

2« ­

+ ° ¬2

exp( ¯ « ¢ + ° ¬ £ ).

ª ª 2 ¡ . If ~ (¢ , £ ) = « © ¬ © ®« ¬ sin( ¯ « ¢ + ² « ) sin( ° ¬ £ + ³ ¬ ), the equation admits solutions of the form =1 =1 ­ ±   (¢ , £ ) = © «

± ¬

=1

7.2.2-2. Domain: − ´

=1

< ¢ < ´ , −´

Solution:   (¢ , £ ) =

1 2Ž µ − ¶

«®¬ ©

+ ° ¬2

sin( ¯ « ¢ + ² « ) sin( ° ¬ £ + ³ ¬ ).

 ≥ 0, ≥  >  ≥ 0, 5 5

≥  >  ≥ 0, ≥  >  ≥ 0.  



) sin(7 + * 28



2

(2

for for

(  −  )] for (  −  )] for

8

 8

7

(2( + 1) , 2

= 7

sin(7 

) sin( * 8  ) cosh( * 8  ) 9 

( 5 −  )] ( 5 −  )]

The Green’s function can be written in form of a double series: 

8

 

sinh(7 sinh(7 0

sin( * 

 



= 7

2

8 

)

,

.

7.2.3. Problems in Polar Coordinate System The two-dimensional Poisson equation in the polar coordinate system is written as 1=

 =



 =

 

= 

+

=

1 2

=

7.2.3-1. Domain: 0 ≤

≤ >

,0≤



2 



2



=

=

+ ( , ) = 0,

= 4



2

+  2.

 

≤ 2 . First boundary value problem.

A circle is considered. A boundary condition is prescribed: =



=  ( ) at

= >

.



© 2002 by Chapman & Hall/CRC Page 485

486

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES Solution: =

= 

1

2

2?

( , )=



=

( ) 

0

= >

− 2>

2

2

− 2 cos( −  ) + >



where

2

2?

+

 



= 



0

( ,  )  ( , ,  ,  )

0@

=

1 1 1

− ln ln 2 |r − r0= | 2 r = { ,  },  = cos ,

r0 = { 0 ,  0 },

0 |( >



0)

= 

− r|

2r

0

,

=  sin   .

0

=

2

The magnitude of a vector difference is calculated as |  r − 5 r 0 | = (  and 5 are any scalars). Thus, we obtain =

=



=

2 2 

− 2 5

cos( −  ) + 5 



2

− 2> = −2

=



4

cos( −  ) + > . cos( −  ) +  2 ] 

2[ 2 >

2 2



=

2 2

1

ln 4

( , , , ) = 

,

= sin , 

=  cos  ,

0 

= > =

( , , , ) = 

 

 



 







References: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964),  A. G. Butkovskiy (1979). =

7.2.3-2. Domain: 0 ≤

≤



,0≤ >

≤ 2 . Third boundary value problem.

A circle is considered. A boundary condition is prescribed: 

=

A

+ 

=  ( ) at 

Solution: =

=

2? >

( )  ( , , > ,  )  



0



where =

( , , , ) =

Here, the



1





( ,  )  ( , ,  ,  )

0@

0

= 

8

=0

0

(!

28

(!

=1

>

2 B

= 1,

+

(!

=

7.2.3-3. Domain:

≤ >






(! 8  ) 2 − ( 2 )[ (! 8 8

DEDED

)]2 >

cos[( ( −  )], 

).

are positive roots of the transcendental equation (! >

) = 0.



,0≤ 

2>

C  =

=2 (( = 1, 2, 

B B ( ) are the Bessel functions the ! C and F !



C 



 







C 

2?

+ 

 



. >



( , )= 

=

≤ 2 . First boundary value problem.

The exterior of a circle is considered. A boundary condition is prescribed: =



Solution: 1 ( , )=

2

=

. >

 =

=



=  ( ) at



2? 

0



( )

=

=

2

− 2> =

2

−> 2 cos( −  ) + >

2

 

+

2? 0





=

( ,  )  ( , ,  ,  )

 

 

,



where the Green’s function  ( , ,  ,  ) is defined by the formula@ presented in Paragraph 7.2.3-1.



Reference: A. G. Butkovskiy (1979). 

© 2002 by Chapman & Hall/CRC Page 486

7.2. POISSON EQUATION =

7.2.3-4. Domain:

≤

1 >

≤



= − (x) 

487



0≤

2, >

2 G

≤ 2 . First boundary value problem.

An annular domain is considered. Boundary conditions are prescribed:  = = = 

1( 

Solution:

2( 

( , )=

1 >

1 ( 



0

2?

+

2



0

@

)$

=

( , , , )

  

= 

=

) at

2. >





2?



&

)

=



2?

−>

 

2

1

2 ( 

0

)$

=



( , , , ) 





)

=

2



( ,  )  ( , ,  ,  )

.

@  

 

@

1

@ =

where

 =

 =

2

2

=

+I

−2 =

I

(> (>



1 =

ln

>

2 >

2 )

1

1)

2J



2

( ) = for for





=

2



2 J +2

 > = 1H

− ln

=0  = H

cos( −  ), I

0





 

=

2

1

2

( , , , ) = 

H

(

 =

2

+ (I

) − 2

cos( −  ),  >

= I

H

I

H

= 2 , = 2  + 1, (

, 

2 1

.

I



Reference: B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). =

7.2.3-5. Domain: 0 ≤

≤



,0≤ >

≤ . First boundary value problem.

A semicircle is considered.= Boundary conditions are prescribed:  = = 



=

= 0,

2 ( ) at 

= 



= .

3 ( ) at 

  ?

( , ) =−>

1 ( 



0

)$



−

0@







( , , , ) 



=

$ 





=



1

3 ( )



)

=



@ &

'

+

 

=?

2 ( 

0@



( , , , ) 

+

  &

)

?

0

1

=

$ 



( , , , ) 



= 

= =

( , , , ) =

=

>

=0

  

 

,

=

− 2 > 2=  cos( −  ) + > 4 1 − ln 2 [ 2 − 2  cos( −  ) +  2 ] 4



'



=

2 2

1

ln 4

  &

( ,  )  ( , ,  ,  )

0@





where 

= 

, >



= 

=

1 ( ) at

Solution:



  &



Here,



= 

1, >



= 

=

) at

2 2

=



>



− 2 > 2=  cos( +  ) + > 4 . 2 [ 2 − 2  cos( +  ) +  2 ] =



 See also Example 2 in Paragraph 7.2.4-2.  

References: V. S. Vladimirov, V. P. Mikhailov, A. A. Vasharin, et al. (1974), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). =

7.2.3-6. Domain: 0 ≤

≤

,0≤ >

≤



2. First boundary value problem. 

A quadrant of a circle is considered. Boundary conditions are prescribed:=  = = = 



Solution:

1(

= 

, >



2(

= 0,

) at

 ? K

( , ) =−>



2 

0

1 (



−

= 



3(

=

) at



= 

=

) at



0@



3 ( )

1 

)$  $ 

( , , , ) 





= 

  & )

=

+



&

@ '

=? K

 

2

+

? K

0

2 ( 

0@



( , , , )



2.



=





2

)

1  

0@

 $  

= 

( , , , ) =

  &

'

=0



( ,  )  ( , ,  ,  )

 

 

,



© 2002 by Chapman & Hall/CRC Page 487

488

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES

where =

( , , , ) =

=

, , , )− 

=

=

, ,  =, 2 −  ) − = 1 ( , ,  , −  ) +  = 2 2 2 1  = − 2> =  cos( − )+> 4    1 ( , ,  ,  ) =  ln 2 2 . 4 > [ − 2  cos( −  ) +  2 ]   See also Example 3 in Paragraph 7.2.4-2.  



=

1(



1(



, ,  , +  ),

1(



References: V. S. Vladimirov, V. P. Mikhailov, A. A. Vasharin, et al. (1974), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980). =

7.2.3-7. Domain: 0 ≤

≤

,0≤ >

≤ L . First boundary value problem.

A circular sector is considered. Boundary conditions are prescribed:  = = = 



1(

Solution:



( , )=−>

1 ( 

0M

)$



− 1 1 . For L =



0@



 =

$  

+

  &

)

=



0@

2 (



( , , , ) 



=

@ &

'

=



0M





$ 



 =

( , , , ) 



  &

'

=0



( ,  )  ( , ,  ,  )

0@

 

 

.



−1

=

1( , ,  , 2 

=0 J

=

=

+

= 

1( , ,  , 2

=

2 2

1

ln 4

1( , ,  ,  ) =

+ )− L



>







2 1 . For arbitrary L , the Green’s function is given by =

= R SUT

R S Q

,

1

ln 2

( , , , ) = 

Q  , ¯= '

=

O

R S

'

−

−  ), , L

− 2 > 2=  cos( −  ) + > 4 . 2 [ 2 − 2  cos( −  ) +  2 ] =

Reference: B. M. Budak, A. A.  Samarskii, and A. N. Tikhonov (1980).

=



1

=L .

) at

= 

+

 

)

3( 





( , , , ) = 

where

=

= 

, where ( is a positive integer, theM Green’s function is expressed as

 (





= 0,

) at

( , , , ) 





2( 

=



1

3 ( ) 

= 

, >



= 

=

) at

Q

− ¯? Q − ?

? K N

NPO ? K M N N O M

K > N N

K M

, and V 2 = −1.

N N > N N N N M

Q

K

2?

K M

− ( ¯ )? Q − ( O )?

2?

K N N

O M

, N

K M N M

=

7.2.3-8. Domain: 0 ≤


0, 0 ( ì í ) if û = ì 4    where í = ä 2 + å 2 , 0 (  ) is the modified Bessel function of the second kind, 0(1) (  ) and 0(2) (  ) are the Hankel functions of the first and second kind of order 0, ä 0 and å 0 are arbitrary constants, and ô 2 = −1. The leading term of the asymptotic expansion of the fundamental solutions, as í î 0, is given by 21 ln 1ð .

(ä , å ) = −

3 Ø . Suppose ü = ü (ä , å ) is a solution of the homogeneous Helmholtz equation. Then the functions ü ü 1 = ( ä +  1 ,  å +  2 ), ü ü 2 = (− ä +  1 ,  å +  2 ),

= ü (ä cos  + å sin  +  1 , −ä sin  + å cos  +  2 ), where  1 ,  2 , and  are arbitrary constants, are also solutions of the equation. ö÷ ü

3

Reference: A. N. Tikhonov and A. A. Samarskii (1990).

7.3.2-2. Domain: − ï

< ä


0: ë ü

0: (2) 0 ( 

1)

+



(2) 0 ( 

2)

+



(2) 0 ( 

3)

+



(2) 0 ( 

4 )

.

© 2002 by Chapman & Hall/CRC Page 496

7.3. HELMHOLTZ EQUATION

7.3.2-7. Domain: − ï




2−  2

=

|1 − % | , 2−%

@ + " 2 A < =?> 

2−  2

@ B

for

1

> 0,

@ + " 2D

2−  2

@ B

for

1

< 0,

>

=

< =?> 

2 2−% F

|

1|



,

where " 1 and " 2 are arbitrary constants, ; < ( G ) and A < ( G ) are the Bessel functions, and C < ( G ) and D < ( G ) are the modified Bessel functions. The solution of equation (4) is expressed as 

1−  2



1−  2

8

3 ( ) = 67 79

: " 1 ; H =I 

2−  2

: " 1 CH =?I 

2−  2

J = |1 − & | , 2−&

@ + " 2 A H =?I 

2−  2

@ B

for

1

< 0,

@ + " 2 D H =?I 

2−  2

@ B

for

1

> 0,

I =

2 2−&

| F

1|



,

where " 1 and " 2 are arbitrary constants. The sum of solutions of the form (2) corresponding to different values of the parameter 1 is also a solution of the original equation; the solutions of some boundary value problems may be obtained by separation of variables. 4  . See equation 7.4.3.3, Item 4  , for K = 0.

© 2002 by Chapman & Hall/CRC Page 519

520

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES 

2. L

L M

N O

M P

+ L

L Q L M



N

=R .

L Q

 

L 

L 



This is a two-dimensional equation of the heat and mass transfer theory with constant volume release of heat in an inhomogeneous anisotropic medium. Here,  1 ( ) =    and  2 ( ) =   are the principal thermal diffusivities. 1  . For % ≠ 2 and &

≠ 2, there are particular solutions of the form #

The function

#

#

=

1+ 2 ( = )  (2 − & )2  2−  +  (2 − % )2  2−  * .

= # (( ),

(1)

(( ) is determined by the ordinary differential equation # ./,-, . +

# ., =! , (

(2)

where

4−% & 4K , ! = . (2 − % )(2 − & )   (2 − % )2 (2 − & )2 The general solution of equation (2) is given by =

8

#

(( ) = 67 "

1

and "

2

1

" 1( 79

where "

" 1 ( 1− 1 + " 2 + 2( 1 S +1) ( 2

ln ( + " 2

+"

2

2

+

+ 1 2!

1 4!

(

for

≠ T 1,

(

2

for

= 1,

2

ln (

for

= −1,

are arbitrary constants.

2  . The substitution #

( ,  ) = U ( ,  ) +

K  (2 − % )

leads to a homogeneous equation of the form 7.4.3.1: V

V 

3. L

L M

N O

M P

(3)

L Q

+ L W

L M

L [

N

V

   V 

 W

N [ 

L Q L [

V

U

W

X Z

+ V X

V

U V X

N Y

=R

 2− 

W

= 0.

. Q

This is a two-dimensional equation of the heat and mass transfer theory with a linear source in an inhomogeneous anisotropic medium. 1 \ . For ] ≠ 2 and &

≠ 2, there are particular solutions of the form ^

= ^ (_ ),

X Z c 1d 2 _ = ` (2 − & )2 a 2− b +  (2 − ] )2 2− .

Y ordinary differential equation The function ^ = ^ (_ ) is determined by the ^ f/-e e f + g _

^ fe =h ^ ,

(1)

where

4−] & 4K , h = . (2 − ] )(2 − & )  (2 − ] )2 (2 − & )2 The general solution of equation (1) is given by Y g

=

^

(_ ) = _

1− i 2

jlk 1 m n o _ p

^

(_ ) = _

1− i 2

jlk 1 t?n o _ u h

|h | q + k

2r

n o_ p

q + k 2v n o _ u h

| h | q s q s

for h

< 0,

for h

> 0,

where w = 12 |1 − |; k 1 and k 2 are arbitrary constants; m n ( x ) and r n ( x ) are the Bessel functions; and g t?n ( x ) and v n ( x ) are the modified Bessel functions.

© 2002 by Chapman & Hall/CRC Page 520

521

7.4. OTHER EQUATIONS

2 . There are multiplicatively separable particular solutions of the form


( ,  ) =  ( ) ( ),

where  ( ) and  ( ) are determined by the following second-order linear ordinary differential equations (  1 is an arbitrary constant): (     ) = 

(     )  = (  − 

,

1

1 )

.

(2)

The solutions of equations (2) are expressed in terms of the Bessel functions (or modified Bessel functions); see equation 7.4.3.1, Item 3 . 3 . There are additively separable particular solutions of the form


( ,  ) =  ( ) +  ( ),

where  ( ) and  ( ) are determined by the following second-order linear ordinary differential equations (  2 is an arbitrary constant): (     ) −  = 

(     )  −   = − 

2,

2.

(3)

The solutions of equations (3) are expressed in terms of the Bessel functions (or modified Bessel functions). 4 . The transformation (specified by A. I. Zhurov, 2001) 2− 2



where 

2

 2
  2

=  (2 −  )2 and 

2

=   cos  ,

2− 2



=  (2 −  )2 , leads to the equation 





4−  1
1 2
2 (   + 2  2 − 2 (2 −  )(2 −  )     

+

=   sin  ,

−  −  ) cos 2 + ( −  )
 (2 −  )(2 −  ) sin 2 

which admits separable solutions of the form
( ,  ) =  

4.

 !

 



)  2 ( ).

 !

$ #

( + s)%

( + )   " +  #

 

1 (

= 4
,

=& .

 #

"

The transformation ' =  + ( , ) =  + * leads to an equation of the form 7.4.3.2:   

5.



 ! 

' 

+ $

 ' ,

+  )

+

 )

 ! #

( + s)%

( + )   " +  #

 


 '  

 #

=& "

!


 )

,

=.

.

The transformation ' =  + ( , ) =  + * leads to an equation of the form 7.4.3.3:   

6.

  +

 - . /  !   ,



'

+  #

+


 '  

 ' ,

+  )

+

 )


 )

,

=
.

+

$ - 0 1  !  # ,

= 0.

This is a two-dimensional equation of the heat and mass transfer theory in an inhomogeneous 

anisotropic medium. Here,  1 ( ) =  243 and  2 ( ) = 5 2 6 are the principal thermal diffusivities.

© 2002 by Chapman & Hall/CRC Page 521

522

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES

1 . Particular solutions (  ,  , 7 are arbitrary constants):


(  ,  ) =  2 −3


( ,  ) =


( ,  ) =





+  2 −6

+7 ,



(8  + 1) 2 −3  8 2 − −  6 + .  2 3

 (9   9 2

−

+ 1) 2 − 6



+ ,

2 . There are multiplicatively separable particular solutions of the form


( ,  ) =  ( ) ( ),

(1)

where  ( ) and  ( ) are determined by the following second-order linear ordinary differential equations (  1 is an arbitrary constant):

(  2 3  ) = −  1  ,  ( 52 6   )  =  1  .

(2) (3)

The solution of equation (2) is given by ( ) = : 

; 2
( 2 −3

; 2
( 2 −3

+7

2@ 1 > ( 2

+7

2C

− ; 2 ?5A

3

1> ( 2

− ; 2 ?5A

3

for 

1

> 0,

for 

1

< 0,

where ( = −(2 D 8 ) E |  1 | D  ; 7 1 and 7 2 are arbitrary constants; = 1 ( F ) and @ 1 ( F ) are the Bessel functions; and B 1 ( F ) and C 1 ( F ) are the modified Bessel functions. The solution of equation (3) is given by 

( ) = :

 ; 2<  ; 2? 2 −6 7 1 = 1 > *G2 − 6  ; 2<  ; 2? 2 −6 7 1 B 1 > *G2 − 6

+7

2 @ 1 > *G2

+7

2C

−  ; 2 ?5A

6

1 > *G2

−  ; 2 ?5A

6

for 

1

< 0,

for 

1

> 0,

where * = −(2 D 9 ) E |  1 | D  ; 7 1 and 7 2 are arbitrary constants. The sum of solutions of the form (1) corresponding to different values of the parameter  also a solution of the original equation.

1

is

3 . See equation 7.4.3.8, Item 3 , for  = 0. 

7.

 - . /  !

 



+

$ - 0 1  !

+  #

  ,

=& .

 # +

,

This is a two-dimensional equation of the heat and mass transfer theory with constant volume release 

of heat in an inhomogeneous anisotropic medium. Here,  1 ( ) =  2 3 and  2 ( ) = 52 6 are the principal thermal diffusivities. The substitution 


( ,  ) = H ( ,  ) − (8  + 1) 2 −3 2  8

leads to a homogeneous equation of the form 7.4.3.6:   

8.

 



+

 - . /  !  



,

+  #

+

 2 3





 H





,

+ 

+

$ - 0 1  !  # ,

=&

!



+

52 6



 

H 

,

= 0.

.

This is a two-dimensional equation of the heat and mass transfer theory with a linear source in an inhomogeneous anisotropic medium.

© 2002 by Chapman & Hall/CRC Page 522

523

7.4. OTHER EQUATIONS

1 . For 8 9 ≠ 0, there are particular solutions of the form


=
(I ),





;

= >  9 2 2 −3 +  8 2 2 − 6 )1 2 . The function
=
(I ) is determined by the ordinary differential equation 1
K 4
KLK
J − = ,  = . 2 2 I

I

  8

9

For the solution of this equation, see 7.4.3.3 (Item 1 for  = −1). 2 . The original equation admits multiplicatively (and additively) separable solutions. See equation 

7.4.3.12 with  ( ) =  2 3 and  ( ) = 52 6 . 3 . The transformation (specified by A. I. Zhurov, 2001) − ; 2

2

where 

2

= 8

2

and 

=   cos  ,

3

2

−  ; 2

6

2

=   sin  ,

=  9 2 , leads to the equation  2
  2

 1


 2

2   − cot 2    2  2  2   which admits separable solutions of the form
( ,  ) =  1 ( )  2 (   !   $ - . 1  !  ! 9.   +  # =& .     + , + ,

−

+



1

= 4
, ).

1 . For  ≠ 2 and 8 ≠ 0, there are particular solutions of the form


=




2

 2 −3

 2− 

+ . (2 −  )2  8 2 The function
=
( ) is determined by the ordinary differential equation ( ),

 2
  2



+

=



 1




= 4
.



2−   For the solution of this equation, see 7.4.3.3 (Item 1 ).

2 . The original equation admits multiplicatively (and additively) separable solutions. See equation  7.4.3.12 with  ( ) =    and  ( ) = 52M3 . 3 . The transformation (specified by A. I. Zhurov, 2001) 1  1− 2 

2

where 

2

2

=  (2 −  ) and   2
  2

1  2 −23

=   cos  ,

=   sin  ,

2

=  8 , leads to the equation

 2 (1 −  ) cos 2 + 1
 + + − 2 2−    (2 −  ) sin 2  which admits separable solutions of the form ( ,  ) =  1 ( )  2 ( ). 

10.

ON 

 !

where V




=7

1




=7

1




= [7

1R

( ) + 7

2 ]

= [7

1R

( ) + 7

2 ]

7 1 + 7 3   ( ) Q

2

+7

2

−2P

3

+7

2

− 6 P

1, V 2, V 3, V 4,

and V

5

= 4
,

= 0.

2

1 . Particular solutions:




 2
  2  2


1



 2!

( )  " +  #

 

 1




+7 2

+7

+7

4,

7 1 + 7 3   ( ) Q

+7

( ) + 7

R

( ) = P

−2P S

3R 3R

( ) + 7

4,

4

4,



Q



( )

1 P [7  ( )

, 1R

( ) + 7

2]

Q

 T

Q U

,

are arbitrary constants.

© 2002 by Chapman & Hall/CRC Page 523

524

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES

2 W . Separable particular solution: X = (V

+V

1Y Z [

2Y

−

Z [ )\

( ), U

where V 1 , V 2 , and ] are arbitrary constants, and the function \ differential equation [ ^ ( ) \ ` _ ] _` + ] 2 \ = 0.

= \ ( ) is determined by the ordinary U

U

X 3 W . Separable particular solution:

= [V

sin( ] a ) + V

1

2

cos( ] a )] b ( ), U

where V 1 , V 2 , and ] are arbitrary constants, and the function b = b ( ) is determined by the ordinary U differential equation [ ^ ( ) b ` _ ] _` − ] 2 b = 0. U

4 W . Particular solutions with even powers of a : X

e

e

where h

e g

U

( )+j e

i

− 2 m (2 m − 1) k

U

5 W . Particular solutions with odd powers of a : X

e

where h 11.

r

sOt

U

=0

( )=h U

−1 (

( )q u q

q r

2 +1

( )a )

q r

l U

T

l U

,

,

( )+j U

e

U

c i

)=h

e

, ( ) U 1 k ^ ( )n

U

i c

( )+j

^

− 2 m (2 m + 1) k

U

i

( )=k

,

l U

+ q v

( )q u

q w

w

syx

q w

e )

( ) U

U

are arbitrary constants ( m = o , ppp , 1).

and j

U

U

c

)

( ) g

U

= ) ( ) are defined by the recurrence relations

) e

de

e

e e

= c e

where the functions )

e

l U

^

are arbitrary constants ( m = o , ppp , 1).

and j

e

, ( ) U 1 k ^ ( )n

U

c

( )+j i

( )=k

,

U

e c i

)=h U

,

U

( )=h

−1 (

e

U

( ) are defined by the recurrence relations g

g c

2

( )a =0 g

= g

dfe

= c e

where the functions

e e

l U

T

l U

,

= 0. v

This is a two-dimensional sourceless equation of the heat and mass transfer theory in an inhomogeneous anisotropic medium. The functions ^ = ^ ( ) and z = z (a ) are the principal thermal U diffusivities. 1 W . Particular solutions: X

X e

e

( ,a ) = h

1 k

( ,a ) = h

2 k

( ,a ) = h

3 k

U

X

U U

l U

( )

^

U

U l U ^ ( ) U ^

l U

( ) U

+j

1k

−h

2 k

k z

l

l z

a z a

(a )

a

(a ) l

+V

1,

+j

2,

a

(a )

+j

3,

where the h , j , and V 1 are arbitrary constants. A linear combination of these solutions is also a solution of the original equation.

© 2002 by Chapman & Hall/CRC Page 524

525

7.4. OTHER EQUATIONS

2 W . There are multiplicatively separableX particular solutions of the form ( , a ) = { ( )| (a ), U

(1)

U

where { ( ) and | (a ) are determined by the following second-order linear ordinary differential U equations ( h is an arbitrary constant): ( ^ { `_ ) _` = h { , (z | _ ) _ = − h | , ^

[ [

= ^ ( ), U z = z (a ).

(2)

The sum of solutions of the form (1) corresponding to different values of the parameter h in (2) is also a solution of the original equation (the solutions of some boundary value problems may be obtained by separation of variables). 12.

( )q u q

q r

sOt

r

q r

+ q v

( )q u

q w

w

syx

=}

q w

v

. u

This is a two-dimensional equation of the heat and mass transfer theory with a linear source in an inhomogeneous anisotropic medium. The functions ^ = ^ ( ) and z = z (a ) are the principal thermal U diffusivities. 1 W . There are multiplicatively separableX particular solutions of the form ( , a ) = { ( )| (a ), U

(1)

U

where { ( ) and | (a ) are determined by the following second-order linear ordinary differential U equations ( h is an arbitrary constant): ( ^ { _` ) _` = h { , (z | _ ) _ = ( ~ − h ) | ,

= ^ ( ), U z = z ( a ). ^

[ [

(2)

The sum of solutions of the form (1) corresponding to different values of the parameter h in (2) is also a solution of the original equation; the solutions of some boundary value problems may be obtained by separation of variables. solutions of the form 2 W . There are additively separable particular X ( , a ) = ( ) +  (a ), U

i

U

where ( ) and  (a ) are determined by the following second-order linear ordinary differential U equations i ( V is an arbitrary constant): ( ^ `_ ) _` − ~ =V , (z  i _ ) _ − ~  i = − V , [ [

^

= ^ ( ), U z = z (a ).

In the special case ~ = 0, the solutions of these equations can be represented as ( )=V k i 

where h

1, h

2, j

1,

and j

2

U

(a ) = − V

U l U +h 1k ^ ( ) U a a k l +h 2k z (a ) ^

l U

( )

z

U

l

a

(a )

+j +j

1, 2,

are arbitrary constants.

© 2002 by Chapman & Hall/CRC Page 525

526

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES

7.4.4. Other Equations Arising in Applications 1.

2

q w

u

2

+ q

2

q r

u

= 0.

2

q w

Tricomi equation. It is used to describe near-sonic flows of gas. 1 W . Particular solutions: X X

=h

+j a

U

= h (3 where h , j , V , and €

2

U

+V a +€ , U

3

−a )+j (

are arbitrary constants.

3

U

U

2 W . Particular solutions with even powers of : X

(a )e = h { c

and j

where h

,

c

{

)=h

−1 ( a

+j a

e

where h

=|

(a )e = h

and j

− 2 m (2 m − 1) k

[

e

(a −  )‚{

0

( )  , l

a

= c |

(a )

=0

2 +1

U

,

(a ) are e defined bye the recurrence relations e

+j

c

e e

U ƒd e

e

where the functions |

c

,

are arbitrary constants ( m = o , ppp , 1). X

|

2

U

c

3 W . Particular solutions with odd powers of :

e

− a 4 ),

(a ) are e defined bye the recurrence relations e

+j a

(a ) {

=0

={

where the functions { e

de

e

2

U

e e

U

= c

e

− a 3 ) + V (6a

, |

)=h

−1 ( a

a

+j

− 2 m (2 m + 1) k

c

[ 0

e

(a −  )„|

( )  , l

are arbitrary constants ( m = o , ppp , 1).

X 4 W . Separable particular solutions:

( , a ) = …yh sinh(3 ] X

U

U

) + j cosh(3 ]

)† ‡ a …yV ˆ 1 ‰ 3 (2 ] a 3 ‰ 2 ) + € Š 1 ‰ 3 (2 ] a 3 ‰ 2 )† , U

)† ‡ a …yV ‹ 1 ‰ 3 (2 ] a 3 ‰ 2 ) + € Œ 1 ‰ 3 (2 ] a 3 ‰ 2 )† , where h , j , V , € , and ] are arbitrary constants, ˆ 1 ‰ 3 ( F ) and Š 1 ‰ 3 ( F ) are the Bessel functions, and ‹ 1 ‰ 3 ( F ) and Œ 1 ‰ 3 ( F ) are the modified Bessel functions. 5 W . For a > 0, see also equation 7.4.4.2 with o = 1. For a < 0, the change of variable a = − leads to an equation of the form 4.3.3.11 with o = 1. ( , a ) = …yh sin(3 ] U

yŽ

2.

U

) + j cos(3 ]

U

Reference: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964). 2

w 

q

u q r

2

+ q

2

u

2

= 0. X

q w

1 W . Particular solutions:

+V a +€ , 2h =h 2− a +2 , U (o + 1)(o + 2) c X 6h +2 =h 3− a , U (o + 1)(o + 2) U c X 2h =h a 2− a +3 , U (o + 2)(o + 3) c are arbitrary constants. X

where h , j , V , and €

=h

U

a

+j

U

© 2002 by Chapman & Hall/CRC Page 526

527

7.4. OTHER EQUATIONS

2 W . Particular solutions with even powers of : U

X

dfe

=  e

e

(a ) = h {

e



where h

e

,



e

)=h

−1 ( a

{

e

+j a

U

X e

dƒe

e

(a ) = h e

and j

(a −  ) {

( )  , l

c

2 +1

U

,

(a ) are defined by the recurrence relations e

+j a

(a ) |

=0

=|

where the functions |

where h

‘

e e

= 



e [

− 2 m (2 m − 1) k



3 W . Particular solutions with odd powers of :

|

,

are arbitrary constants ( m = ’ , ppp , 1), “ is any number.

and j

e

2

U

(a ) are defined by the recurrence relations

+j a

e

(a ) {

=0

={

where the functions {

e e

,



e

)=h

−1 ( a

|

a

e

+j

e [

− 2 m (2 m + 1) k ‘



(a −  ) | c

( )  , l

are arbitrary constants ( m = ’ , ppp , 1), “ is any number.

particular solutions: 4 W . Separable X ( , a ) = … h sinh( ] ” X

U

( , a ) = … h sin( ] ” U

U

U

) + j cosh( ] ” ) + j cos( ] ”

U

)† ‡ a … V ˆ

)† ‡ a … V ‹ U

1 2

•

1 2

•

(] a – ) + € Š

(] a – ) + € Œ

( ] a – )† ,

1 2

•

1 2

•

= 12 (o + 2), ”

( ] a – )† ,

where h , j , V , € , and ] are arbitrary constants, ˆ — ( F ) and Š — ( F ) are the Bessel functions, and ‹ — ( F ) and Œ — ( F ) are the modified Bessel functions. 5 W . Fundamental solutions (for a > 0): X



o

1 (˜

, a , ˜ 0 , a 0 ) = m 1 (™ 12 )−š › (~ , ~ , 2~ ; 1 − œ ),

2 (˜

, a , ˜ 0 , a 0 ) = m 2 (™ 12 )−š (1 − œ )1−2š › (1 − ~ , 1 − ~ , 2 − 2~ ; 1 − œ ).

~

=

2(o + 2)

,

= œ

™ 22

, ™ 12

Here, › ( “ , ž , Ÿ ; œ ) is the hypergeometric function and ™ 1

2

= ( ˜ − ˜ 0 )2 +

4 a (o + 2)2   c

+2 2

™ 22

= ( ˜ − ˜ 0 )2 +

4 (o + 2)2   ¤ ¥

+2 2

=

1 4   4¢ o +2¡

2š

m 1

=

1 4 4¢   o + 2 ¡

2š

m 2

+2

+ a 0c 2 ¡ , +2

− ¤ 0¥ 2 ¡ ,

£ 2 (~

) , (2~ )

£ £

£ 2 (1 − ~

) , (2 − 2~ )

£

where (~ ) is the gamma function; ˜ 0 and ¤ 0 are arbitrary constants. The fundamental solutions satisfy the conditions ¦ § 

yŽ

§

1 ¨ =0 ¨

= 0,



§

2 ¨ =0 ¨

=0

(˜ and ˜

0

are any, ¤

0

> 0).

The solutions of some boundary value problems can be found in the first book cited below. Reference: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), A. D. Polyanin (2001a).

© 2002 by Chapman & Hall/CRC Page 527

528

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES 2

3. q

u

2

+ ª

2

+ q

q u

u

= 0.

2

q © © q © q « Elliptic analogue of the Euler–Poisson–Darboux equation.

1 ¬ . For ­ = 1, see Subsections 8.1.2 and 8.2.3 with  =  (™ , F ). For ­ ≠ 1, the transformation ˜ = (1 − ­ ) F , ¤ = ™ 1− ® leads to an equation of the form 7.4.4.1: ¦ 2 ¦ ˜ 2

2® 1− ®

¤

¦ 2

+ ¦

2

¤

= 0.

2 ¬ . Suppose  ® =  ® (™ , F ) is a solution of the equation in question for a fixed value of the parameter ­ . Then the functions  ¯ ® defined by the relations ¦ 

  

¯ ®

® = ¦ ,

¯ ®

=

¯ ®

=

¦ F ¦  ® ® ™ ¦ +F ¦ ™¦ F  ® 2™ F ¦ + (F 2 ™

,

¦ 

® − ™ 2) ¦ +­ F  ® F

are also solutions of this equation. 3 ¬ . Suppose  ® =  ® (™ , F ) is a solution of the equation in question for a fixed value of the parameter ­ . Using this  ® , one can construct solutions of the equation with other values of the parameter by the formulas        yŽ

4.

−1  ® ¦  ® ™ ¦ ¦ ™ ® ™ F ¦ ™

2− ®

=™ ®

,

® −2

=

+ ( ­ − 1) ® ,

® −2

=

® −2

= ™ (™

® +2

=

® +2

=

® +2

=

2

F ™ ™ 2

−™

¦ 

+ ( ­ − 1) F  ® , ¦  ® ¦ F

® − F 2) ¦ + 2™ 2 F

¦ 1  ® ¦ ™

¦  ® ¦ F

2

¦  ™ ® ¦ ™

−F

™

,

+ …°™

2

− ( ­ − 1) F 2 †  ® ,

¦ 

® − ¦ , F

2 ¦ 

¦

™

® ™

¦  ® + 2F ¦ F

+­  ® .

Reference: A. V. Aksenov (2001). 2

q

u

2

2

+ ( )q

q r

t

r

u

2

= 0.

q w

1 ¬ . Particular solutions:

where ±



=± 

=± ¸

=± ¸

= (±

1˜ ¤ 1¤ 1¤ 1´

2

3

+±

+±

2¤

+±

2˜ ¤

+±

+±

2´ ¤

+±

+±

2 )¤

1, ± 2, ± 3, ± 4, ± 5,

2

+±

and ±

+±

3˜ 3¤ 3¤

3´ ¤ 6

4,

+±

4˜

− 2±

+±

4´

− 6±

+±

4¤

+±

1² 1¤ 5´

²

‘

(´ − µ ) ¶ (µ ) · µ + ± ³

‘

³

(´ − µ ) ¶ (µ ) · µ + ±

− 2 ² ³ (´ − µ )( ± ‘

1µ

5, 5,

+±

2)¶

(µ ) · µ + ±

6,

are arbitrary constants, “ is any number.

© 2002 by Chapman & Hall/CRC Page 528

529

7.4. OTHER EQUATIONS

2 ¬ . Separable particular solution: ¸

§

= (±

+±

1¹ º

2¹

−

§

) » (´ ), º

= » (´ ) is determined by the ordinary

where ± 1 , ± 2 , and ¼ are arbitrary constants, and the function » differential equation » ½J½ + ¼ 2 ¶ (´ ) » = 0. 3 ¬ . Separable particular³ ³ solution: ¸

= [±

1

sin( ¼ ¤ ) + ±

2

cos( ¼ ¤ )] ¾ (´ ),

where ± 1 , ± 2 , and ¼ are arbitrary constants, and the function ¾ = ¾ (´ ) is determined by the ordinary differential equation ¾ ½J½ − ¼ 2 ¶ (´ ) ¾ = 0. ³ even powers of ¤ : 4 ¬ . Particular solutions³ with ¿fÀ

¸ À

,

Ã

À

)=Ä

−1 ( ´

Ã

À

´

+Å

À

1(

q r

ÊOË

r

À

+Å

,

Ã

À

)=Ä

−1 ( ´

Ã

À

´

+Å

− 2 Æ (2 Æ + 1) ² ³ (´ − µ ) ¶ (µ ) ‘

É

À

(µ ) · µ , É

are arbitrary constants ( Æ = Ç , ppp , 1), and “ is any number.

and Å

q

´

À

,

=0 É

É

(´ ) = Ä

2 +1

(´ )Â

(´ ) are defined by the recurrence relations

É É Ã

= Ã

À

=

where the functions

5.

(µ ) · µ , Á

À À

¿À

¸

where Ä

À

− 2 Æ (2 Æ − 1) ² ³ (´ − µ ) ¶ (µ ) ‘

Á

5 È . Particular solutions with odd powers of  :

À

,

are arbitrary constants ( Æ = Ç , ppp , 1), “ is any number.

and Å

where Ä

À

+Å ´

À

2

(´ ) are defined by the recurrence relations Á

(´ ) = Ä Á Ã

(´ )Â =0 Á

= Á

À

= ¥

À

where the functions

À À

)q Ì q r

+ q Í

2(

q Î

ÊOË

Î

)q Ì

+ Ï ÐÒÑ 1 ( ) + Ñ 2 ( )Ó

q Î

r

Í

Î

Ì

= 0.

This equation is encountered in the theory of vibration of inhomogeneous membranes. Its separable solutions are sought in the form ¸ (´ , Â ) = (´ ) (Â ). The article cited below presents an algorithm Á É for accelerated convergence of solutions to eigenvalue boundary value problems for this equation.

ÔyÕ

Reference: L. D. Akulenko and S. V. Nesterov (1999).

7.4.5. Equations of the Form Ö

2Ù

(× ) Ø Ø

× 2

+ Ø

2Ù

Ø Ú

2

Ù

+ Û (× ) Ø Ø

×

+ Ü ( × )Ù

= – Ý (× , Ú )

7.4.5-1. Statements of boundary value problems. Relations for the Green’s function. Consider two-dimensional boundary value problems for the equation Þ

2¸

(´ ) ß ß

´ 2

2¸

+ ß ß

 2

¸

+ à (´ ) ß ß

´

+ á (´ )¸ = − â (´ , Â )

(1)

© 2002 by Chapman & Hall/CRC Page 529

530

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES

with general boundary conditions in ´ , ã

¸

1

− Æ 1 ¸ = ¶ 1 (Â ) at + Æ 2 ¸ = ¶ 2 (Â ) at

ã ß ³ ¸ 2 ß ³

= ´ 1, ´ = ´ 2, ´

(2)

and different boundary conditions in  . We assume that the coefficients of equation (1) and the boundary conditions (2) meet the requirement Þ

(´ ), à (´ ), á (´ ) are continuous functions (´

Þ

≤ ´ ≤ ´ 2 );

1

| ã 1 | + | Æ 1 | > 0,

> 0,

| ã 2 | + | Æ 2 | > 0.

In the general case, the Green’s function can be represented as ä

¿

(´ )

(´ , Â , å , æ ) = ç (å ) è =1 é

Ã

Here,

1 (´ ) = Þ exp (´ ) ç

) í (í ) î

ÊOì

à é

à (í Þ

ê

Ã

ê é

ê

, Í

(å )

ê 2

).

Ã

2

ç

2

(í )

1

ì

(3)

Ã

=

à é

(Â , æ ; ¼ ë

Ã2

ï

é

(í ) í ,

(4)

î

Ã

and the ð and (í ) are the eigenvalues and eigenfunctions of the homogeneous boundary value ï à problem for differential equation à the ordinary é

Þ

(í ) ½J½ + à (í ) ½ + [ ð + á (í )] = 0, ã

−Æ

½ é 1 ã ï ï ½ 2é ï

1 é = 0 at + Æ 2 é ï = 0 at

(5)

=é í 1 , í

(6)

= í 2. í

(7)

é The functions ë for various boundaryé conditions in  are specified in Table 25. ï Equation (5)à can be rewritten in self-adjoint form as

[ñ (í ) ½ ] ½ + [ ð ç (í ) − ò (í )] = 0, é where the functions ñ (í ) and ò (í ) are given by

é

ï ï

ñ

(í ) = exp Þ

à (í

) í (í ) î

ÊOì

(8)

, ò (í ) = − Þ

á (í

Í

) exp (í )

Þ ÊOì

à (í

) í (í ) î

, Í

and ç (í ) is defined in (4). The eigenvalue problem (8), (6), (7) possesses the following properties: 1 È . All eigenvalues ð 1 , ð 2 , ppp are real and ð 2 È . The system of eigenfunctions { 1 (í ), weight ç (í ), that is, é

2

ç

3 È . If the conditions

(í )

1

ì

ï

ï

2 (í

é

Ã

) ≥ 0,

ó

.

ô

ó

ô

), ppp } is orthogonal on the interval í

1

≤í ≤í

2

with

(í ) í = 0 for Ç ≠ ö .

(í ) é

ò (í

as Ç Ã

î

é õ ã

1Æ 1

≥ 0,

ã

2Æ 2

≥0

(9)

are satisfied, there are no negative eigenvalues. If ò ≡ 0 and Æ 1 = Æ 2 = 0, then the least eigenvalue is ð 0 = 0 and the corresponding eigenfunction is 0 = const; in this case, the summation in (3) must start with Ç = 0. In the other cases, if conditions (9) are satisfied, all eigenvalues are positive; for é example, the first inequality in (9) holds if á (í ) ≤ 0.

© 2002 by Chapman & Hall/CRC Page 530

531

7.4. OTHER EQUATIONS

The functions ë Domain −

|ù | < ô

Ã

1

ó

= 0 for ù

Function ë

for  ô

0≤
æ , cosh(÷  ) for æ >

−û üGÿ

[÷  cosh(÷  æ ) +  3 sinh(÷  æ )] for > æ , [÷  cosh(÷  ) +  3 sinh(÷  )] for æ > sinh(÷  æ ) sinh[÷  (  − )] for > æ , sinh(÷  ) sinh[÷  (  − æ )] for æ > cosh(÷  æ ) cosh[÷  (  − )] for > æ , cosh(÷  ) cosh[÷  (  − æ )] for æ >

1 û ü sinh(û ü )

= 0, =

−û üGþ

(Â , æ ; ð

sinh(÷  æ ) cosh[÷  (  − )] for > æ , sinh(÷  ) cosh[÷  (  − æ )] for æ >

Subsection 1.8.9 presents some relations for estimating the eigenvalues ð  and eigenfunctions  (í ). é Green’s function of the two-dimensional third boundary value problem (1)–(2) augmented The by the boundary conditions ù ß

−  3 ù = 0 at

= 0,

ù ß

ß

+  4 ù = 0 at

=

ß

is given by relation (3) with ë 



( , æ ; ð  ) = 



÷ 

cosh(÷  )+  3 sinh(÷  )  ÷  cosh[÷  (  − )]+  4 sinh[÷  (  − )]  ÷  ÷  (  3 +  4) cosh(÷   )+(÷ 2 +  3  4) sinh(÷   )

for

> ,

÷ 

cosh(÷  )+  3 sinh(÷  )  ÷  cosh[÷  (  − )]+  4 sinh[÷  (  − )]  ÷  ÷  (  3 +  4) cosh(÷   )+(÷ 2 +  3  4) sinh(÷   )

for

< .

7.4.5-2. Representation of solutions to boundary value problems using the Green’s function. 1  . The solution of the first boundary value problem for equation (1) with the boundary conditions ù

=  1( ) ù

=  3 (í ) at

at í

= í 1, = 0, ù

ù

=  2( )

at í

=  4 (í ) at

* For unbounded domains, the condition of boundedness of the solution as   omitted.

= í 2, =

 

is set; in Table 25, this condition is

© 2002 by Chapman & Hall/CRC Page 531

532

ELLIPTIC EQUATIONS WITH TWO SPACE VARIABLES

is expressed in terms of the Green’s function as ù



(í , ) =  (í 1 ) +!



2

ì

 1 ( )  

0

(  , , , )  

  

=

−  ( 1



 3( )   (  , " , , ) # −!  

 1  =0     2  +! ! ( , ) ( , " , , ) .    1 0 $  

2

2)

!



 2 ( )  

0

(  , " , , )  

  

 4( )   (  , " , , ) #  

=  

1

=

2



2  . The solution of the second boundary value problem for equation (1) with boundary conditions 

=  1 (" )

 %  &

 =

at

=  3 ( ) at %

1,

" = 0,

=  2 (" )

 % 

%

 &

at

 =

=  4 ( ) at

" =

2,

is expressed in terms of the Green’s function as % ( , " ) = −  (

−!



1) 2

!

 0

 1 ( ) ( , " , 

1,

) +  ( 2 ) !



 3 ( ) ( , " , , 0) +!     1    2  +! ! ( , ) ( , " , , )   1 0 $  



2

 0

 2 ( ) ( , " , 

2,

)



 4 ( ) ( , " , ,  )    

1



.

3  . The solution of the third boundary value problem for equation (1) in terms of the Green’s function is represented in the same way as the solution of the second boundary value problem (the Green’s function is now different).

© 2002 by Chapman & Hall/CRC Page 532

Chapter 8

Elliptic Equations with Three or More Space Variables =0 3( The three-dimensional Laplace equation is often encountered in heat and mass transfer theory, fluid mechanics, elasticity, electrostatics, and other areas of mechanics and physics. For example, in heat and mass transfer theory, this equation describes stationary temperature distribution in the absence of heat sources and sinks in the domain under study. A regular solution of the Laplace equation is called a harmonic function. The first boundary value problem for the Laplace equation is often referred to as the Dirichlet problem, and the second boundary value problem, as the Neumann problem. Extremum principle: Given a domain ) , a harmonic function % in ) that is not identically constant in ) cannot attain its maximum or minimum value at any interior point of ) . 8.1. Laplace Equation

'

8.1.1. Problems in Cartesian Coordinates The three-dimensional Laplace equation in the rectangular Cartesian system of coordinates is written as 2

 



%

2

2

+  

"

%

2

+ 

2

 *

%

2

= 0.

8.1.1-1. Particular solutions and some relations. 1  . Particular solutions: +) , % ( , " , ) = +  + , " + * * 2 2 (  , " , ) = +  + , " − ( + +, ) 2+-  " +)  +. " , % * * * * % ( , " , ) = cos(/ 1  + / 2 " ) exp(0 / ), * * % ( , " , ) = sin(/ 1  + / 2 " ) exp(0 / ), * * % ( , " , ) = exp(/ 1  + / 2 " ) cos(/ + + ), * * % ( , " , ) = exp(0 /  ) cos(/ 1 " + + ) cos(/ 2 + , ), * * % ( , " , ) = cosh(/ 1  ) cosh(/ 2 " ) cos(/ + , ), * * % ( , " , ) = cosh(/ 1  ) sinh(/ 2 " ) cos(/ + , ), * * % ( , " , ) = cosh(/  ) cos(/ 1 " + + ) cos(/ 2 + , ), * * % ( , " , ) = sinh(/ 1  ) sinh(/ 2 " ) sin(/ + , ), * * % ( , " , ) = sinh(/  ) sin(/ 1 " + + ) sin(/ 2 + , ), * * where + , , , - , ) , . , / 1 , and / 2 are arbitrary constants, and / = 1 / 21 + / 22 . 2 2 . Fundamental solution:

3

( , " , ) = *

1 44 1 

2

+"

2

+ *

2

.

© 2002 by Chapman & Hall/CRC Page 533

534

ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES

3 2 . Suppose % = % ( , " , ) is a solution of the Laplace equation. Then the functions % %

1

%

*

= + % (0 5  + +6

6

2

=

3

= 9 : % 7

%

7

2

,

6

6

, *28 , 6

2

2

 −; :

+

0 5 " +-

1,

6"

,

6

" −< :

0 5  +-

2,

=1  2

2

6

−: =

, *

3 ),

+" 2

+

2

*

2,

:

8 ,

= 1 − 2( ;  + " + = ) + ( ; *

2

+
( $ − % )( & − % ) . % (% −/ ) ,





−1

% −& ) =  −1 # ( $ −% (% % )(−1) ; 1 < " < ) < ! , 0 ≤  < 2

( $ −1)( & −1) % −1

Coordinates ! , 3 , " , $ 45&  = 1+ ' %5/

−1

( ) =  1   (cos

(  ) = 1 cosh  (  + 12 )   (  ) =  1 cos(   = 0, 1, 2,  ;

cos  , sin  ,  =  −1 ' −! " ( ) ; ! >) >1, " < 0, 0≤  < 2

( $ −1)(1− & ) % −1

+#

−1

sinh cos  , sinh sin  ,  =  −1 sin  ;

≥ 0, −  ≤  ≤  , 0≤  < 2

Coordinates ! , " ,  , ( $ − % )( % − & )  =# % ( % −1) −#

=  −1 sin cos  , =  −1 sin sin  ,  =  −1 sinh  ; 0 ≤ ≤  ,  is any, 0 ≤  < 2 =  = 



3

Functions , ,  (equations for , ,  )

Transformations of coordinates 



541

+  (x) = 0

3


*

8.2.1-1. First boundary value problem. The solution of the first boundary value problem for the Poisson equation 3



+ 6 (r) = 0

(1)

in a domain 7 with the nonhomogeneous boundary condition 

= (r) for r 8 9

© 2002 by Chapman & Hall/CRC Page 541

542

ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES TABLE 28 The volume elements and distances occurring in relations (2) and (5) in some coordinate systems. In all cases, : = {; , < , = } Gradient, ∇&? (|i@ | = |i A | = |i B | = 1)

Coordinate system

Volume element, > 7 &

Cartesian r = { ,  ,  }

> ; > < > =

i @ C D @ + i A C DA + i B C D B

Cylindrical r = {E ,  ,  }

; > ; > < > =

i@ C D@ + i A 1@ C DA + i B C DB

Spherical ; r = {E , F ,  }

2

C

C

C

1 sin < > ; > < > = i@ C D @ + i A @ C DA + i B @

C

>

C

1 sin A

C DB C

Distance, = |r − : |

= ' (  − ; )2 + (  − < ) 2 + (  − = ) 2 >

C

C

C

>

= ' E 2 + ; 2 − 2E ; cos(  − < ) + (  − = )2

= ' E 2 + ; 2 − 2E ; cos G , where cos G = cos F cos < + sin F sin < cos(  − = ) >

can be represented in the form 

(r) = H I 6 ( : ) J (r, : ) > 7 & − H K ( : ) L L M

J &

> 9 &

.

(2)

Here, J (r, : ) is the Green’s function of the first boundary value problem, C N is the derivative C O P of the Green’s function with respect to ; , < , = along the outward normal N the boundary 9 of the domain 7 . Integration is everywhere with respect to ; , < , = . The volume elements in solution (2) for basic coordinate systems are presented in Table 28. In addition, the expressions of the gradients are given, which enable one to find the derivative along the normal in accordance with the formula C N = (N ⋅ ∇& J ). C O P The Green’s function J = J (r, : ) of the first boundary value problem is determined by the following conditions: 1 Q . The function J satisfies the Laplace equation with respect to  ,  ,  in the domain 7 everywhere except for the point (; , < , = ), at which it can have a singularity of the form 41R |r−1S | . 2 Q . The function J , with respect to  ,  ,  , satisfies the homogeneous boundary condition of the K first kind at the boundary, i.e., the condition J T = 0. The Green’s function can be represented as 1 1 (3) +? , 4 |r − : | where the auxiliary function ? = ? (r, : ) is determined by solvingK the first boundary value problem for the Laplace equation 3 ? = 0 with the boundary condition ? T = − 41R |r−1S | ; the vector quantity : in this problem is treated as a three-dimensional free parameter. The Green’s function possesses the symmetry property with respect to their arguments: J (r, : ) = J ( : , r). The construction of Green’s functions is discussed in Paragraphs 8.3.1-4 and 8.3.1-6 through 8.3.1-8 for , = 0. U VW X Y[Z \ ] For outer first boundary value problems for the Laplace equation, the following condition is usually set at infinity: |^ | <  ( |r| (|r| _ ` , a = const). (r, : ) =

J

8.2.1-2. Second boundary value problem. The second boundary value problem for the Poisson equation (1) is characterized by the boundary condition ^ L = b (r) for r 8 9 . L M

© 2002 by Chapman & Hall/CRC Page 542

+ e (x) = 0

543

(r) > r + H K b (r) > 9 = 0.

(4)

8.2. POISSON EQUATION c

3d

Necessary condition solvability of the inner problem: H I

6

The solution of the second boundary value problem can be written as ^

(r) = H I 6 ( : ) J (r, : ) > 7 & + H K b ( : ) J (r, : ) > 9 & + f ,

(5)

where f is an arbitrary constant, provided that the solvability condition is met. The Green’s function J = J (r, : ) of the second boundary value problem is determined by the following conditions: 1 Q . The function J satisfies the Laplace equation with respect to g , h , i in the domain 7 everywhere except for the point (; , < , = ) at which it has a singularity of the form 41R |r−1S | . 2 Q . The function J , with respect to g , h , i , satisfies the homogeneous condition of the second kind at the boundary, i.e., the condition j J jj 1 L , jK = where 9

0

9 0

L M

is the area of the surface 9 .

The Green’s function is unique up to an additive constant. U VW X Y[Z k ] The Green’s function cannot be identified with condition 1 Q and the homogeneous jK boundary condition C N j = 0; this problem for J has no solution, because, on representing J in the form (3), for ? weC O obtain a problem with a nonhomogeneous boundary condition of the second kind, for which the solvability condition (2) is not met. U VW X Y[Z l ] Condition (4) is not extended to the outer second boundary value problem (for infinite domain). 8.2.1-3. Third boundary value problem. The solution of the third boundary value problem for the Poisson equation (1) in a bounded domain 7 with the nonhomogeneous boundary condition L

^

+ m ^ = b (r) for r 8 9

L M

is given by relation (5) with f = 0, where J = J (r, : ) is the Green’s function of the third boundary value problem; the Green’s function is determined by the following conditions: 1 Q . The function J satisfies the Laplace equation with respect to g , h , i in 7 everywhere except for the point (; , < , = ) at which it has a singularity of the form 41R |r−1S | . 2 Q . The function J , with respect to g , h , i , satisfies the homogeneous boundary condition of the K third kind at the boundary, i.e., the condition n C N + m J o = 0. C O

The Green’s function can be represented in the form (3), where the auxiliary function ? is determined by solving the corresponding third boundary value problem for the Laplace equation p 3 q = 0. The construction of Green’s functions is discussed in Paragraphs 8.3.1-4 and 8.3.1-6 through 8.3.1-8 for r = 0.

sut

References for Subsection 8.2.1: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), N. S. Koshlyakov, E. B. Gliner, and M. M. Smirnov (1970).

© 2002 by Chapman & Hall/CRC Page 543

544

ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES

8.2.2. Problems in Cartesian Coordinates The three-dimensional Poisson equation in the rectangular Cartesian system of coordinates has the form 2

L L

8.2.2-1. Domain: − `

sut

g 2

(g , h , i ) =

2

+ L L

< g < ` , −`

Solution: ^

^

^

h 2

^

+ v (g , h , i ) = 0.

i 2 L

< h < ` , −`

1 4w x − y x − y x − y

Reference: R. Courant and D. Hilbert (1989). y y

8.2.2-2. Domain: − `

2

+ L

< g < ` , −`

Ì ≥ 0.

(Ì , Å ) = Ñ

An alternative representation of the Green’s function: Â

µ

Ç 1 Ç (¿ , À , Ì , Ã , Ä , Å ) = 4 ¹ =− Æ Æ º =− Î

·

where

¹ º · (1)

Æ

1

−

(1) ¹ º

·

1 (2) ¹ º

− ·

1 (3) ¹ º

1

+ ·

(4) ¹ º

− ·

1 (5) ¹ º

+ ·

1 (6) ¹ º

+ ·

1 (7) ¹ º

−

1 (8) · ¹ º ¶

,

Æ

= Ò ( ¿ − à − 2 ¼ )2 + ( À − Ä − 2 ž ) 2 + ( Ì − Å )2 , Í

Ï

Í

Ï

Í

Ï

Í

Ï

Í

Ï

Í

Ï

Í

Ï

Í

Ï

(2) · ¹ º

= Ò ( ¿ + à − 2 ¼ )2 + ( À − Ä − 2 ž ) 2 + ( Ì − Å )2 ,

(3) · ¹ º

= Ò ( ¿ − à − 2 ¼ )2 + ( À + Ä − 2 ž ) 2 + ( Ì − Å )2 ,

(4) · ¹ º

= Ò ( ¿ + à − 2 ¼ )2 + ( À + Ä − 2 ž ) 2 + ( Ì − Å )2 ,

(5) · ¹ º

= Ò ( ¿ − à − 2 ¼ )2 + ( À − Ä − 2 ž ) 2 + ( Ì + Å )2 ,

¹ º · (6)

= Ò ( ¿ + à − 2 ¼ )2 + ( À − Ä − 2 ž ) 2 + ( Ì + Å )2 ,

¹ º · (7)

= Ò ( ¿ − à − 2 ¼ )2 + ( À + Ä − 2 ž ) 2 + ( Ì + Å )2 ,

(8) · ¹ º

= Ò ( ¿ + à − 2 ¼ )2 + ( À + Ä − 2 ž ) 2 + ( Ì + Å )2 .

© 2002 by Chapman & Hall/CRC Page 550

8.2. POISSON EQUATION Ó

551

+ Õ (x) = 0

3Ô

8.2.2-14. Domain: 0 ≤ ¿ ≤ ¼ , 0 ≤ À ≤ ž , 0 ≤ Ì < Ö . Third boundary value problem. A semiinfinite cylindrical domain of a rectangular cross-section is considered. Boundary conditions « « are prescribed: × Ø « × Ø « × Ù « × Ú «

«

× Ù «

− » 1 = Á 1 (À , Ì ) at ¿ = 0, « − » 3 = Á 3 (¿ , Ì ) at À = 0, − » 5 = Á 5 (¿ , À ) at Ì = 0.

+» +»

2

«

= Á 2 (À , Ì ) at = Á 4 (¿ , Ì ) at

4

=¼ , À =ž , ¿

Green’s function: Â

Ç

(¿ , À , Ì , Ã , Ä , Å ) =

Ç

¹ º

(¿ , À ) ¹ º (Ã , Ä ) ª

¹ Æ =1 º Æ =1 Û

¹ º

«

2

Ë

¹ º

( Ì , Å ),

Û «

where

ª

Û

(¿ , À ) = (Ü ¹ cos Ü ¹ ¿ + » 1 sin Ü ¹ ¿ )( Ý º cos Ý º À + » 3 sin Ý º À ), 1 ( » 1 + » 2 )(Ü 2¹ + » 1 » 2 ) ( » 3 + » 4 )( Ý º2 + » 3 » 4 ) = (Ü 2¹ + » 12 )( Ý º2 + » 32 ) Þß¼ + Þ ß ž + , 4 (Ü 2¹ + » 12 )(Ü 2¹ + » 22 ) à ( Ý º2 + » 32 )( Ý º2 + » 42 ) à

ª

¹ º

¹ º

ª 2

È

È

È

È

exp(− ¹ º Ì ) å ¹ º cosh( ¹ º Å ) + » 5 sinh( ¹ º Å )æ È È for Ì > Å , ¹ º ( ¹ º + » 5) È È È È Ë ¹ º ( Ì , Å ) = âáâ exp(− ¹ º Å ) å ¹ º cosh( ¹ º Ì ) + » 5 sinh( ¹ º Ì )æ È È for Å > Ì , ¹ º ( ¹ º + » 5) âäâ Here, the Ü ¹ and Ý º are positive roots of the transcendental equations ã

1 + » 2 )Ü 2 Ü − » 1» 2

(»

tan(Ü ¼ ) =

tan( Ý ž ) =

,

3 + » 4)Ý 2 Ý − » 3» 4

(»

È

¹ º

= Ð Ü 2¹ + Ý º2 .

.

8.2.2-15. Domain: 0 ≤ ¿ ≤ ¼ , 0 ≤ À ≤ ž , 0 ≤ Ì < Ö . Mixed boundary value problems. 1 ç . A semiinfinite cylindrical domain of a rectangular cross-section is considered. « « conditions are prescribed: «

= Á 1 (À , Ì ) at

× Ú «

= Á 3 (¿ , Ì ) at = Á 5 (¿ , À ) at

«

= 0, ¿ À

= Á 2 (À , Ì ) at

= 0, Ì = 0.

= Á 4 (¿ , Ì ) at

Boundary

=¼ , ¿ À

=ž,

Green’s function: Â

(¿ , À , Ì , Ã , Ä , Å ) =

4 Ç ¼ ž É ¹

Ë

¹ º

Ç ¹ Æ =1 º Æ =1

= Í Î , ¼

(Ì , Å ) = Ñ

È

1

sin(É ¹ ¿ ) sin( Ê º À ) sin(É ¹ Ã ) sin( Ê º Ä ) Ë ¹ º ( Ì , Å ),

¹ º

= Ï Î ,

Ê º È

ž

È

¹ º

= Ð É 2¹ + Ê º2 ,

È

exp(−È ¹ º Ì ) cosh(È ¹ º Å ) for Ì > Å ≥ 0, exp(− ¹ º Å ) cosh( ¹ º Ì ) for Å > Ì ≥ 0.

2 ç . A semiinfinite cylindrical domain of a rectangular cross-section is considered. conditions are prescribed: × Ø « × Ø « × Ù « «

= Á 1 (À , Ì ) at ¿ = 0, = Á 3 (¿ , Ì ) at À = 0, = Á 5 (¿ , À ) at Ì = 0.

× Ù «

Boundary

= Á 2 (À , Ì ) at ¿ = ¼ , = Á 4 (¿ , Ì ) at À = ž ,

© 2002 by Chapman & Hall/CRC Page 551

552

ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES Green’s function: Â

(¿ , À , Ì , Ã , Ä , Å ) = É ¹

1 Ç ¼ ž

= Í Î , ¼

Ç

¹ È

º

¹ Æ =0 º Æ =0 è

È

= Ï Î ,

Ê º

cos(É ¹ ¿ ) cos( Ê º À ) cos(É ¹ Ã ) cos( Ê º Ä ) Ë ¹ º ( Ì , Å ),

¹ º è

= Ð É 2¹ + Ê 2º ,

¹ º

ž

= Ñ ¹

1 for = 0, Í 2 for ≠ 0,

È È exp(−È ¹ º Ì ) sinh(È ¹ º Å ) è for Ì > Å ≥ 0, Ë ¹ º (Ì , Å ) = Ñ exp(− ¹ º Å ) sinh( ¹ º Ì ) for Å > Ì ≥ 0.

Í

8.2.2-16. Domain: 0 ≤ ¿ ≤ ¼ , 0 ≤ À ≤ ž , 0 ≤ Ì ≤ é . First boundary value problem. A rectangular parallelepiped is considered. Boundary « « conditions are prescribed: « «

= Á 1 (À , Ì ) at = Á 3 (¿ , Ì ) at = Á 5 (¿ , À ) at

«

= 0, À = 0, Ì = 0, ¿

= Á 2 (À , Ì ) at = Á 4 (¿ , Ì ) at = Á 6 (¿ , À ) at «

=¼ , À =ž , Ì =é . ¿

1 ç . Representation of the Green’s function in the form of a double series: Â

4 Ç

(¿ , À , Ì , Ã , Ä , Å ) =

ê ¹ º

¼ ž

sin(É ¹ ¿ ) sin( Ê º À ) sin(É ¹ Ã ) sin( Ê º Ä ) ê ¹ º ( Ì , Å ),

¹ Æ =1 º Æ =1 È sinh( ¹ º Å È ¹ º È ¹ sinh( º Ì È ¹ º

ã

( Ì , Å ) = âáâ É ¹

Ç

ââä

= Î Í ,

Ê º

¼

È

) sinh[ È sinh( ¹ È ) sinh[ È sinh( ¹ = Î Ï

(é º é ) ¹ º (é º é ) È

,

ž

− Ì )]

¹ º

− Å )] ¹ º

for é ≥ Ì > Å ≥ 0, for é ≥ Å > Ì ≥ 0,

= Ð É 2¹ + Ê 2º .

This relation can be used to obtain two other representations of the Green’s function by means of the following cyclic permutations: (¿ , à , ¼ ) ì ë ( Ì , Å , é ) í î (À , Ä , ž ) 2 ç . Representation of the Green’s function in the form of a triple series: Â

(¿ , À , Ì , Ã , Ä , Å ) =

8 Ç ¼ žïé

Ç

Ç

¹ Æ =1 º Æ =1 ¸ Æ =1 É ¹

sin(É ¹ ¿ ) sin( Ê º À ) sin( ð ¸ Ì ) sin(É ¹ Ã ) sin( Ê º Ä ) sin( ð ¸ Å ) , É 2¹ + Ê 2º + ð 2¸

= Î Í ,

= Î Ï

Ê º

¼

ž

,

ð ¸

= Î

» é

.

3 ç . An alternative representation of the Green’s function in the form of a triple series: Â

Ç 1 Ç (¿ , À , Ì , Ã , Ä , Å ) = Æ 4 ¹ =− Æ º =−

¸ =− Æ

1

ò (1) ó ô õ

Æ

Î

Æ

ñ Ç

Æ

1 1 1 − ò (2) − ò (3) + ò (4) ó ô õ ó ô õ ó ô õ

1 1 1 1 − ò (5) + ò (6) + ò (7) − ò (8) , ó ô õ ó ô õ ó ô õ ó ô õ ö

© 2002 by Chapman & Hall/CRC Page 552

8.2. POISSON EQUATION Ó

553

+ Õ (x) = 0

3Ô

where ò (1) ó ô õ

= Ò ( ÷ − ø − 2 ¼ )2 + ( ù − ú − 2 ž )2 + ( Ì − û − 2 ü é )2 ,

ò (2) ó ô õ

= Ò ( ÷ + ø − 2 ¼ )2 + ( ù − ú − 2 ž )2 + ( Ì − û − 2 ü é )2 ,

ò (3) ó ô õ

= Ò ( ÷ − ø − 2 ¼ )2 + ( ù + ú − 2 ž )2 + ( Ì − û − 2 ü é )2 ,

ò (4) ó ô õ

= Ò ( ÷ + ø − 2 ¼ )2 + ( ù + ú − 2 ž )2 + ( Ì − û − 2 ü é )2 ,

ò (5) ó ô õ

= Ò ( ÷ − ø − 2 ¼ )2 + ( ù − ú − 2 ž )2 + ( Ì + û − 2 ü é )2 ,

ò (6) ó ô õ

= Ò ( ÷ + ø − 2 ¼ )2 + ( ù − ú − 2 ž )2 + ( Ì + û − 2 ü é )2 ,

ò (7) ó ô õ

= Ò ( ÷ − ø − 2 ¼ )2 + ( ù + ú − 2 ž )2 + ( Ì + û − 2 ü é )2 ,

ò (8) ó ô õ

= Ò ( ÷ + ø − 2 ¼ )2 + ( ù + ú − 2 ž )2 + ( Ì + û − 2 ü é )2 .

Í

Ï

Í

Ï

Í

Ï

Í

Ï

Í

Ï

Í

Ï

Í

Ï

Í

Ï

8.2.2-17. Domain: 0 ≤ ÷ ≤ ¼ , 0 ≤ ù ≤ ž , 0 ≤ Ì ≤ é . Third boundary value problem. A rectangular parallelepiped is considered. Boundary conditions are prescribed: × Ø ý × Ù ý

ý

× Ø ý

− ü 1 ý = þ 1 (ù , Ì ) at − ü 3 ý = þ 3 (÷ , Ì ) at − ü 5 = þ 5 (÷ , ù ) at

× Úý

ý

× Ù ý

= 0, ù = 0, Ì = 0, ÷

+ ü 2 ý = þ 2 (ù , Ì ) at + ü 4 ý = þ 4 (÷ , Ì ) at + ü 6 = þ 6 (÷ , ù ) at

× Úý

=¼ , ù =ž , Ì =é . ÷

Green’s function: ÿ


(÷ , ù , Ì , ø , ú , û ) =




ó 






ó =1 ô =1 =1

(÷ ) = cos( ó ÷ ) + ó







(ù ) = cos( ô

ü 3 ô

)+

ô ù

ü 1 ó







( ) = cos( 



 

)+

ü 5

where the ó , ô , and 



tan(

) 







ó





2 





sin(

sin(



ô ù

),





ô







2

=





), 

2 





=



ü 6

2





2







2 







2 1 2 2

+ü 2ó + ü







2ó 

ü 4 2 2ô













ü 2 2 2ó

=









2ô

+ü 2ô + ü

+ü 2+ü

2 5 2 6



ü 1 2 2ó

+

+



2 3 2 4

ü 3 2 2ô

+

+

2

2



2

ñ

2

2

2ô



ü 52

1+







2

ö

, ö

ü 32

1+

ñ

+

2ó



ž

+

ü 12

1+





ü 5





ñ

ö

,

,

are positive roots of the transcendental equations +ü 2 , − ü 1ü 2

tan( ž )

ü 1

=



sin( ó ÷ ),











(÷ ) ó (ø ) ô (ù ) ô (ú ) ( ) ( û ) , 2 ó 2 ô ( û ) 2 ( 2ó + 2ô + 2 ) 



2

8.2.2-18. Domain: 0 ≤ ÷ ≤ , 0 ≤ ù ≤ ž , 0 ≤ 

2



+ü 4 , − ü 3ü 4

ü 3

=





tan( )



+ü 6 . − ü 5ü 6

ü 5

=





2

≤ . Mixed boundary value problem.

A rectangular parallelepiped is considered. Boundary conditions are prescribed: ý

× Ø ý

ý

= þ 1 (ù , ) at ý = þ 3 (÷ , ) at = þ 5 (÷ , ù ) at 



= 0, ù = 0, = 0, ÷



×

ý

= þ 2 (ù , ) at = þ 4 (÷ , ) at = þ 6 (÷ , ù ) at 

×

ý





= , ù =ž , = . ÷







© 2002 by Chapman & Hall/CRC Page 553

554 ý

ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES ÿ

Solution: (÷ , ù , ) = 

+ 













,û )





(÷ , ù , , ø , ú , û )

×ø



, ) × 









( , , , , , ) 



0

, ) 

ø 





















 

+ 





( , , , , , ) 

 











 









2( 

0

0





4( 























, ) ( , , , ,ž , ) 

























6( 



0





0



, ) ( , , , , , ) 





0

+ 

=0 



+ 

=0 





=0

× 5( 

ú 

× ×



0

(× ø ,ÿ ú , û ) (÷ , ù , , ø , ú , û ) û 

0

3( 

0





þ 1 (ú



0





0

0



+





0

+



0

0

, ) ( , , , , , ) 























.



1 . A double-series representation of the Green’s function: !

#

4

( , , , , , )= 














$#


sin( &

ž

#

=0 "



% 

#

( , )= %



% ,

sinh(



() )

% ,

#



% ,

&

%

#

#

) sin( 

≥

for





%

#

) sin( 

( − )] ) ( − )] ) + 1) ,

# ,

,

#

% '

#

) cosh[ cosh( ) cosh[ cosh( (2 =

# ,

*

ê

) sin( 

=0 "

%

sinh( #

#

% '

)ê 

> 

( , ), %





≥ 0, 





%

,

≥

for





%

,

#

> 

≥ 0, 

#

(2 + 1) 2 + 2 . , = 2 2ž This relation can be used to obtain two other representations of the Green’s function by means of the following cyclic permutations: ( , , ) ) )+

=

&

.



/

-

%

'

-

%

,

0

&

%

'







1

( , , ) 

2 



3

( , ,ž ) 4







2 . A triple series representation of the Green’s function: 5

#

!



8

( , , , , , )= 












$#


&

ž 

= &

=0

"

#

sin(


65

=0

" %

=0

"

) sin( #

% '



7

2 &

5

(2 + 1) , 2

(2

=

.

-

) sin( 

% '

+ 1) /

2ž -

,

5 #

) sin( + 2 + 

%

'

= 7

) sin(

5

&



% '



) sin(

2 7

7



)

,

(2 + 1) . 2 8

-



8.2.3. Problems in Cylindrical Coordinates The three-dimensional Poisson equation in the cylindrical coordinate system is written as 1

1

9

9

:

9

;

:

:

< 9

+ =

9

2

:

2 :

+


0. This equation governs mass transfer phenomena with volume chemical reaction of the first order for " < 0. Any elliptic equation with constant coefficients can be reduced to the Helmholtz equation.

8.3.1. General Remarks, Results, and Formulas 8.3.1-1. Some definitions. The Helmholtz equation is called homogeneous if # = 0 and nonhomogeneous if # ≠ 0. A homogeneous boundary value problem is a boundary value problem for a homogeneous equation with homogeneous boundary conditions; = 0 is a particular solution of a homogeneous boundary value problem. The values "  of the parameter " for which there are nontrivial solutions (i.e., not identically zero solutions) of a homogeneous boundary value problem are called eigenvalues. The corresponding solutions, =  , are called eigenfunctions of this boundary value problem. In what follows, we consider simultaneously the first, second, and third boundary value problems for the three-dimensional Helmholtz equation in a finite three-dimensional domain $ with a sufficiently smooth surface % . It is assumed that  > 0 for the third boundary value problem with the boundary condition & &

where

) * ) +

'



+

= 0 for r (

%

,

is the derivative along the outward normal to the surface % , and r = { ,  ,  }.

© 2002 by Chapman & Hall/CRC Page 561

562

ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES

8.3.1-2. Properties of eigenvalues and eigenfunctions. 1 , . There are infinitely many eigenvalues { " value problem.

}; they form a discrete spectrum of the boundary 

2 , . All eigenvalues are positive, except for one eigenvalue " 0 = 0 of the second boundary value problem (the corresponding eigenfunction is 0 = const). The eigenvalues are assumed to be ordered so that " 1 < " 2 < " 3 < -.-.- . 3 , . The eigenvalues tend to infinity as the number  increases. The following asymptotic estimate holds:  $ 3 , lim 3 1 2 =  / 0 2 6   " where $ 3 is the volume of the domain under consideration. 4 , . The eigenfunctions are defined up to a constant multiplier. Any two eigenfunctions,

that correspond to different eigenvalues "  ≠ " 2 are orthogonal, that is, 3

and



2

,

4





2

5

= 0. $

5 , . Any twice continuously differentiable function  =  (r) that satisfies the boundary conditions of a boundary value problem can be expanded into a uniformly convergent series in the eigenfunctions of this boundary value problem, specifically, = 

0







=1



,

where

= 

6

1 7



6

3 7

4



2

5 

7

, $



3 7



2

4

=

2

5

. $

If  is square summable, then the series is convergent in mean. 6 , . The eigenvalues of the first boundary value problem do not increase if the domain is extended. 8 9;: < =?> @ A In a three-dimensional problem, to each eigenvalue "  finitely many linearly inde pendent eigenfunctions (1) , B.B.B , ( C ) generally correspond. These functions can always be replaced by their linear combinations ¯ ( D ) =



(1) 



,1

E D

(1) 

+

+E

-.-.-

D ,D

−1

( D −1)

( D )

+

,

= 1, 2, F

B.B.B

,G ,

( C )



such that ¯ , B.B.B , ¯ are now pairwise orthogonal. Therefore, without loss of generality, we assume that all eigenfunctions are orthogonal.



Reference: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1984).

8.3.1-3. Nonhomogeneous Helmholtz equation with homogeneous boundary conditions. Three cases are possible. 1 , . If " is not equal to any one of the eigenvalues, then the solution of the problem is given by =

0



 

=1

 "

H

−"



, 

where

= 

1 7

H



2 , . If " coincides with one of the eigenvalues, the function # to the eigenfunction 2 , 3

3 7

#

2

= "

4

2 "



5

7

, $



7 

3

2

4

=

2

5 $

.

, then the condition of the orthogonality of

4

#



2

5

= 0, $

is a necessary condition for a solution of the nonhomogeneous problem to exist. The solution is then given by 2



=

−1  

=1

"

  H

−" 2



+

 

0

=

 2

where I is an arbitrary constant and

+1 7



 " 

7

H

−" 2

=

2



J

4





+I 2 5 $



2

.

, 

= 7



1 

7

3

2

4 #



5 $

,

H

© 2002 by Chapman & Hall/CRC Page 562

8.3. HELMHOLTZ EQUATION 4

+

3


563

= −  (x)


3 , . If " = " 2 and J # 2 5 $ ≠ 0, then the boundary value problem for the nonhomogeneous equation has no solution. 8 9;: < =?> K A If to each eigenvalue "  there are corresponding G  mutually orthogonal eigenfunctions ( D ) ( F = 1, B.B.B , G  ), then the solution is written as

=

0  

C.L 

=1 D

=1

( D ) "

provided that " ≠



H

−" 

,

( D )

where

= 7

3

1

4

( D ) 2

( D )

#

7

5

7

, $

( D ) 7 2

3 4

=

M

( D ) N 2 5

$

,

H

. 

"

( D )

Reference: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1984).

8.3.1-4. Solution of nonhomogeneous boundary value problems of general form. 1 , . The solution of the first boundary value problem for the Helmholtz equation with the boundary condition

=  (r) for r ( % can be represented in the form 3

4

3

(r) =

( O ) (r, O ) 5 #

$

& Q

− P

(O ) 

&

' P

(r, O ) 5 %

P

.

(1)

Here, r = { ,  ,  }, O = { ,  ,  } (r ( $ , O ( $ ); ) denotes the derivative along the outward ) + R normal to the surface % with respect to ,  ,  . The Green’s function is given by the series

0 

(r, O ) =

7



where the  and value problem. "





=1





(r)

7 2("

 

(O ) , −" ) "

≠ "



,

(2)

are the eigenfunctions and eigenvalues of the homogeneous first boundary

2 , . The solution of the second boundary value problem with the boundary condition &

&

'



=  (r) for r (

%

can be represented in the form 3

(r) =

4

3

#



( O ) (r, O )

5 $

P

Q

+ 

( O ) (r, O ) 5 %

P

.

(3)

Here, the Green’s function is given by the series

(r, O ) = − $

1 3"

+

0 





7

=1





(r)

7 2("

 

(O ) , −" )

(4)

where $ 3 is the volume of the three-dimensional domain under consideration, and the "  and  are the positive eigenvalues and corresponding eigenfunctions of the homogeneous second boundary value problem. For clarity, the term corresponding to the zero eigenvalue " 0 = 0 ( 0 = const) is singled out in (4). It is assumed that " ≠ 0 and " ≠ "  . 3 , . The solution of the third boundary value problem for the Helmholtz equation with the boundary & condition

& ' +  =  (r) for r ( % is given by relation (3) in which the Green’s function is defined by series (2) with the eigenfunctions  and eigenvalues "  of the homogeneous third boundary value problem.

© 2002 by Chapman & Hall/CRC Page 563

564

ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES

boundary conditions of various types be set on different portions 4 , . Let nonhomogeneous 2 the surface % = S

T %

=1

, S

%

S

of

U S

[ ] =

(r) for r

 S

(

%

. S

Then the solution of the corresponding mixed boundary value problem can be written as 3

2

4

(r) =

( O ) (r, O ) #

5 $

+ P

W

Q V

( O ) W S (r, O ) 5

 S S

where

3



% P

(S )

,

=1

S

&

(r, O ) =

Y[Z X

− \

&

' P

(r, O ) \

if a first-kind boundary condition is set on % S ,

(r, O )

if a second- or third-kind boundary condition is set on % S .

The Green’s function is expressed by series (2) that involves the eigenfunctions ] ues " ^ of the homogeneous mixed boundary value problem.

and eigenval^

8.3.1-5. Boundary conditions at infinity in the case of an unbounded domain. Below it is assumed that the function # is finite or sufficiently rapidly decaying as _ `

. a

1 , . If " < 0 and the domain is unbounded, the additional condition that the solution must vanish at infinity is set: ] ` 0 as _ ` a . 2 , . If " > 0, the radiation conditions (Sommerfeld conditions) are often used at infinity. In threedimensional problems, these conditions are expressed as lim0

= const,

_ ]

b;/

lim0

b;/

_

c

] d d

_

+ egf h

]

= 0, i

where e 2 = −1. The principle of limit absorption and the principle of limit amplitude are also employed to separate a single solution.

j

Reference: A. N. Tikhonov and A. A. Samarskii (1990).

8.3.1-6. Green’s function for an infinite cylindrical domain of arbitrary cross-section. Consider the three-dimensional Helmholtz equation k

+h

3]

= − l (r) ]

(5)

inside an infinite cylindrical domain m = {(n , o ) p q , − a < r < a } with arbitrary cross-section q . On the surface of this domain, let s = {(n , o ) p t , − a < r < a }, where t is the boundary of q , the homogeneous boundary condition of general form u

' d

+w v

d

v

= 0 for r p

(6) s

be set, with u w ≥ 0. By appropriately choosing the constants u and w in (6), one can obtain boundary conditions of the first ( u = 0, w = 1), second ( u = 1, w = 0), and third ( u w ≠ 0) kind. The Green’s function of the first or third boundary value problem can be represented in the ~ ~ form* x ~ ~ 1 (n , o , r , y , z , { ) = 2

~

 |

=1

~

(n , o )

}~

€



€

2 f

€

‚

(y , z ) −h ƒ

−f

* In Paragraphs 8.3.1-6 through 8.3.1-8, the cross-section

„ …

− † |‡ −ˆ |

,



Œ

€



2

= ‰

2 Š

(n , o ) ‹ n

‹ o

,

(7)

€

is assumed to have finite dimensions.

© 2002 by Chapman & Hall/CRC Page 564

8.3. HELMHOLTZ EQUATION ~

~



+

3Ž

565

= −  (x) Ž

where the ‚ and are the eigenvalues and eigenfunctions of the corresponding two-dimensional boundary value problem in q , €

k

u ~

+‚

2 ‘ ‘

+w

’

€ €

€

=0

for (n , o ) p

q

=0

for (n , o ) p

t

,

€

Recall that all ‚ are positive. In the second boundary value problem, the zero eigenvalue summation in (7) must start with “ = 0. In this case, 0 = 1 and of the cross-section q . j

(8)

.

€

‚

0

0

= 0 appears, and hence the = q 2 , where q 2 is the area

2

€

References: B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980), A. N. Tikhonov and A. A. Samarskii (1990).

8.3.1-7. Green’s function for a semiinfinite cylindrical domain. 1 ” . The Green’s function of the three-dimensional first boundary value problem for equation (5) in a semiinfinite cylindrical domain m = {(n , o ) p q , 0 ≤ r < • } with arbitrary cross-section q is ~ ~ x given by ~ (n , o , r , y , z , { ) = where

~

(n , o )

}~



|

=1 €



(y , z ) 2

( r , { ), –

(9)

€ €

~

–

~

1

(r , { ) =

˜

~

| r − { | ) − exp(−—

exp(−—

2—

~

—

exp(−—

1

š›

exp(−—

— ›

) sinh(— r

~

~

| r + { | )™

~

~

1

œ

=

~

{

) for {

~

) sinh(—

) for r

> r

> {

~

≥ 0, {

= —

≥ 0, r

~ ž

(10) −Ÿ .

‚

~ ~

Relations (9) and (10) involve the eigenfunctions and eigenvalues first boundary value problem (8) with u = 0 and w = 1.

of the two-dimensional ‚

€

2 ” . The Green’s function of the three-dimensional second boundary value problem for equation (5) in a semiinfinite cylindrical domain m = {(n , o ) p q , 0 ≤ r < • } with arbitrary cross-section q is ~ ~ x given by ~ 1

(n , o , r , y , z , { ) = where

2 q

–

0 (r

,{ )+

~

(n , o )

}~



|

=1 €



(y , z ) 2

( r , { ), –

(11)

€ €

–

~

=

~

1

(r , { ) =

˜

~

| r − { | ) + exp(−—

exp(−—

2—

~

1 —

exp(−—

~

1

š› —

exp(−—

| r + { | )™

~

~

œ

›

~

~

) cosh(— r

{

) cosh(—

{ ~ r

) for ) for ~

> r

{

> r

{

~

≥ 0, ≥ 0,

~

= —

ž ‚

(12) −Ÿ .

~

Relations (11) and (12) involve the eigenfunctions and eigenvalues ‚ of the two-dimensional second boundary value problem (8) with u = 1 and w = 0. Note that in (11) the term corresponding € to the zero eigenvalue ‚ 0 = 0 is specially singled out; q 2 is the area of the cross-section q .

© 2002 by Chapman & Hall/CRC Page 565

566

ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES

3 ” . The Green’s function of the three-dimensional third boundary value problem for equation (5) with the boundary conditions ‘ ‘

v

−w r

1

= 0 for v

r

= 0, ‘

in a semiinfinite cylindrical domain m = {(n , o ) p and lateral surface s is given by relation (9) with ~ ~ ~ exp(−—

~

(r , { ) = –

œ š› ›

exp(−—

~

) r

—

{

cosh(— { ) + w ~ (— + w 1 ) ~ ~ cosh(— r ) + w — (— + w 1 ) ~

˜

~

2

’

, 0≤ q

< r

1

sinh(—

1

sinh(—

)™ {

~

—

› ›

p

s

} with arbitrary cross-section •

for

—

)

= 0 for r v

q

~

˜

~

+w v

‘

~

≥ 0, {

~

= —

)™ r

> r

for {

>

ž ‚

−Ÿ .

≥ 0, r

~ ~

Relations (9) and (13) involve the eigenfunctions and eigenvalues third boundary value problem (8) with u = 1 and w = w 2 .

(13) of the two-dimensional

‚

€

4 ” . The Green’s function of the three-dimensional mixed boundary value~ problem for equation (5) ~ with a second-kind boundary condition at the end face and a first-kind boundary condition at the lateral surface is given by relations (9) and (12), where the ‚ and are the eigenvalues and eigenfunctions of the two-dimensional first boundary value problem (8) with u = 0 and w = 1. € The Green’s functions of other mixed boundary value problems can be constructed likewise. 8.3.1-8. Green’s function for a cylindrical domain of finite dimensions. 1 ” . The Green’s function of the three-dimensional first boundary value problem for equation (5) in a cylindrical domain of finite dimensions m = {(n , o ) p q , 0 ≤ r ≤   } with arbitrary cross-section q ~ is given by relation (9) with~ ~

sinh(— ~ œ

(r , { ) = –

— š› ›

~

) sinh[— (   − r )] ~ sinh(—   ) ~ ) sinh[— (   − { )] sinh(—   ) {

~ ~

sinh(—

r

— › ›

for  

≥

for

 

≥

> r

~ ~

≥ 0, {

= —

~

> r ≥ 0, {

ž

−Ÿ . ‚

(14)

~

and eigenvalues Relations (9) and (14) involve the eigenfunctions first boundary value problem (8) with u = 0 and w = 1. € Another representation of the Green’s~ function: ~

of the two-dimensional ‚

x

2

(n , o , r , y , z , { ) =

 

}~

~

(n , o ) }

|



=1

¡ |

=1 €

€

~

(y , z ) sin( ¢ ¡ r ) sin( ¢  2( + ¢ 2¡ − Ÿ ) ‚ ¡

) {

,

=

¢ ¡

£  

¤

.

€

It is a consequence of formula (2).

2 ” . The Green’s function of the three-dimensional second boundary value problem for equation (5) in a cylindrical domain of finite dimensions ¥ = {(¦ , § ) ¨ © , 0 ≤ ª ≤   } with arbitrary cross-section © ~ ~ is given by relation (11) with

–

~

cosh(— ~

(ª , { ) =

œ — ›

š›

~

cosh(— › ›

{

~

—

ª

~

) cosh[— sinh(— ~ ) cosh[— sinh(— ~

 

 

(   − ª )] ) (   − { )] )

for  

≥

for

 

≥

> ª

{

~

≥ 0, —

~

{

> ª

≥ 0,

Relations (11) and (15) involve the eigenfunctions and eigenvalues ‚ second boundary value problem (8) with « = 1 and ¬ = 0.

~

= ž

‚

−Ÿ .

(15)

~

of the two-dimensional

€

© 2002 by Chapman & Hall/CRC Page 566

8.3. HELMHOLTZ EQUATION 

+

3Ž

Another representation of the Green’s function: ~ x (¦ , § , ª , y , z , { ) = =

¢ ¡

£  

1

~ ~

(¦ , § ) ¡

~



|

, ¤

}

 

¡ |

=0

= ¡

®

=0 ­

€

 €

1 for 2 for

­

It is a consequence of formula (4).

¯

}~

567

= −  (x) Ž

(y , z ) cos( ¢ ¡ ª ) cos( ¢ 2( + ¢ 2¡ − Ÿ ) ‚

= € 0, ≠ 0, ¤ ¤

= 0,

0 ‚

¡

) {

,

= 1.

0 €

Reference: B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

3 ” . The Green’s function of the three-dimensional third boundary value problem for equation (5) with the boundary conditions ‘

−¬

‘

° ª

= 0 at

1°

= 0, ª

‘

+¬

° ª ‘

= 0 at

2°

[— –

²

(ª , ³ ) =

cosh(— ²

› ›

[—

²

cosh(— ²

› ›

ª ²

— ›

)+ ¬ 1 sinh(— ² ³ )] ´ — [— ² ( ¬ 1 + ¬ 2) cosh(— )+ ¬ 1 sinh(— ² ª )] ´ — [— ² ( ¬ 1 + ¬ 2) cosh(— ³

² —

š›

²

—

= ²

ž

¶

+¬ °

’

= 0 for r

3°

² ²

(0 ≤

≤  , 0≤

ª

±

³

>³ ,

for

  ²

¨

, 0 ≤ ª ≤   } with arbitrary cross-section © ©

cosh[— ² (   − ª )]+ ¬ 2 sinh[— ² (   − ª )] µ )+(— 2 + ¬ 1 ¬ 2) sinh(— ²   )] ² cosh[— ² (   − ³ )]+ ¬ 2 sinh[— ² (   − ³ )] µ 2 + ¬ ¬ ) sinh( —   )+( —   )] 1 2 ² ² ²

−Ÿ ²

‘ ‘

in a cylindrical domain of finite dimensions ¥ = {(¦ , § ) ¨ and lateral surface ± is given by relation (9) with œ

=  , ª

ª

(16) for ª

ª

³

≥ 0, —

³

> ª

≥ 0,

²

= ž

¶ ²

−Ÿ .

(17)

Relations (9) and (17) involve the eigenfunctions · ² and eigenvalues ¶ ² of the two-dimensional first boundary value problem (8) with « = 0 and ¬ = 1. The Green’s functions of other mixed boundary value problems can be constructed likewise.

8.3.2. Problems in Cartesian Coordinates The three-dimensional nonhomogeneous Helmholtz equation in the rectangular Cartesian system of coordinates has the form 2 ‘ ‘

¦

°

2

+

2 ‘ ‘

§

°

2

+

2 ‘ ‘

ª

°

2

+Ÿ °

= − ¹ (¦ , § , ª ).

© 2002 by Chapman & Hall/CRC Page 567

568

ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES

8.3.2-1. Particular solutions of the homogeneous equation ( ¹ ≡ 0): = (º °

1

+º

cos ¬ ¦

2 sin

)( »

¬

¦

1 cos

+» §

¤

°

= (º

1

cos ¬ ¦

+º

2 sin ¬

¦

)( »

1 cosh

°

= (º

1

cos ¬ ¦

+º

2 sin ¬

¦

)( »

1 cos

°

= (º

1

cosh ¬ ¦

+º

2 sinh ¬

¦

)( »

1 cos

°

= (º

1

cosh ¬ ¦

+º

2 sinh ¬

¦

)( »

1 cosh

= (º

1

cosh ¬

+º

2

)( »

1 cosh

°

¦

sinh ¬ ¦

where º 1 , º 2 , » 1 , » 2 , ¼ 1 , and ¼ Fundamental solutions:½

ž

¾

¦

2

+§

< 2

1 ” . Solution for Ÿ = − ¬ 1 4

(¦ , § , ª ) =

− Â

2 Í . Solution for Î = °

, −•

< ¦

•

Ç

Ã

2 Ã

− Ã

+»

2 sinh

1 4

)( ¼ § ¤ §

¤

= Ÿ

2 ),

+¼ ª

1 cos “

2 ¬

+

= Ÿ

+¼ ª

)( ¼

1 cos “

)( ¼

1 cosh “

2 ¤

2 ¬

2 sin “

ª

; 2

− ¤

), ª

2 sin “

ª

4

Ÿ

= −¬

1

exp(¿

Ÿ

=

¾

£

£

exp(− ¬ ), ¾

ÀÁ¬

0, À 2 = −1.

2

8.3.2-2. Domain: − •

°

+»

)( ¼

§ ¤

)( ¼ §

¤

+¼

1ª

are arbitrary constants.

2

(¦ , § , ª ) = where =

2 sin

§ ¤

)( ¼ §

¤

2 sinh

+» §

¤

(¦ , § , ª ) =

½

+» §

¤

2 sin

Ì

Å Ì

Ä Ì

Å

. ³

Ì

Ì

³

.

This solution was obtained taking into account the radiation condition at infinity (see Paragraph à à à 8.3.1-5, Item 2 Í ).

ÐÑ

Reference: A. N. Tikhonov and A. A. Samarskii (1990).

8.3.2-3. Domain: − Ò

< É


, ) A triple series representation of the Green’s function: 

1  (ÿ , / , 0 ,  , > ,  ) = ? 

1 &

 =0

&

  8&

=0 =0

@

 @

8 cos(  @

9    = ? ,

(*)

 

=

ÿ

) cos(   / ) cos( ' 8 0 ) cos(   ) cos(   > ) cos( ' 8  ) , 7  2 +  2 + ' 28 − 9  

,

9 ' 8 = : . 1

References: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964), B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

 8.3.2-19. Domain: 0 ≤ ÿ ≤ ? , 0 ≤ / ≤ , 0 ≤ 0 ≤ 1 . Third boundary value problem.

A rectangular parallelepiped is considered. Boundary conditions are prescribed: 2 3  2 E 

− 1  = 4 1 (/ , 0 ) at : − 3  = 4 3 (ÿ , 0 ) at : 2 5  − 5  = 4 5 (ÿ , / ) at :

ÿ

= 0, / = 0, 0 = 0,

2 3  2 E 

+ 2  = 4 2 (/ , 0 ) at : + 4  = 4 4 (ÿ , 0 ) at : 2 5  + 6  = 4 6 (ÿ , / ) at :

ÿ

=? ,  / = , 0 =1 .

© 2002 by Chapman & Hall/CRC Page 577

578

ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES Eigenvalues of the homogeneous problem:

7   F = G 2 + H 2 + I F2 ;

Here, the G



,H



F

, and I

are positive roots of the transcendental equations 1+ 2−

( tan(G ? ) = : G

Eigenfunctions:

2 )G

:

1 2

:

N = !

 ( 3 + 4)H tan( H ) = : 2 : , H − 3 4

,

:

1    F = K  L  M F (G N cos G N K

 ,  , J = 1, 2, 3, 

:

+ ÿ

G 2N +

sin G N

1

:

L

2 1,

:

( tan( I 1 ) = : I

:

)( H O cos H O / + ÿ

= ! H O2 + O

3

I F2 + :

The square of the norm of an eigenfunction is defined as ;QP (*)

N O

F ;

2

: :

6 )I 5 6

.

:

sin H O / )( I F cos I F 0 +

M F = !

2 3,

:

:

5+ 2−

:

5

sin I F 0 ),

2 5.

1 ? ( 1 + 2 )(G 2N + 1 2 ) ( 3 + 4 )( H O2 + 3 4 ) ( 5 + 6 )( I F2 + 5 6 ) 1 + : F2 : 2 F2 : : 2 + : 2 : 2 2 : :2 + : 2 : 2 2 : :2 . 8 R (G N + 1 )(G N + 2 ) S RUT ( H O + 3 )( H O + 4 ) S R ( I + 5 )( I + 6 ) S

=

:

:

:

:

Reference: B. M. Budak, A. A. Samarskii, and A. N. Tikhonov (1980).

:

:

8.3.2-20. Domain: 0 ≤ ÿ ≤ ? , 0 ≤ / ≤ , 0 ≤ 0 ≤ 1 . Mixed boundary value problems. T

1 6 . A rectangular parallelepiped is considered. Boundary conditions are prescribed: P

= 4 1 (/ , 0 ) at P

= 4 3 (ÿ , 0 ) at = 4 5 (ÿ , / ) at

2 5 P

P

= 0, ÿ

= 4 2 (/ , 0 ) at P

/ = 0,

= 4 4 (ÿ , 0 ) at = 4 6 (ÿ , / ) at

2 5 P

0 = 0,

=? , ÿ

/ = , T 0 =1 .

Eigenvalues of the homogeneous problem: 7

N O 8 =9

2

2

V W ?

2

Eigenfunctions:

2

+ X

2

T

P

2

+ :

2

1

Y ;

9 N O 8 = sin V W ?

= 1, 2, 3, ZZZ ;

ÿ

W ,X 9

V Y sin

/ X T

V Y cos

9

0 :

= 0, 1, 2, ZZZ :

Y .

1

The square of the norm of an eigenfunction is defined as ;QP

8 ;

?

1

1 for = 0, (1 + < 8 0 ), < 8 0= = : 8 0 for ≠ 0. : 2 6 . A rectangular parallelepiped is considered. Boundary conditions are prescribed: N O

2

=

T

P

= 4 1 (/ , 0 ) at 2 E P = 4 3 (ÿ , 0 ) at 2 5 P = 4 5 (ÿ , / ) at ÿ

P

= 0, / = 0, 0 = 0,

= 4 2 (/ , 0 ) at = 4 4 (ÿ , 0 ) at 2 5 P = 4 6 (ÿ , / ) at ÿ

2 E P

= / = 0 =

? ,

, 1 . T

Eigenvalues of the homogeneous problem: 7

N O 8 =9

2

V W

Eigenfunctions:

2

?

2

P

2

+ X

2

T

+ : 1

2 2

W = 1, 2, 3, ZZZ ;

Y ;

9 N O 8 = sin V W ? ÿ

V Y cos

9

/ X T

V Y cos

, X

9

0 : 1

:

= 0, 1, 2, ZZZ

Y .

The square of the norm of an eigenfunction is defined as ;QP

; N O 8

2

=

?

T

8

1

(1 + s

)( n cosh s p + o sinh s p ),

X

+ l sin

X c

2

X c

j

c

,

2

,

j > −s

)( n cosh s p + o sinh s p ),

j > −s

)( n cos s p + o sin s p ),

j <s

)( n cos s p + o sin s p ),

X

2

2

2

,

2

,

,

2

<s ,

g r

where = 0, 1, 2, ZZZ ; t , l , n , o , n 1 ,r n 2 , o 1 , o 2r , and s are arbitrary constants; the (v ) and r X m (v ) are the Bessel functions; and the u (v ) and w (v ) are the modified Bessel functions. b

8.3.3-2. Domain: 0 ≤

≤x ,0≤

c

≤ 2y , − z

< p < z . First boundary value problem.

An infinite circular cylinder is considered. A boundary condition is prescribed: b

c

q

= { ( , p ) at

=x .

Solution: b q ( ,c , ) =−x p

+|

2}

| …

0

0

|

b |

{ (~ ,  ) € 

−&

2} 0



|& ‡

(v , ~ ,  ) (‰ , Š , p , v , ~ ,  )v ‚

−

Here,

‡

Œ

v ‚

c

( , , p , v , ~ ,  )ƒ „

Œ

ˆ

†

=…





†

† ~

~

†

†

v .

Œ

Œ Œ Œ r g r k Œ Œ  ‹ Œ ‹ Œ 1 t (s ‰ ) (s v ) ‘ r ‡ ‡ (‰ , Š , p , v , ~ ,  ) = cos[ ’ ( Š − ~ )] exp − | −  | , Žg  h p r r r 2‘ ‚ 2y x 2Œ (s x Œ ) =0 =1 Œ j r ‘ r 1 for ’ = 0, =“ s 2 − , t =” 2 for ’ ≠ 0, Œ Œ Œ g r where the (v ) are the Bessel functions and the s are positive roots of the transcendental equation g (s x ) = 0. •*– g

Reference: A. N. Tikhonov and A. A. Samarskii (1990).

8.3.3-3. Domain: 0 ≤ ‰ ≤ x , 0 ≤ Š ≤ 2y , − z

< p < z . Second boundary value problem.

An infinite circular cylinder is considered. A boundary condition is prescribed:  —

q

= { ( Š , p ) at

‰ =x .

© 2002 by Chapman & Hall/CRC Page 580

8.3. HELMHOLTZ EQUATION

3

+




581

= − (x) 




Solution: 2

( , , )= 













( , ) ( , , , , , )



0



+

−

2



























( , , ) ( , , , , , ) 

0







0

− 























.







Here, exp − − | − |

( , , , , , )= 













2



+

1 2





2 







− 





=0

 

= 

!







( 





2

(

=1 











) cos[ ( − )] exp − )





 











 !

| − | , 







!

1 for 2 for

=



#







$

(



2) 2 (

− ,

2 "

where the ( ) are the Bessel functions and the ( ) = 0. 





 





) 



−

2 









2

















are positive roots of the transcendental equation 



= 0, ≠ 0,

%'&



Reference: A. N. Tikhonov and A. A. Samarskii (1990).

8.3.3-4. Domain: 0 ≤

≤ 

,0≤ 

≤2 ,− 



< (

< 

. Third boundary value problem. (

An infinite circular cylinder is considered. A boundary condition is prescribed: )

+

* 

= ( , ) at 

+





= 

. 



Solution: 2



( , , )= 











( , ) ( , , , , , )



0



+

2



− 







−

















0

Here,





( , , ) ( , , , , , ) 

0



































.



 















1 ( , , , , , )= 2 











 

2 

=0



=1 



= 

!

 



2

(



( 2+







 



2 +

2

− , 

=





8.3.3-5. Domain: 0 ≤

≤ 

,0≤ 







!



| − | , 







!

 

( $



)+ 

≤2 ,0≤ 



+



( 


=



A

+

A

. 

;

A

Eigenfunctions possessing the axial symmetry property: @

(1) 0 



=
=





1)



( I





1) 

1) 









)] cos(

 



) sin 

+



)] sin( 



) sin 

+

A

The square of the norm of an eigenfunction is defined as 

 B 

(1) 

B


=

= 1, 2, 3, ?

=>=>=



( I

1) O

= 0.

Eigenfunctions: 





(1) T

=[




”>”

The solution was written out under the assumption that

+1 2 (

„



–'—

U

„

‰

‚

„



) ≠ 0 for €

= 0, 1, 2, 

”>”>”

ˆ

References: M. M. Smirnov (1975), A. N. Tikhonov and A. A. Samarskii (1990). 

Paragraphs 8.3.4-3 through 8.3.4-6 present only the eigenvalues and eigenfunctions of homogeneous boundary value problems for the homogeneous Helmholtz equation (with ≡ 0). The solutions of the corresponding nonhomogeneous boundary value problems ( 0) can be constructed by the relations specified in Paragraph 8.3.1-4. ˜

™

™

8.3.4-3. Domain: 0 ≤

≤ ~

š

. Second boundary value problem. ‚

A spherical domain is considered. A boundary condition is prescribed: = 0 at = . Eigenvalues: ›

œž

~

€

= 0,

00

2

= €

‚

Here, the

;

2

 



‚

= 0, 1, 2, 

;

”>”>”

= 1, 2, 3, p

”>”>”

Ÿ

are roots of the transcendental equation 2 +1 2 ( ) = 0. +1 2 ( ) − Eigenfunctions: 1 (1) (1) = (cos ) cos +1 2 000 = 1, 

Ÿ

 



Ÿ

‰

‰

 

ˆ

 

¡

 

ˆ









~





=

¢

  ˆ

£

+1 2 ‰

(cos ) sin …

“ ¢   ‚

£



¤ 

(1) 2 000 ¤

=

4 3

Ÿ

„

3 Œ

‚

,





where

= ¥

•

2 for 1 for

¤

Ÿ

2

Ž

Ž

= 0, ≠ 0.



Ÿ

(1) 

„

„

Ÿ

¤

2

=

Ž

, †

Œ

¤

”>”>”

‚



;



(2) „

”>”>”

§

¥

Ž



„



= 1, 2, 3, Ž

( + )! ( + 1) 1− 2 (2 + 1)( − )!

=

= 0, 1, 2, Ž





¤

–'—

2

(1)

¤

,

„

The square of the norm of an eigenfunction: 



†



~

ˆ

Ž

„

Ÿ



~ 

‚ 

1

Ÿ „

(2)

…

“

‰

~ 

¤

2



Ž

,

2



= 1, 2, 3, 

Ÿ ”>”>”

),  

ˆ



 

Ž

+1 2 ( ‰

¦



Ÿ

,

Ÿ

Reference: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964). „

© 2002 by Chapman & Hall/CRC Page 589

590

ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES

8.3.4-4. Domain: 0 ≤

≤ ~

. Third boundary value problem. ‚

A spherical domain is considered. A boundary condition is prescribed: ›

Eigenvalues:

2

= €

‚

Here, the

;

2

 



+

œž

= 0 at 

¨

= 0, 1, 2, 

= ~

;

”>”>”

. ‚

= 1, 2, 3, p

”>”>”

Ÿ

are positive roots of the transcendental equation 

Ÿ

 

2



Ÿ

‰

ˆ

1

=

(2)

‚ 

£

‰

(cos ) sin …

“ ¢  

ˆ

‚ 



, †

= 0, 1, 2, Ž

;

”>”>”



~

+1 2

~

Ž

„

Ÿ



Ÿ „

¢  

1

= 



 

…

“

‰

Ÿ „

‰

(cos ) cos

~

+1 2

) = 0.

+1 2 ( ˆ 

ˆ ~





) ¨



(1) 

‚

  ¡

Eigenfunctions: 

) − (1 − 2

+1 2 (  

£

Ž

, †

= 1, 2, 3, Ž

”>”>”

„

Ÿ 

Here, the ( ) are the associated Legendre functions. The square of the norm of an eigenfunction: n

“

„



2

(1)

¤ 

¤

2

( + )! ( 1+ (2 + 1)( − )!

=

Œ





¤

–'—





Ÿ

(1) „

2 ¤

= ¤

Ž



„

(2) 

+ )(

¥

‚

¤

2

Ž

‚

¨



‚

§

2

+1 2 (

),

‰

 



= ¥

2 for 1 for ©

  ˆ

 

= 1, 2, 3, Ž

− 1) 

2

¦

,

− ¨

Ÿ

Ž

Ž

= 0, ≠ 0,

„

Ÿ

”>”>”

Reference: V. M. Babich, M. B. Kapilevich, S. G. Mikhlin, et al. (1964). 

Ÿ

„



Ÿ

„

≤

8.3.4-5. Domain: ‚

< ~

. First boundary value problem. ª

A spherical cavity is considered and the dependent variable is prescribed at its surface: = ( , ) at 

«

…

†

= ¬

, ‚

and the radiation conditions are prescribed at infinity (see Paragraph 8.3.1-5, Item 2 ). Solution for = 2 > 0: ­

€

p

( , , )= 

¬

…

¯

¯

†

 ®

=0

° «



=− °

±

( ( p

p

¬

‚

) )

³

´

µ

1

²

¸

µ

·

 

(2) +1 2 (

¶

±

± 

( )=

( , ), °

µ

),





(2) +1 2 (





where ) is the Hankel function of the second kind and the other quantities are defined just as in Paragraph 8.3.4-2, Item 2 . µ

¸

·

­



–'—

Reference: A. N. Tikhonov and A. A. Samarskii (1990).

8.3.4-6. Domain:

1 ‚

≤

≤ ¬

2. ‚

First boundary value problem.

A spherical layer is considered. Boundary conditions are prescribed: = 0 at 

Eigenvalues:

= ¬

‚

1,

= 0 at 

= ¬

2. ‚

¹

2

=

;

= 0, 1, 2, 

;

”>”>”

= 1, 2, 3, º

”>”>”

 

Here, the

are positive roots of the transcendental equation 

Ÿ



Ÿ

  

Ÿ

+1 2 ( »

‚

¸

  

1)

+1 2 ( ‚

¸

2)

−

+1 2 ( »



‚

¸

  ²



 

2)

+1 2 ( ‚

¸

1)

= 0.

  ² 

© 2002 by Chapman & Hall/CRC Page 590

8.3. HELMHOLTZ EQUATION

3 ¼

+ ½

¾

591

= − (x) ¿

½

Eigenfunctions: (1) 

1

=

°

+1 2 (

¶



=

°



Here, the



³

Ÿ



+1

¶

¬

(cos ) cos °

“

Á

¸

) ¬

(cos ) sin °

³

“

Á



Ÿ

Ÿ

+1 2 (

)= ¬

+1 2 ( »

1) ‚

¸

 

+1 2 (

)− ¬

¸

 



2

(1) ¤

+1 2 (

¥

Ä

°

Ã

Ã

2 »

+1 2 (

Ã

2



Á

2 »



Ÿ

2 ¤

=

°



 

¤

(2) 

¤

2

Ÿ

°

Ÿ



,

= 1, 2, 3, Á



Ÿ

−

+1

2 »

  

2) ‚

¸

2( ¸

+1 2 (



,

 

2) ‚

). ¬

¸



1) ‚

¸

 



”>”>”

+1 2 ( »

  ²



Á

1) ‚

¸

  ²



4 ( + )! (2 + 1)( − )!

=

°

(1)

= 1, 2, 3, Á



The square of the norm of an eigenfunction:



, ´

  À

Â

¸

¤

;

”>”>”

( ) are the associated Legendre functions and °

“

À



= 0, 1, 2, Á



2(





¤

, ´

  À

1

Ÿ

(2) 

) ¬

¸

¬



Ÿ

¥

= °

2 for 1 for ©

= 0, ≠ 0, Á

Á

  



Ÿ

”>”>”

Ÿ

8.3.5. Other Orthogonal Curvilinear Coordinates The homogenous three-dimensional Helmholtz equation admits separation of variables in the eleven orthogonal systems of coordinates listed in Table 29. For the parabolic cylindrical system of coordinates, the multipliers and are expressed in terms of the parabolic cylinder functions as «

( )= «

Â

1 Æ

Ç

−1 2 ( È

¸

É

)+ Â

2 Æ

Ç

−1 2 (− È

É

¸

), Â

Å

( )= Ê

1 b

− −1 2 ( Ç

È

¸

¹

1 2

=  

(

2

Ë

¹

− )

º

−1 2 ¸

,

= 4( É

Ì

2 º

− )

É

Å

)+ Ê

1 4 ¸

Í

b

Æ

2

− −1 2 (− Ç

È

¸

É

), Ê

,

where 1 , 1 , 2 , and 2 are arbitrary constants. For the elliptic cylindrical system of coordinates, the functions and modified Mathieu equation and Mathieu equation, respectively, so that Æ

b

b

«

«

Ce ( , ), Se ( , ),

( )=

Î

Î

©

Î



( )=

Ï

Å

ce ( , ), se ( , ), Ð

Ð

©

Ï

¹

1 2 ( 4

=

Ï

Ð



are determined by the Å

Ï

2

−

Ñ

Ï

º

),

where Ce ( , ) and Se ( , ) are the modified Mathieu functions, and ce ( , ) and se ( , ) are the Mathieu functions; to each value of the parameter there are certain corresponding eigenvalues = ( ) [see Abramowitz and Stegun (1964)]. In the prolate and oblate spheroidal systems of coordinates, the equations for and are different forms of the spheroidal wave equation, whose bounded solutions are given by 

Î

Ï

Î





Ï

Ð

Ï



Ï

Ð



Ï



Ï

Ë

Ë

«



¹

( ) = Ps| | (cosh , Î

¹

2

Ò

«

( ) = Ps| | (cos ,

),

Î

Ñ

2

Ò

Å

Î

)

Ð

Ñ

¹ 

Ò

Î

Ó

Î

Å

º

2

Ò

Î

Ð

Ñ



for prolate spheroid,

¹

( ) = Ps| | (− sinh , 2 ), ( ) = Ps| | (cos , − is an integer, = 0, 1, 2, , 

«

Å



Ã

Ô>Ô>Ô

Ñ

) for oblate spheroid, − ≤ ≤ , Ã

º

Ã

where Ps ( , ) are the spheroidal wave functions; see Bateman and Erd´elyi (1955, Vol. 3), Arscott (1964), and Meixner and Sch¨afke (1965). The separation of variables for the Helmholtz equation in modified prolate and oblate spheroidal systems of coordinates, as well as the spheroidal wave functions, are discussed in Abramowitz and Stegun (1964). In the parabolic coordinate system, the solutions of the equations for and are expressed in terms of the degenerate hypergeometric functions [see Miller, Jr. (1977)] as follows: Ò

Õ

Ñ



«

( )= «

Å

( )= Ê

exp Ò

Â

Â

Ò Ê

exp

ÖØ×

ÖØ×

1 2 1 2

2 Ú

Û

− Ü

 Ù

2 Ê Ù

Ú

Û

Ü

+

Ë

4

+

Ë

4

Ù

Ù

º

º

+1 , + 1; 2 +1 , + 1; 2 º

º

Å

¹

2 Ý

, Þ

 Ù

2

Ý

Ê Ù

Þ

Ù

=

¶

− ,

.

© 2002 by Chapman & Hall/CRC Page 591

592

ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES TABLE 29 Orthogonal coordinates ¯ , ¯ , ¯ that allow separable solutions of the form = ( ¯ ) ( ¯) ( ¯) for the three-dimensional Helmholtz equation 3 + =0 ß

à

¹

Õ

á

ß

à

â

ã

Coordinates

Õ

ä

Transformations

Particular solutions (or equations for , , ) â

= , = , =

ß

Cartesian , , ß

ß

à

= cos( á

+ 1 ) cos( 2 + 2 ) cos( where 12 + 22 + 32 = ; see also Paragraph 8.3.2-1 1

º

à

à

Õ

Õ

Õ

Cylindrical , ,

æ

æ

à

=[ where ( á

´

Õ

»

º

3 º

ã

+ Õ

3 ), å

º

æ ç

b

¹

Ã

Ë

´

è

º

Õ

å

Ë

²

º

»

Õ

¹

å

º

Å

( )+ ( )] cos( + ) cos( + ), 2 + 2 = , see also Paragraph 8.3.3-1 and are the Bessel functions) ç

Æ

´

Õ

à

å

æ

= cos , = sin , =

ß

ß

º

æ

´

á

á

Å

Ë

ç

ç

²

Parabolic cylindrical , , Â

Ê

Â

Â

Õ

Ð

2 Ê

),

Î

Ð

à

³

Prolate spheroidal , , Ð

³

Ð

á

Ð

Ð

¹

+

é’é

= cosh sin cos , = cosh sin sin , = sinh cos Ð

Parabolic , ,

+1 2 (

ç

Â

Ê

Â

Ð

¹

2

´

²

´

Õ

ß

Î

Å

é

é’é

+

Î

Ð

´

Î

General ellipsoidal , , í

î

Conical , , î

ó

ô

ò

= õ

ñ

2 â

2

+ +

é’é

òžó

ò

= õ

ñ

ð

=

ó

ò

ï

,

( −1)( −1) , −1 ò

ï

−1)

÷

ý

í

î

´

å

º

Î

Å

é’é

º

Ð

â

Å

Å ¹

é

Ê

¹

º

Â

´

å

Â

º

â

Ë

Ê

Å

é

Ê

Ê

º

Å

Ë

¹

+ (− − +( + + (− +

é’é â

Î

Ñ

Å

ã

é’é

é’é

4 4 4 ÷

Ë

Ð

Ñ

Ë

÷

÷

ø

ø

¹

´

ï

÷

ø

=

í

( ) ] ( ) ] ( ) ] ( )= î

ø

ã

Ñ

â

÷

â

Å

Ñ

Ë

ø

î

Ð

´

Ñ

í

( )[ ( )[ ( )[

Î

¹

Ñ

º

º

ø

÷

cosh 2 + 12 2 cosh 4 ) = 0, cos 2 − 12 2 cos 4 ) = 0, cosh 2 − 12 2 cosh 4 ) = 0

º

ã

é

é

é

é

ü

−1 2

é

û

ï

óžô

−1)( 1−

º

Î

Ñ

Â

â

ò

òþñ

Ð

= ( ) ( ) cos( + ), + ( 4 − 2 − 2 ) = 0, + ( 4 + 2 − 2) = 0 Â

ø

ð ï

â

Å

Ñ

Ë

ö

(

= à

ò

ò

ñ

í

ï

ô

( −1)( −1)( −1) , 1−

ð

ï

ß

ò

ò

= à

ó

¹

Ð

´

ò

Å

Ð

í

ñ

Î

Ð

ì

´

( − )( − )( − ) , ( −1) ð

å

º

º

¹

é

Å

Î

=

´

Ñ

Ë

é’é Å

Ñ

ß

º

Î

Ð

Î

´

Ð

Ð

= ( ) ( ) cos( + ), tanh + (− + 2 cosh2 + 2 cosh2 ) = 0, + cot + ( − 2 sin2 − 2 sin2 ) = 0 é

Ñ

Õ

å

ì

â

Ê

Ð

å

´

Ñ

â

Ð

Î

¹

Ð

â

â

Â

= 2 cosh cos sinh , = 2 sinh sin cosh , = 12 (cosh 2 + cos 2 − cosh 2 )

Å

¹

á

Ñ

à

´

Á

Ë

Ë

é’é

Ê

Î

Paraboloidal , ,

Á

³

ì

´

Â

³

ë

Ë

´

Ê

ë

ç

ê

¸

Ë

Ð

Ê

Â

Å

ç

ê

æ

Ë

ç

Î

= cos , = sin , = 12 ( 2 − 2 ) à

Ð

= ( ) ( ) cos( + ), coth + (− + 2 sinh2 − 2 sinh2 ) = 0, + cot + ( + 2 sin2 − 2 sin2 ) = 0

´

Î

â

) (cos ) cos( + ), ) (cos ) cos( + ), +1 2 ( = 2 ; see also Paragraph 8.3.4-1 »

Ñ

ß

å

Ë

¸

é

â

Ñ

Õ

Õ

Î

á

Ñ

Î

º

ì

Å

Ð

Î

Ð

Ë

Ñ

´

Å

º

â

â

Ñ

à

Î

á

´

Î

2

¸

´

Î

ß

Å

º

¸

³

Õ

Ê

æ

æ

´

Î

Oblate spheroidal , , Î

é’é

= −1 = −1 where á

Ñ

´

å

â

Ñ

= sinh sin cos , = sinh sin sin , = cosh cos à

Õ

= ( ) ( ) cos( + ), + [ 12 2 ( − 2 ) cosh 2 − ] = 0, − [ 12 2 ( − 2 ) cos 2 − ] = 0 ¹

Å

´

Õ

ß

º

æ

æ

´

Î

é’é

Õ

æ

º

Â

Ñ

= sin cos , = sin sin , = cos

ß

é’é

â

â

æ

æ

Ê

º

á

Ð

Ñ

Spherical , ,

Å

Ë

Î

Õ

Â

¹

Ë

Å

Ñ

à

é’é

Õ

= cosh cos , = sinh sin , =

ß

¹

â

â

Ê

Õ

Õ

³

= ( ) ( ) cos( + ), + [( − 2 ) 2 + ] = 0, + [( − 2 ) 2 − ] = 0

á

à

Elliptic cylindrical , , Î

= 12 ( 2 − = , = ß

ü

í

+( 2+ 1 + +( 2+ 1 + +( 2+ 1 + ( − 1)( − ) ù

ú

é

ù

î

ù

ú

ï

ú

ü

ü

ú

î

ï

ú

ú

= 0, = 0, 2 ) = 0, 2)

2)

â

û

ã

ý



 ( )  ( ), + [ − ( + 1) 2 sn2  ] = 0,   + [ − ( + 1) 2 sn2  ]  = 0, where  = sn2 ( , ), = sn2 ( , ), = á

û

ÿ

ï




ç

( +1 2)



ÿ

ú

ï

÷

ù

þû

û

ú

÷

ý

© 2002 by Chapman & Hall/CRC Page 592

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8.4. OTHER EQUATIONS WITH THREE SPACE VARIABLES

In the case of the paraboloidal coordinate system, the equations for , , and  the Whittaker–Hill equation  

+

  

+ 

1 2 8

1 2 8  cos 4 

+  cos 2 −

are reduced to 

û

= 0. 

Denote by gc  ( ;  ,  ) and gs  ( ;  ,  ), respectively, the even and odd 2 -periodic solutions of the Whittaker–Hill equation, which is a generalization of the Mathieu equation. The subscript = 0, 1, 2,  labels the discrete eigenvalues  =   . Each of the solutions gc  and gs  can be represented in the form of an infinite convergent trigonometric series in cos  and sin  , respectively; see Urvin and Arscott (1970). The functions , , and  can be expressed in terms of the periodic solutions of the Whittaker–Hill equation as follows [Miller, Jr. (1977)]: û



( )=

Î



gc 

;2 Î

gs 

 

;2 Î

 ý

ý



1 2 1 2

, ,

 ú

 ú







, 



( ) = û

,



gc

 

gs 



;2 ;2 ý

, 

ý

, 

1 2 1 2

 ú

  ú



, 



, 

( )= 



gc

ø

+ ø

 

gs 

+ ø





2

2

;2 ;2

 ý

 ý

, ,

1 2 1 2



ú

ú



 





, ,

where  = and =   − 12 2 . For the general ellipsoidal coordinates, the functions , , and  are expressed in terms of the ellipsoidal wave functions; for details, see Arscott (1964) and Miller, Jr. (1977). For the conical coordinate system, the functions and  are determined by the Lam e´ equations that involve the Jacobian elliptic function sn = sn( , ). The unambiguity conditions for the transformation yield = 0, 1, 2,  It is known that, for any positive integer , there exist exactly 2 +1 solutions corresponding to 2 +1 different eigenvalues . These solutions can be represented the form of finite series known as Lam e´ polynomials. For more details about the Lam´e equation and its solutions, see Whittaker and Watson (1963), Arscott (1964), Bateman and Erd´elyi (1955), and Miller, Jr. (1977). Unlike the Laplace equation, there are no nontrivial transformations for the three-dimensional Helmholtz equation that allow the -separation of variables. ÷

ù

ý

ù

û

û

õ

õ

ú

!#"

References for Subsection 8.3.5: F. M. Morse and H. Feshbach (1953, Vols. 1–2), P. Moon and D. Spencer (1961), A. Makarov, J. Smorodinsky, K. Valiev, and P. Winternitz (1967), W. Miller, Jr. (1977).

8.4. Other Equations with Three Space Variables 8.4.1. Equations Containing Arbitrary Functions 2%

1. $

2 $

2%

+ $

&

2 $

'

+

2% $ $

2

+ )

+ *

+,

,

= 0, %

-

2

=

2 &

2

+ '

+

2 (

.

(

Schr¨odinger’s equation. It governs the motion of an electron in the Coulomb field of a nucleus ( > 0). The desired solutions must satisfy the normalizing condition ý

.

/

.

/

.

/

|0 (1 , 2 , )|2 3 õ

−/

−/

−/

Eigenvalues:

2

3

= 1. õ

2

; = 1, 2, 3,  4 2 Normalized eigenfunctions (in the spherical coordinate system 4 ,  , ): ù



=−

3 1

ý

ø

0



5

6

=

3 27

2 )

ÿ



-

:

(2 8 + 1)( 8 − 9 )! (: − 8 − 1)! 4 ; < : (: + 8 )! (9 + 8 )!

= 1, 2, 3,



; 9

= 0, E 1, E 2,

= 

> : ?

, E

@ 8

A

;

exp − 8

=

2> : ? @

B

= 0, 1, 2

2 +1  − −1 A A

, :

=

(< ) > : ?

@

− 1;

C

( , D ),

A

© 2002 by Chapman & Hall/CRC Page 593

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ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES

where ;

=
12 )

exp F − `

0

a

>

a

c

a

b a

References: E. A. Novikov (1958), D. E. Elrick (1962). 2e

3. d

2 d

2e

+ d

2

&

d

2e

+ d

2

'

d

e

+X

1 &

W

e

+

d d

2 X

'

&

d

e

+

d

3 X

W d

>

2 2)

.

a

= 0.

d

'

O a 1 3(1 + 12

W

This equation is encountered in problems of convective heat and mass transfer in a straining flow. Fundamental solution: Y

(f , Z , [ , T , \ , ] ) = _

(f , T , ; ) = g

4.

2e d d

&

2 1

+

2e d d

&

a

2 2

+

2e d d

&

g a

2^ h

2 ij

−1

L

1)

(Z , \ , ; g

>

−1 S 2

Rk

a

>

3 l

m X

m

P e m n

([ , ] , ; g

exp h −

(f > 2(

a

2 ij

L

, O

− T )2 . − 1) k

ij L

3) >

a

.

d

&

, n =1

2) >

>

=

2 3

(f , T , ; `

0

n d

&

This equation is encountered in problems of convective heat and mass transfer in an arbitrary linear shear flow. The solution that corresponds to a source of unit power at the origin of coordinates is given by o

(f 1 , f 2 , f 3 ) =

1 (4^ )3 p

2 _

0

l

exp h − `

q

3 q

, r =1

( )f q f 4t a ( ) r

s

r v k

tu

a

a

()

.

a

Here, t = t ( ) is the determinant of the matrix B = { w q r }; the q r = q r ( ) are the cofactors of the entries w a q r = w q r ( ); the w q r are determined by solving the following a system of ordinary s s differential equations with a constant coefficients: q w

r

u w

where x

~#

q

q

= 1 and x q

r

=

x q r

l

+

q y

q r

u

a {

x q

= 0 if | ≠ } .

3

a

w r

y

l

+

{

3 r

y

=1 z

as r

y

y w

q

y

,

=1 z

0 (initial conditions),

a

Reference: G. K. Batchelor (1979).

© 2002 by Chapman & Hall/CRC Page 594

595

8.4. OTHER EQUATIONS WITH THREE SPACE VARIABLES

5. d d

h‚

1(

e

)d €

€

+ k

d

d

€

d

2(

h‚

e

)d ƒ

+

ƒ

k d

d

ƒ

3(

h‚

d

e

)d „

=

„

k d

. e

…

„

This is a three-dimensional linear equation of heat and mass transfer theory with a source in an inhomogeneous anisotropic medium. Here, † 1 = † 1 (f ), † 2 = † 2 (‡ ), and † 3 = † 3 ( ˆ ) are the principal thermal diffusivities. 1 ‰ . The equation admits multiplicatively separable solutions, o (f , ‡ , ˆ ) = 2 ‰ . There are also additively separable solutions, (f , ‡ , ˆ ) = ‹ 3 ‰ . If form †

1

q

=

,

f

=

2 †

y

, and

‡

=

3 †

(| ≠ 2, Ž r

Œˆ

z s

=

o

( ),

o



2

=4h

o

,

≠ 2,

2− q

1 (f Š

)Š

)+‹

2 (‡

2− y ‡

2− r ˆ

+

u



2

o

+

=

‘ 

u



’

1 1 + 2−| 2−Ž

=2“ ‘

3 (ˆ

)Š

3(ˆ

).

).

,

(2 − | )2 (2 − Ž )2 Œ (2 − } )2 k o z where the function ( ) is determined by the ordinary differential equation s 2o

2 (‡

≠ 2), there are particular solutions of the }

+



)+‹

1 (f

o

+

1 2−}

− 1, ”

u u whose solutions are expressed in terms of the Bessel functions.

8.4.2. Equations of the Form div [ ( , , )∇ ] – ( , , ) •

–

—

˜

™

š

–

—

˜

=– ( , , ) ™

›

–

—

˜

Equations of this sort are often encountered in heat and mass transfer theory. For brevity, the equation is written using the notation o

div[ (r)∇o ] = z œ

 z



œ

o

+

(r) œ œ



(r) œ

œ ‡

 z

œ

ž

o

+ ‡

œ

(r) œ

œ ˆ

 z

œ

ž

r = { , ‡ , ˆ }.

, ˆ

œ



ž

In what follows, the problems for the equation in question will be considered in a bounded domain Ÿ with a sufficiently smooth surface   . It is assumed that (r) > 0 and ¡ (r) ≥ 0. z

8.4.2-1. First boundary value problem. The following boundary condition of the first kind is imposed: o

= † (r) for r ¢

.  

Solution: o

(r) = £

¤

¥

( ¦ ) § (r, ¦ )

(r, ¦ ) =

¬ q ®

q

«

=1 ­

(r) q

q

®

2¯

(¦ ) q

−£ ¨

©

†

(¦ ) (¦ ) z

u

Here, the Green’s function is given by §

Ÿ

,

® q

®

2

= °

œ œ

2q ±

(r, ¦ ) §

¨ ª

(r) ²

.

(1)

= {µ , ¶ , · },

(2)

 

¨

³

u

, ´

­ ­

­

where the ¯ q and q (r) are the­ eigenvalues and eigenfunctions of the Sturm–Liouville problem for the following second-order elliptic equation with a homogeneous boundary condition of the first ­ kind: div ¸#¹ (r)∇ − » (r) + ¯ = 0, = 0­ º for r­ ¼ ½ ­ .

(3) (4)

The integration in (1) is performed with ­ respect to µ , ¶ , · ; ¾ denotes the derivative along the ¾ ¿ À outward normal to the surface ½ with respect to µ , ¶ , · . General properties of the Sturm–Liouville problem (3)–(4):

© 2002 by Chapman & Hall/CRC Page 595

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ELLIPTIC EQUATIONS WITH THREE OR MORE SPACE VARIABLES

1 Á . There are countably many eigenvalues. All eigenvalues are real and can be ordered so that ¯ ¯ ¯ ¯ Ä as Å Ã Ä ; therefore the number of negative eigenvalues is finite. 1 ≤ 2 ≤ 3 ≤  , with q à 2 Á . If ¹ (r) > 0 and » (r) ≥ 0, all eigenvalues are positive, ¯

> 0. q

3 Á . The eigenfunctions are defined up to a constant multiplier. Any two eigenfunctions, q (r) and (r), corresponding to different eigenvalues, ¯ q and ¯ , are orthogonal to each other in ³ : ­

­

Æ °

(r) q

±

­

(r) ² ­

= 0 for Æ

³

≠Ž . Å

Æ

4 Á . An arbitrary function Ç (r) that is twice continuously differentiable and satisfies the boundary condition of the Sturm–Liouville problem ( Ç = 0 for r ¼ ½ ) can be expanded into an absolutely and uniformly convergent series in the eigenfunctions; specifically, ¬

(r) = Ç

È q

®

q Ç

(r), q

= q

Ç

1 ®

®

q

=1

°

2

±

Ç

(r)

(r) ²

, ³

­

­

®

q

­ where the norm squared q 2 is defined in (2). É ÊË Ì ÍÏÎ Ð In a three-dimensional problem, to each eigenvalue ¯ q finitely many linearly inde­ q pendent eigenfunctions (1) , ÑÑÑ , (q ) generally correspond. These functions can always be replaced by their linear combinations Æ ­

¯ (q r ) =

­

Ò r

,1

(1) q

+

+Ò

ÂÂÂ

r

(q r −1)

, r −1

(q r )

+

, Ó

= 1, 2,

ÑÑÑ

, Ô

,

­ ­ ­ q such that ¯ (1) , ÑÑÑ ­ , ¯ (q ) are now pairwise orthogonal. Therefore, without loss of generality, we can assume that all eigenfunctions are orthogonal. Æ ­

­

8.4.2-2. Second boundary value problem. A boundary condition of the second kind is imposed, Õ Õ

It is assumed that » (r) > 0. Solution: Ö (r) = °

Ö

= Ø (r) for r ×

( ´ ) Ú (r, ´ ) ²

±

Ù

³

Û

+° Ü

¼

. ½

( ´ ) ¹ ( ´ ) Ú (r, ´ ) ² Ø

½

Û

.

(5)

Here, the Green’s function is defined by relation (2), where the ¯ q and q (r) are the eigenvalues and eigenfunctions of the Sturm–Liouville problem for the second-order elliptic equation (3) with ­ the following homogeneous boundary condition of the second kind: Õ Õ

= 0 for r ×

¼

½

.

(6)

­

If » (r) > 0, the general properties of the eigenvalue problem (3), (6) are the same as those of the first boundary value problem (see Paragraph 8.4.2-1). 8.4.2-3. Third boundary value problem. The following boundary condition of the third kind is set: Õ Õ

the

×

Ö

+ Ó (r)

Ö

= Ø (r) for r ¼

½

.

The solution of the third boundary value problem is given by relations (5) and (2), where ¯ q and q (r) are the eigenvalues and eigenfunctions of the Sturm–Liouville problem for the ­

© 2002 by Chapman & Hall/CRC Page 596

597

8.5. EQUATIONS WITH Ý SPACE VARIABLES

second-order elliptic equation (3) with the following homogeneous boundary condition of the third kind: Õ Õ × + Ó (r) = 0 for r ¼ ½ . (7) If » (r) ≥ 0 and Ó (r) > 0, the general properties of the eigenvalue problem (3), (7) are the same ­ as those of the first boundary value problem (see Paragraph 8.4.2-1). Let Ó (r) = Ó = const. Denote the Green’s functions of the second and third boundary value problems by Ú 2 (r, ´ ) and Ú 3 (r, ´ , Ó ), respectively. For » (r) > 0, the following limit relation holds: ­

8.5. Equations with

) = lim

2 (r, ´ Ú

0

rÞ

3 (r, ´ Ú

, Ó ).

Space Variables ß

8.5.1. Laplace Equation à

á

=0 â

The Å -dimensional Laplace equation in the rectangular Cartesian system of coordinates ã has the form Õ Ö Õ Ö Õ Ö 2

Õ ã

2

+

2 1

Õ

+

ã

2 2

2

+

ÂÂÂ

Õ

ÑÑÑ

,ã q

= 0.

2q ã

1,

For Å = 2 and Å = 3, see Subsections 7.1.1 and 8.1.1. A regular solution of the Laplace equation is called a harmonic function. In what follows we use the notation: x = {ã 1 , ÑÑÑ , ã

} and |x| = q

ä

2 1 ã

+

ÂÂÂ

+ã

2q .

8.5.1-1. Particular solutions. 1 Á . Fundamental solution: å

q

2ç p 2 (x) = − , æ = è q (Å − 2) æ q |x| −2 (Å é 2) 2 Á . Solution containing arbitrary functions of Å − 1 variables: 1

Ö

(ã 1 , ÑÑÑ , ã q

)=

¬

r

where Ø (ã 1 , ÑÑÑ , ã Ö

q

3 Á . Let (ã 1 , ÑÑÑ , ã

−1 )

(−1) r È

ã ê

2q r r

(2 Ó )! ë

=0

and ì (ã 1 , ÑÑÑ , ã q

(Å ≥ 3).

q

−1 )

(ã 1 , ÑÑÑ , ã Ø

−1 ) q

+

2q r +1 ã

(2 Ó + 1)! ë

r

ì

(ã 1 , ÑÑÑ , ã q

−1 )í

,

are arbitrary infinitely differentiable functions.

) be a harmonic function. Then the functions Ö Ö (î ¯ ã 1 + ï 1 , ÑÑÑ , î ð ã q + ï q ), 1 = Ò q

Ö

Ö

Ò

=

ñ

1 , |x|2 ã

q

−2

ÑÑÑ

,

ã

q

, |x| |x|2 ò are also harmonic functions everywhere theyÖ are defined; Ò , ï 1 , ÑÑÑ , ï q , and ð are arbitrary constants. The signs at ð in the expression of 1 can be taken independently of one another. ó#ô 2

References: A. V. Bitsadze and D. F. Kalinichenko (1985), R. Courant and D. Hilbert (1989).

8.5.1-2. Domain: − õ

< ã

1

< õ

,

ööö

, −õ

 ) sin(  3 ) sin( ? > 4 )

= $ B

2

,

@  >

=C

sin( @  > ! ) , @  >

2 2  +? > ) +% .

2 ( 2

9.3.4-3. The first and third derivatives are prescribed at the sides of a rectangle: . / ; . /
4 ) =C

2 2  +? > ) +% ,

2 ( 2

D

sin( @  > ! ) , @  >

1 for = 0, A for ≠ 0.

 = E 2

A

© 2002 by Chapman & Hall/CRC Page 619

620

HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS

9.3.4-4. Mixed boundary conditions are prescribed at the sides of a rectangle: .

= . / ; ;

.

= / ! ) , @  >

1 for B 2 for B

4( ' '

+*

- 4 I

(

= 0, ≠ 0.

= +

(, , - , ) )

9.3.5-1. Domain: 0 ≤  ≤ F 1 , 0 ≤  ≤ F 2 . Solution in terms of the Green’s function. We consider boundary value problems in a rectangular domain 0 ≤  ≤ F 1 , 0 ≤  ≤ F 2 with the general . . initial conditions

= ( ,  ) at ! = 0, / 0 = 1 ( ,  ) at ! = 0 and various homogeneous boundary conditions. The solution can be represented in terms of the Green’s function as .

+J

0

J

0

22

21

/

( ,  , ! ) = ! J /

0

0

21 J

J

0

22



7

0

21

(3 , 4 ) 5 ( ,  , 3 , 4 , ! ) 6 4 6 3 + J

0

J

22 0

1 (3 , 4 ) 5 ( ,  , 3 , 4 , ! ) 6 4 6 3

(3 , 4 , 8 ) 5 ( ,  , 9 , 3 , 4 , : , ! − 8 ) 6 4 6 3 6 8 .

Paragraphs 9.3.5-2 through 9.3.5-4 present the Green’s functions for three types of boundary conditions. 9.3.5-2. The function and its first derivatives are prescribed at the sides of a rectangle: .

.

. = / ; . = 0 at = / < = 0 at

 = 0,  = 0,

.

.

. = / ; . = 0 at = / < = 0 at

 = F 1,  = F 2.

Green’s function:

@  >

Here,

16 



L L  4 ? 4> sin( @  > ! ) > ( )  (3 )P > (4 ) ( ) P , KML NON ON N   2 F 1F 2 G > G @  >  =1 =1  ( F 1 ) P > ( F 2 )Q L ON N L NON 6 2  6 2P > 2 4 4 > =C (  + ? > ) + , ( ) = , P ( ) = .    % 6  2 6  2

5 ( ,  , 3 , 4 , ! ) =

L

K K  ( ) = sinh(  F 1 ) − sin(  F 1 )Q cosh(   ) − cos(  K K − cosh(  F 1 ) − cos(  F 1 )Q sinh(   ) − sin(  K K P > ( ) = sinh( ? > F 2 ) − sin( ? > F 2 )Q cosh( ? >  ) − cos( ? >  K K − cosh( ? > F 2 ) − cos( ? > F 2 )Q sinh( ? >  ) − sin( ?

 )Q  )Q ,

)Q >  )Q ,

where the   and ? > are positive roots of the transcendental equations cosh( F 1 ) cos( F 1 ) = 1,

cosh( ? F 2 ) cos( ? F 2 ) = 1.

© 2002 by Chapman & Hall/CRC Page 620

621

9.4. FOURTH-ORDER STATIONARY EQUATIONS

9.3.5-3. The function and its second derivatives are prescribed at the sides of a rectangle:
 

=
 =

= 0 at = 0 at

Green’s function: 

( ,  , , ,  ) =



4

=
 =

 = 0,

= 0 at = 0 at

 =  1,  =  2.



sin(   ) sin(    ) sin(  ) sin(   )

 1 2  =1  =1

  =   , 1


 

 = 0,

=  

 

,

2

sin(     )   

,

=   2 ( 4 +  4 ) +  .

  

9.3.5-4. The first and third derivatives are prescribed at the sides of a rectangle:




  

=
 =

= 0 at = 0 at

Green’s function: 

1

( ,  , , ,  ) =   =   , 1



=  

2




 = 0,


  

=
 =

= 0 at = 0 at

 =  1,  =  2.



 1 2  =0  =0   




 = 0,

,



cos(   ) cos(    ) cos(  ) cos(   ) 



  

,

 =  1

=   2 ( 4 +  4 ) +  ,

  

sin(     )



for = 0,  2 for ≠ 0. 

9.4. Fourth-Order Stationary Equations 9.4.1. Biharmonic Equation 





=0

The biharmonic equation is encountered in plane problems of elasticity ( is the Airy stress function). It is also used to describe slow flows of viscous incompressible fluids ( is the stream function). All solutions of the Laplace equation  = 0 (see Sections 7.1 and 8.1) are also solutions of the biharmonic equation. 9.4.1-1. Two-dimensional equation. Particular solutions. In the rectangular Cartesian system of coordinates, the biharmonic operator has the form  

≡

2

=



4  4

+2



4
4
2 2 +
4 .   

1  . Particular solutions: ( ,  ) =   3 +  2  + !   2 + "  3 +   2 + #$  + %& 2 + '  + (  + ) , ( ,  ) = (  cosh (  + sinh (  + !  cosh (  + "  sinh (  )(  cos (  + # sin (  ), ( ,  ) = (  cos (  + sin (  + !  cos (  + "  sin (  )(  cosh (  + # sinh (  ), ( ,  ) =  * 2 ln * + * 2 + ! ln * + " , * = + ( −  )2 + ( − # )2 , ( ,  ) = (   +  + ! )( " cosh (  + , sinh (  )(  cos (  + # sin (  ), ( ,  ) = (   +  + ! )( " cosh (  + , sinh (  )(  cos (  + # sin (  ),

where  ,

( ,  ) = (

2

+  2 )( " cosh (  + , sinh (  )(  cos (  + # sin (  ),

( ,  ) = (

2

+  2 )( " cosh (  + , sinh (  )(  cos (  + # sin (  ),

, ! , " , , ,  , # , % , ' , ( , and ) are arbitrary constants.

© 2002 by Chapman & Hall/CRC Page 621

622

HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS TABLE 30 Particular solutions of the biharmonic equation in some orthogonal curvilinear coordinate systems;  , , ! , " ,  , # , and  are arbitrary constants Transformation

Particular solutions

= *

Degenerate bipolar coordinates 1 , 2 : 1 2  = 2 ,  =− 2 1 +2 2 1 +2

2

+ ! * . + " * − . )(  cos  - + # sin  - ),

2− .

+ *

2

ln * + *

+ ! ln * + "

(at  = 0)

1  cosh(  + 1) + sinh(  + 1) cosh − cos / + ! cosh(  − 1) + " sinh(  − 1) 0 (  cos  + # sin  )

Bipolar coordinates , : % sinh % sin

 = ,  = cosh − cos

cosh − cos

2  . Fundamental solution:

2+ .

= ( *

Polar coordinates * , - :  = * cos - ,  = * sin -

=

=

1 (1

2 + 2 2 )2

/

cosh(  1 ) + 

sinh(  1 )

+ ! 1 cosh(  1 ) + " 1 sinh(  1 )0  cos(  2 ) + # sin(  2 )0

2

/

3

1 2 * ln * , * = +  2 +  2. 8  3  . Particular solutions of the biharmonic equation in some orthogonal curvilinear coordinate systems are listed in Table 30. ( ,  ) =

465

Reference: N. N. Lebedev, I. P. Skal’skaya, and Ya. S. Uflyand (1972).

9.4.1-2. Two-dimensional equation. Various representations of the general solution. 1  . Various representations of the general solution in terms of harmonic functions: ( ,  ) =  1 1 ( ,  ) + 1 2 ( ,  ), ( ,  ) =  1 1 ( ,  ) + 1 2 ( ,  ), ( ,  ) = ( where 1

465

1

and 1

2

2

+  2 )1 1 ( ,  ) + 1 2 ( ,  ),

are arbitrary functions satisfying the Laplace equation  1 7 = 0 (  = 1, 2).

Reference: A. N. Tikhonov and A. A. Samarskii (1990).

2  . Complex form of representation of the general solution: ( ,  ) = Re

/8 9

( ) + : ( )0 , 8

8

where ( ) and : ( ) are arbitrary analytic functions of the complex variable =  + ;



−

( ) @ arctan A 9




 = 0,

9

 − 

+ B

= : ( ) at

 = 0.

 2

 ( − )

? + (  − )2 +  2 C

>



where ! is an arbitrary constant.

: ( ) ? (  − )2 + 

−

2

+! ,

Example. Let us consider the problem of a slow (Stokes) inflow of a viscous fluid into the half-plane through a slit of width 2 D with a constant velocity E that makes an angle F with the normal to the boundary (the angle is reckoned from the normal counterclockwise). With the stream function G introduced by the relations HJI = − KML and H = KML I ( H&I and H are the fluid velocity N N KMN K components), the problem is reduced to the special case of the previous problem with

O

E cos F

(P ) = Q

for |P | < D , for |P | > D ,

0

R

(P ) = Q

E sin F

for |P | < D , for |P | > D .

0

Dean’s solution:

ET

G (P , S ) = 465

U

S ET − P −D X

( P − D ) cos F + S sin F V arctan W

U

S +Y . P +D X

( P + D ) cos F + S sin F V arctan W

Reference: I. Sneddon (1951).

9.4.1-4. Two-dimensional boundary value problem for a circle. Domain: 0 ≤ * ≤  , 0 ≤ - ≤ 2Z . Boundary conditions in the polar coordinate system: [

= ( - ) at 9

Solution: [ 465

(* , - ) =

1

(*

2Z \

2

− \ 2 )2 @_>

2` 0

* =\ ,

] ^

[

= : ( - ) at

[ \ − * cos(a − - )] (a ) ? a 1 − > 9 2 2 2 [* + \ − 2 \ * cos(a − - )] 2

* =\ . 2`

: (a ) ? a

*

0

2

+\

2

.

− 2 \ * cos(a − - ) C

Reference: A. N. Tikhonov and A. A. Samarskii (1990).

9.4.1-5. Three-dimensional equation. In the rectangular Cartesian coordinate system, the three-dimensional biharmonic operator is expressed as  

≡

2

=

] 4 ] b 4

+

] 4 ] c 4

+

] 4 ] 8

4

+2

] 4 ] b 2] c 2

1 d . Particular solutions in the Cartesian coordinate system: [

(b , c , ) = e f

2

+g f +h + i , f

8

f = j

(b , c , l ) = noe b sin(p b ) + g sin(p b ) + h b cos(p b ) + [

(b , c , l ) = n e b sin(p b ) + g sin(p b ) + h b cos(p b ) + [

(b , c , l ) = n e b sin(p b ) + g sin(p b ) + h b cos(p b ) + [

(b , c , l ) = n e b sin(p b ) + g sin(p b ) + h b cos(p b ) + [

(b , c , l ) = n e b sinh(p b ) + g sinh(p b ) + h b cosh(p b ) + [

(b , c , l ) = n e b sinh(p b ) + g sinh(p b ) + h b cosh(p b ) + i

] 4 ] b 2] 8

2

+2

] 4 ] c 2] 8

2

.

( b − \ )2 + ( c − k ) 2 + ( l − m ) 2 ,

[

where e , g , h ,

+2

, p , and r are arbitrary constants.

i

i

i

i

cos(p b ) q sin(r c ) exp sut l j p

2

+r

2

v ,

cos(p b ) q cos(r c ) exp s t l j p

2

+r

2

v ,

cos(p b ) q sinh(r c ) exp s t l j p

2

−r

2

v ,

cos(p b ) q cosh(r c ) exp s t l j p

2

−r

2

v ,

cosh(p b ) q sinh(r c ) sin s l j p

2

+r

2

i

i

cosh(p b ) q cosh(r c ) cos s l j p

2

+r

v ,

2

v ,

© 2002 by Chapman & Hall/CRC Page 623

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HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS

2 d . Particular solutions in the cylindrical coordinate system s f = j b 2 + c 2 v : [ (f , w , l ) = x y (r f )( e f cos w + g f sin w + h )( \ 1 cos z w + k 1 sin z w )( \ 2 cosh r l + k 2 sinh r l ), [

(f , w , l ) = { y (r f )( e f cos w + g f sin w + h )( \ 1 cos z w + k 1 sin z w )( \ 2 cosh r l + k 2 sinh r l ), (f , w , l ) = |}y (r f )( e f cos w + g f sin w + h )( \ 1 cos z w + k 1 sin z w )( \ 2 cos r l + k 2 sin r l ), [ [

(f , w , l ) = ~ y (r f )( e f cos w + g f sin w + h )( \ 1 cos z w + k 1 sin z w )( \ 2 cos r l + k 2 sin r l ), (f , w , l ) = x y (r f )( e cos z w + g sin z w )( \ 1 cosh r l + k 1 sinh r l + \ 2 l cosh r l + k 2 l sinh r l ), [ [

(f , w , l ) = { y (r f )( e cos z w + g sin z w )( \ 1 cosh r l + k 1 sinh r l + \ 2 l cosh r l + k 2 l sinh r l ), (f , w , l ) = |}y (r f )( e cos z w + g sin z w )( \ 1 cos r l + k 1 sin r l + \ 2 l cos r l + k 2 l sin r l ), [ [

(f , w , l ) = ~ y (r f )( e cos z w + g sin z w )( \ 1 cos r l + k 1 sin r l + \ 2 l cos r l + k 2 l sin r l ), where z = 0, 1, 2, MM ; e , g , h , \ 1 , \ 2 , k 1 , k 2 , and r are arbitrary constants; the x y (€ ) and { y (€ ) are the Bessel functions; and the | y (€ ) and ~ y (€ ) are the modified Bessel functions. 3 d . Particular solutions in the spherical coordinate system s f = j b 2 + c 2 + l 2 v : [

2

+ g f + h + f −1 , i y y y y [ (f ,  ) = s e f +2 + g f + h f 1− + f −1− vM‚ y (cos  ), i y y y y [ (f ,  , w ) = s e f +2 + g f + h f 1− + f −1− vM‚ y ƒ (cos  )( \ cos „ w + k sin „ w ), i where z = 0, 1, 2, MM ; „ = 0, 1, 2, MM , z ; e , g , h , , \ , and k are arbitrary constants; the i ‚ y (€ ) are the Legendre polynomials; and the ‚ y ƒ (€ ) are the associated Legendre functions defined (f ) = e f

by

y

ƒ 1 y ‚ y (b ). (b 2 − 1) , ‚ y ƒ (b ) = (1 − b 2 ) ƒ † 2 … y … y z !2 b b ƒ … ‡ … 4 d . Fundamental solution: 1 j b 2 + c 2 + l 2. (b , c , l ) = − 8Z 5 d . Representations of solutions to the biharmonic equation in terms of harmonic functions: [ (b , c , l ) = b ˆ 1 (b , c , l ) + ˆ 2 (b , c , l ), ‚ y (b ) =

[

(b , c , l ) = (b 2 + c 2 + l 2 )ˆ 1 (b , c , l ) + ˆ 2 (b , c , l ), where ˆ 1 and ˆ 2 are arbitrary functions satisfying the three-dimensional Laplace equation ‰ (6Œ  ‹ = 1, 2). The coefficient b of ˆ 1 in the first formula can be replaced by c or l .

3ˆ

Š =0

Reference: A. V. Bitsadze and D. F. Kalinichenko (1985).

9.4.1-6. z -dimensional equation. 1 d . Particular solutions: [

‘

(x) =

y

Ž



, , Š =1 2

‘

(x) = e f

+g +h f

(x) = ( e + g f ‘

2− y

)“

4− y

‘

(x) = “

y

b  +˜

+

y Ž 

i



y

, =1

g  b  b  +

y f 2− ,

h b  +

=1

(x) = exp s t b y j – y v “ Ž

Ž

e  Š b  b  b Š +

Ž

y 

=1 — ƒ ™ −1

i ”

b  +˜

f 2=

Ž

y Ž  y

h b  +

=1

Š =1

• 2=

,

”

y −1 ™ Š =1

i

,

( b Š − ’ Š )2 , Ž

y Š =1

( b Š − ’ Š )2 ,

sin( ’ Š b Š + p Š ), ™

y

– y =

y Ž −1 Š =1 ƒŽ −1

’ 2Š , 2

Ž

y

sin( ’ Š b Š + p Š )› sinh(œ Š b Š )› , ’ Š − š Š š Š Š =1 Š =ƒ =1 — =1 =ƒ where the   Š , ˜   ,  ,   , , ˜ ,  , ž , ’ Š , Ÿ Š , and œ Š are arbitrary constants. — — — 

”

œ Š2 = 0,

© 2002 by Chapman & Hall/CRC Page 624

625

9.4. FOURTH-ORDER STATIONARY EQUATIONS

2 d . Fundamental solution: ¤

(¥ ¦ 2)|x|4− § 2 § © (¥ − 2)(¥ − 4) (x) = ¡  4¨

‡

for ¥ = 3, 5, 6, 7, ªMªMª ;

¢

¡£

−

1 ln |x| 8¨ 2

for ¥ = 4.

For ¥ = 2, see Paragraph 9.4.1-1, Item 2 d .

«6¬

Reference: G. E. Shilov (1965).

3 d . Various representations of solutions to the biharmonic equation in terms of harmonic functions: ‘

(x) = b ­¯® 1 (x) + ® 2 (x), ‘

(x) = |x|2 ® 1 (x) + ® 2 (x),

° = 1, 2, ªMªMª Ž²± § |x|2 = b

±

,¥ ;

2

,

=1

where ® 1 and ® (µ = 1, 2).

«6¬

are arbitrary functions satisfying the ¥ -dimensional Laplace equation ³ § ® ´

2

=0

Reference: A. V. Bitsadze and D. F. Kalinichenko (1985).

9.4.2. Equations of the Form ¶

¶

= ·

(¹ , º ) ¸

Nonhomogeneous biharmonic equation. It is encountered in plane problems of elasticity and hydrodynamics. 9.4.2-1. Domain: − »

< ¼ < » , −»

0, Ö

Ý

( ) and ú ( ) = 0, 1, 2, üMüMü

are arbitrary functions satisfying the Helmholtz equations ë ý 1 + þ é ý 1 = 0,

ë ý 2 − þ é ý 2 = 0.

For solutions to these equations, see Section 7.3.

© 2002 by Chapman & Hall/CRC Page 626

627

9.4. FOURTH-ORDER STATIONARY EQUATIONS

Û

9.4.3-2. Domain: 0 ≤

≤ ÿ 1 , 0 ≤ Ü ≤ ÿ 2 . Boundary value problem.

¾

¾

¾

¾

Û A rectangle is considered. Boundary conditions are prescribed: ¾

¾

= = î î

¾

ïï

Solution:

= 0 at

= 0,

= 0 at

Ü = 0,










1

2

= ÿ 1,

= 0 at

Ü = ÿ 2.

Ý

ç ( ,Þ ) ( , , ,Þ ) Þ 

0

= 0 at

ïï Ý

0





Ý





, 

Ý



4

( , , ,Þ , ) = 

=

Ý


Û

¾

= î î

Û

( ,Ü ) = where

¾







sin(

ÿ 1ÿ 2 

=1

9.4.3-3. Domain: 0 ≤ • ≤







=1 



) sin( ) sin( ( 2 + 2 )2 −

) sin(











Þ )



,

, 0 ≤ ö ≤ 2 . Eigenvalue problem with 

= ß



ÿ1





= ß



ÿ2

.

≡ 0. 

ß

,

The unknown and its normal derivative are zero on the boundary of a circular domain: 

= 

Eigenvalues:

• =

. 





4

=



= 0, 1, 2, üMüMü ,

,



4





where the

= 0 at •

= 1, 2, 3, üMüMü ,





are positive roots of the transcendental equation











( )ù ( ) − ù ( ) ÷ ÷





Numerical values of some roots: = = = =



01 

11 

21 

31

Eigenvalues:

(c)

02 

12 

22 

32

(• , ö ) = ù (







(s)

(• , ) = 

!



03 

13 

23 

33



)

= = = =



−





(



14 

24 

34

−









•



cos(





( 



12.58; 14.11; 15.58, 17.01.





• 

04 

)



= = = =











9.439, 10.96, 12.40, 13.79,





( 















6.306, 7.799, 9.197, 10.54,



( ) = 0. 



•

)





= = = =









3.196, 4.611, 5.906, 7.144,



)









•



 



sin(





), 

). 



Reference: V. V. Bolotin (1978).

9.4.3-4. Domain: ( 

"

#

)2 + ( 

"

$

)2 ≤ 1. Eigenvalue problem with

≡ 0. 

The unknown and its normal derivative are zero on the boundary of an elliptic domain: 



=

= 0 on ( %



"

#

)2 + ( 

"

$

)2 = 1

( ≥ ). #

$

Eigenvalues and eigenfunctions (approximate formulas): %

&





4 01

01 =



(s) 11 =

4

 #

4 11

(c) 11 = 

3

8 

8 

5 

4



4 11

#

8

#

1 4

3

+

4 $

1

+ $

4

+

5 $

4

2

+

,

2 2 #

$

#

2 2 $



'

)=

0 ( 01 ) 0 ( 01 

, )=





()

, 



(s) 11 ( '

, )= 



0 ( 01 ) 0 ( 01 





1 ( 11 ) 1 ( 11 



(

)− '





2

+

(c) 11 (

,

2 2 #

'



2

+

01 ( 



$

)− 

1 ( 11 ) 1 ( 11 



1 ( 11 ) 1 ( 11 



'





'

)− 

), '



1 ( 11 ) 1 ( 11 



'

) cos , *





'



) sin , *



© 2002 by Chapman & Hall/CRC Page 627

628

HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS

where = equations '

( +



"

)2 + ( #



"

$

01 = 3.196 and

)2, and

11 = 4.611 are the least roots of the transcendental





) 1( ) + ( 1 ) 1( ) − 0( 

) 0 ( ) = 0, ( 1 ) 1 ( ) = 0. 1(



































The above formulas were obtained with the aid of generalized (nonorthogonal) polar coordinates , defined by '



= 

#

cos , '

= 

(0 ≤

sin

$,'



≤ 1, 0 ≤ '

≤2 )



ß



and the variational method. The maximum error in the eigenvalue 1 for = 1 − ( )2 ≤ 0,86 is less than 1%. The errors (s) in (c) 11 and 11 for ≤ 0,6 do not exceed 2%. In the limit case = 0 that corresponds to a circular domain, the formulas provide exact results. 



!



-

+

$."

#

-

-

Reference: L. D. Akulenko, S. V. Nesterov, and A. L. Popov (2001).

4

9.4.4. Equations of the Form

/

4

+

0

4 /

1

=

0

æ 4 /

( ,æ ) 2

1

/

9.4.4-1. Homogeneous equation ( ≡ 0). 

1 . Particular solutions: 3

( , )= 





( , )= 




=

= 0, 

≡ =






=0

2


+

2




2 2

.



Particular solutions:





( , )=



.

( , ), B



=1

.



are solutions of the Helmholtz equations

where the





− 5



characteristic equation

= 0.

M



')(



B



L

=0





= 0 and the L







are roots of the L



Reference: A. V. Bitsadze and D. F. Kalinichenko (1985). ? ?

9.

N

@? >

[ ] = 0. 



=0

Here, is any constant coefficient linear differential operator with arbitrarily many independent variables 1 , , . Particular solutions: O



!"!"!







C



( 1,



!"!"!

,

)=

C



=1

C

( 1,

are solution of the equations [ ] − 

M

B



=0

L

= 0, and the

,

),

C 

C

O



!"!"!

C

P



where the equation

B  C



C

DC

L

= 0 the L

are roots of the characteristic

are arbitrary constants. P

© 2002 by Chapman & Hall/CRC Page 643

644

HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS

9.6. Higher-Order Linear Equations with Variable Coefficients 9.6.1. Equations Containing the First Time Derivative 9.6.1-1. Statement of the problem for an equation with two independent variables. 

Consider the linear nonhomogeneous partial differential equation 

− 



where O

, 

O

, 

[ ] = ( , ), 





is a general linear differential operator of order

with respect to the space variable ,



;











O 

, 

(1)



 

[ ]≡ 



( , ) 







=0 

, 

(2)



whose coefficients = ( , ) are sufficiently smooth functions of both arguments for ≥ 0 and ≤ 2 . The subscripts and indicate that the operator , is dependent on the variables 1 ≤ and . We set the initial condition = ( ) at =0 (3) 













O













and the general nonhomogeneous boundary conditions 





(1)

[ ]≡

J

B

B

(2)

()

H

=

( ) at



= 1,

(

1

A

, ),

!"!"!

Q

(4) B



= 

B

,







=0

(1)

= 

 

−1



J 

() 

H

=0

[ ]≡



(1)





(2)



−1





(2) ,

( ) at

=

(

2

= + 1, A

Q

, ),

!"!"!





B

B

where ≥ 1 and ≥ + 1. We assume that both sets of the boundary forms (1) [ ] ( = 1, (2) [ ] ( = + 1, , ) are linearly independent, which means that for any nonzero the following relations hold: Q



Q

, ) and = ()

A

!"!"!

J

A

Q

!"!"!



B

C

J

R





B



(1)

() R

[ ]



0, S

C



J

=1 B

[ ]

B

B

0. S

J

= +1 B

(2)

() R

Q

R

B B

In what follows, we deal with the nonstationary boundary value problem (1), (3), (4). B

B

9.6.1-2. The case of general homogeneous boundary conditions. The Green’s function. 

The solution of equation (1) with the initial condition (3) and the homogeneous boundary conditions 

J

(1)

[ ] = 0 at

(2)

[ ] = 0 at

B

=

=



1

(



2

(

= 1, A

, ),

!"!"!

Q

= + 1, A

Q

!"!"!

(5)

, ) 

J

can be written as B





( , )=

#

2



( ) ( , , , 0) .



Here,



9



.

&

+ .

#

0

1



2

( , ) ( , , , )

#

.



T

( , , , ) is the Green’s function that satisfies, for > ;

9



.

T



− 9



O



, 

[ ]=0 9

9



.

T

&

.

&

T

.

(6)

1 T

≥ 0, the homogeneous equation (7)

© 2002 by Chapman & Hall/CRC Page 644

645

9.6. HIGHER-ORDER LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS

with the special nonhomogeneous initial condition = ( − ) at 9



.

=

(8) T



and the homogeneous boundary conditions J

(1)

[ ] = 0 at

(2)

[ ] = 0 at

B

=

9



9

=



1

(



2

(

= 1, A

, ),

!"!"!

Q

= + 1, A

Q

(9)

, ).

!"!"!



J

The quantities and appear in problem (7)–(9) as free parameters ( 1 ≤ ≤ 2 ), and ( ) is the Dirac delta function. It should be emphasized that the Green’s function is independent of the functions ( , ), ( ), (1) ( ), and (2) ( ) that characterize various nonhomogeneities of the boundary value problem. If the coefficients , (1) , and (2) determining the differential operator (2) and boundary conditions (4) are independent of time , then the Green’s function depends only on three arguments, ( , , , ) = ( , , − ). B

.

T



.









9



,



,







;



B

B

H

H

9

B

9

')(



.



.

T

B

T

Reference: Mathematical Encyclopedia (1977, Vol. 1).

9.6.1-3. The case of nonhomogeneous boundary conditions. Preliminary transformations. To solve the problem with nonhomogeneous boundary conditions (1), (3), (4), we choose a sufficiently smooth “test function” = ( , ) that satisfies the same boundary conditions as the unknown function; thus, (1) [ ] = (1) ( ) at = 1 ( = 1, , ), (10) (2) (2) [ ]= ( ) at = 2 ( = + 1, , ). U

U

U

,

U

,









A





A

!"!"!

Q

J

B

B

Q

!"!"!



J

Otherwise the choice of the “test function” is arbitrary and is not linked to the solution of the equation in question; there are infinitely many such functions. B

Let us pass from

= V

B

U

( , ) to the new unknown V

W

X

= ( , ) by the relation Y

Y

W

X

( , ) = ( , ) + ( , ). V

W

X

Y

W

X

Z

W

(11)

X

Substituting (11) into (1), (3), and (4), we arrive at the problem for an equation with a modified right-hand side, [

[

− Y

[

\

[ ]=

,^ ]

Y

( , ), _

W

X

( , )= _

W

X

( , )− _

W

+ Z

X

[

X

\

,^ ]

[ ],

(12)

Z

X

subject to the nonhomogeneous initial condition = ( ) − ( , 0) Y

`

W

Z

at

W

=0 X

(13)

and the homogeneous boundary conditions a

a

(1) b

[ ] = 0 at

(2) b

[ ] = 0 at

=

Y

W

Y

= W

W

1

(

W

2

(

= 1, c

, ),

deded

f

= + 1, c

f

dedgd

(14)

, ). h

The solution of problem (12)–(14) can be found using the Green’s function by formula (6) in which one should replace by , ( , ) by ( , ), and ( ) by ( ) = ( ) − ( , 0). Taking into account relation (11), for we obtain the representation V

Y

_

W

X

_

W

X

`

W

2 ]

( , )= V

W

X

i

j

2

i

k

W

j

X

Z

+

lmj

j

k

W

j

X

W

2 ]

^ i

( , , , ) k

W

j

X

n

\

o

2 ]

i

Z

i

n

k

W

j

X

n

lmj

+ ( , )

lmn

Z

W

X

[ Z

( , ) ( , , , ) j

[

0 j

2 ]

^ i

j

1 ]

]

[ ( , )]

,p

( , ) ( , , , ) _

0

−

lmj

V

^ i

1 ]

0

W

( , 0) ( , , , 0) Z

i

`

1 ]

+

W

( ) ( , , , 0) `

]

−

`

n

1

n

k

W

j

X

n

lmj

lmn

lmj

n

lmn

.

(15)

1 ]

Changing the order of integration and integrating by parts with respect to , we find, with reference to the initial condition (8) for the Green’s function, n

^ i

[ Z k [

0 n

lmn

= ( , ) ( − ) − ( , 0) ( , , , 0) − Z

j

X

q

W

j

Z

j

k

W

j

X

[ ^

i

Z

0

( , ) j

n

k

( , , , ) W

[

j

X

n

lmn

.

(16)

n

© 2002 by Chapman & Hall/CRC Page 645

646

HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS

We transform the inner integral of the last term in (15) using the Lagrange–Green formula [see Kamke (1977)] to obtain 2 ]

k

1 ]

\

w

[ ]

,p o

w

Z

k

‚r 

,ƒ

Z

y

\

[ ]

,p

or

1

w

z8{

(−1) u

( , ) j

k

n

k

, |

}

−1

vx~

k

„

t o

y ~

v

y

(−1)

+€ = 

„

}

}

j

~

z7{ 

}

, ) n

k

|

,

j

and …

+1 ( j 

€ j

n

(17) y

y €



= ( , ); and

[ ] of (2);

,ƒ ‚



k

u

=0

[ ] is the differential form adjoint with

Z

=] 2 , =] 1

o

[ , ]

s

t

[ , ]≡

s

+

lmj

j

=0

where

i

y

vxw

[ ]≡

,p

or

=

lmj

] w

\

2 ]

i

are nonnegative integers. †

Using relations (16) and (17), we rewrite solution (15) in the form ‡

2

( , )= ˆ

‰

Š

( ) ( , , , 0)

‹



1

Ž

ˆ



‰



+ 

2 Š



Š

0

Œ

( , ) ( , , , )

‹



1

’

Ž

ˆ



‰

’







+ ’

Š



[ , ]

s

“

0

‘

Ž

= =

” ”8•

2 1 ‹



. (18) ’

‹ •

This formula was derived taking into account the fact that the Green’s function with respect to and satisfies the adjoint equation* ‹

‹

’



–

–

+ Ž

—

, ˜

[ ] = 0. Ž

™

’ •

For subsequent analysis, it is convenient to represent the bilinear differential form œ

œ

C

–

s

“

−1

›œ

[ , ]= Ž

œ



œ 

[ ]=

[ ], “

Ž

Ž

›žC

C

Ÿ¡ 

[ , ] as “

Ž

œ

–

(−1)

š



=0

C

− −1

–

š

s

–

C œ

+ +1 (

, ) 

’

Ž

. ¢

(19)



=0

Note that in the special case where operator (2) is binomial, –

 

  –

[ ]=

, —

 

‡

+

‡

š

0(

= const,

, ) , ‡

ˆ

‰

ˆ  ‹

š

š

š

the differential forms in (19) are written as œ

œ

 

£

› œ

[ , ]= “

(−1) =0

š

œ

− −1 –

“





œ 

 

œ

− −1 –

[ ]=

,

Ž

− −1

š š

– œ

–

− −1

œ

–

œ

−1

š

Ž

œ

–

− −1

(−1)

Ž

 š

š š

Ž

− −1

š

.

š

9.6.1-4. The case of special nonhomogeneous boundary conditions. Consider the following nonhomogeneous boundary conditions of special form that are often encountered in applications: œ¥¤

–

œ¥¤

œ¥¤ – ‡

– ˆ

(1)

=

œ¥¤

¦

–

=

( ) at ‰

ˆ

ˆ

1

(

2

(

= 1, A

!"!"!

, ), §

(20)

œ¥¤

œ¥¤ ‡

=

(2) ¦

( ) at ‰

ˆ

= ˆ

= + 1, A

§

!"!"!

, ). ¨

ˆ

Without loss of generality, we assume that the following inequalities hold: C

¨

−1≥ ©

1

>

2 ©

>

ª"ª"ª

>

C

, ©

¨

−1≥ ©

C

>

+1

©

+2

>

ª"ª"ª

>

. ©

š

The Green’s function satisfies the corresponding homogeneous boundary conditions that can be obtained from (20) by replacing by and setting (1) ( ) = (2) ( ) = 0. œ¥¤

œ¥¤

‡

Ž

¦

‰

¦

‰

* This equation can be derived by considering the case of homogeneous initial and boundary conditions and using arbitrariness in the choice of the test function = ( , ); it should be taken into account that the solution itself must be independent of the specific form of , because does not occur in the original statement of the problem. By appropriately selecting the test function, one can also derive the boundary conditions (21). }

}

}

«

¬

}

© 2002 by Chapman & Hall/CRC Page 646

647

9.6. HIGHER-ORDER LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS

The adjoint homogeneous boundary conditions, with respect to (20), which must be met by the Green’s function with respect to and have the form 

œ®­



[ ] = 0 at

œ®­ 

’

=

Ž

ˆ

[ ] = 0 at Ž

= ˆ

ˆ

1

(

ˆ

2

(

≠

©

¯

©

≠ ¯

= + 1,

, ©

©

°

, B ²



§

= 1, œ

°

, ;

!"!"!

¨

, ;

!"!"!

§

= 1, ±

= + 1, ±

§

, ),

!"!"!

§

(21)

, ).

!"!"!

¨

These conditions involve the linear differential forms [ ] defined in (19). For each endpoint of the interval in question, the set { } of the indices in the boundary operators (21) together with the set { } of the orders of derivatives in the boundary conditions (20) make up a complete set of nonnegative integers from 0 to − 1. Taking into account the fact that the test function must satisfy the boundary conditions (20) and the Green’s function to conditions (21), we rewrite solution (18) to obtain Ž

©

©

¯

²

¨

“

Ž

2

( , )= ‡

ˆ

‰

Š

( ) ( , , , 0) 

‹

1

−

Œ

Š

‰





2

+ 

Š

Š 

0

œ¥¤

( )

¦

[ ]

’

=

Ž

” ”



1

’

Ž

ˆ



š Š









’

œ¥¤

( )

¦

[ ]

’

0

= +1

’



(2)

‹

‰

œ¥¤

›

² •

’

‘

+



1

( , ) ( , , , )

‹

‹

0

=1





(1)

³

²

ˆ

œ¥¤

‹ ›

œ¥¤ 

Ž

Ž

= ” ”

2



, ’

(22)

‹ •

where the [ ] are differential operators with respect to , which are defined in (19). If the Green’s function is known, formula (22) can be used to immediately obtain the solution of the nonhomogeneous boundary value problem (1), (3), (20) for arbitrary ( , ), ( ), (1) ( ) ( = 1, , ), and (2) ( ) ( = + 1, , ). ³

Ž



œ¥¤

œ¥¤

ˆ

‰

ˆ

‘

±

!"!"!

§

¦

‰

±

§

!"!"!

¦

‰

Œ

¨

9.6.1-5. The case of general nonhomogeneous boundary conditions. On solving (4) for the highest derivatives, we reduce the boundary conditions (4) to the canonical form œ¥¤

œ¥¤

´

–

–

œ¥¤

– ‡

−1

›µ´

+

œ¥¤ ˆ –

´

(1)

œ¥¤ – ‡

+

¶

´

‰

ˆ

= 1,

(

1 ˆ

±

, ),

!"!"!

§

(23)

œ¥¤ ´

–

(2)

=

( ) at

·

ˆ

´ ‡

(2)

=

() ‰

²

ˆ

(1)

=

‰

–

−1

›µ´

‡

()

²

œ¥¤

=0

œ¥¤

´

–

( ) at

·

‰

= ˆ

(

2 ˆ

= + 1, ±

§

, ),

!"!"!

¨

ˆ

=0 ¶

where the leading terms in different boundary conditions are different, −1≥ ¨

>

1 ©

2 ©

>

ª"ª"ª

>

, ©

−1≥ ¨

>

+1 ©

The sums in (23) do not contain the derivatives of orders 1 , (for = 2 ); thus, (1) ( ) = 0 at = ( = 1, ³

³

>

+2 ©

,

(for

³

©

!"!"!

©

>

ª"ª"ª

. ©

= ˆ

1)

š ˆ

´

ˆ

ˆ

(2) ¶

‰

¸

( ) = 0 at ‰

²

©

= ¸

¹

!"!"!

( = + 1, E

©

¹

+1 , ©

³

E

´

²

and

§

!"!"!

, ©

š ³

, ), §

, ).

!"!"!

¨

It can be shown that the solution of problem (1), (3), (23) is given by ¶

2

( , )= ‡

ˆ

‰

Š



1

œ¥¤

Š

=1

ˆ



œ¥¤

(1)

³

²

Œ

Ž

‹ ›

− 

( ) ( , , , 0)

‹



0

·



‰



2

+ 

Š

Š 

0

œ¥¤



1

’

[ ] Ž

= ” ”

1



’

‹

Ž

ˆ

+



‰

œ¥¤

›

(2) š

² •

’

‘

‹

( )

( , ) ( , , , )

‹

Š

= +1



·

0



’

( ) ’







’

œ¥¤

[ ] Ž

= ” ”

•

2



’

,

(24)

‹

where the [ ] are differential operators with respect to , which are defined in (19). Relation (24) is similar to (22) but contains the Green’s function satisfying the more complicated boundary conditions that can be obtained from (23) by substituting for and setting (1) ( ) = (2) ( ) = 0. ³

Ž



œ¥¤

œ¥¤

‡

Ž

·

‰

·

‰

© 2002 by Chapman & Hall/CRC Page 647

648

HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS

9.6.2. Equations Containing the Second Time Derivative 9.6.2-1. The case of homogeneous initial and boundary conditions. Consider the linear nonhomogeneous differential equation –

œ

–

2 –

‰

  – ‡

‡

+ ( , )

2

º

ˆ

−

‰

–

œ œ –

› œ

= ( , ).

( , ) š

‰

‡

ˆ

‰

ˆ

(1)

‰

ˆ

=0

‘

We set the homogeneous initial conditions = 0 at = 0 at

‡ –

‡

= 0, =0 ‰

‰

(2)



and the homogeneous boundary conditions (1) »

[ ] = 0 at

»

=

‡

²

(2)

ˆ

[ ] = 0 at

²

=

‡

ˆ

ˆ

1

(

ˆ

2

(

= 1, ±

!"!"!

, ), §

= + 1, ±

§

(3)

, ),

!"!"!

¨

where the boundary operators (1) [ ] and (2) [ ] are defined in Paragraph 9.6.1-1. The solution of problem (1)–(3) can be represented in the form* »

»

‡

‡

²

ˆ

Ž

=

2

( , )= ‡

Here,

²

‰

Š



Š

( , ) ( , , , )

‹

0



1

’

Ž

ˆ



( , , , ) is the Green’s function; for > ˆ



‰

’

–

‰

–

2 –

‰

  – Ž

2

+ ( , ) º

ˆ

’







. ’

(4)

≥ 0, it satisfies the homogeneous equation

‹

Ž

‰

‘

− Ž

‰

‰

œ

’

–

œ œ –

› œ

( , ) š

ˆ

Ž

‰

=0

(5)

ˆ

=0

with the special semihomogeneous initial conditions =0 at = ( − ) at

– Ž

Ž

¼

ˆ

‰



‰

= , = ’

(6)

’



and the corresponding homogeneous boundary conditions (1) »

² »

(2) ²

[ ] = 0 at

=

Ž

[ ] = 0 at Ž

ˆ

= ˆ

ˆ

1

(

ˆ

2

(

= 1, ±

!"!"!

= + 1, ±

§

, ), §

!"!"!

(7)

, ). ¨

The quantities and appear in problem (5)–(7) as free parameters ( 1 ≤ ≤ 2 ), and ( ) is the Dirac delta function. One can verify by direct substitution into the equation and the initial and boundary conditions (1)–(3) that formula (4) is correct, taking into account the properties (5)–(7) of the Green’s function. 

’

ˆ



ˆ

¼

ˆ

9.6.2-2. The case of nonhomogeneous initial and boundary conditions. Consider the linear nonhomogeneous differential equation (1) with the general nonhomogeneous initial conditions = 0 ( ) at = 0, (8) = 1 ( ) at =0 ‡

–



ˆ

‰

ˆ

‰

Œ

‡

Œ

* Problem (1)–(3) is assumed to be well posed.

© 2002 by Chapman & Hall/CRC Page 648

649

9.6. HIGHER-ORDER LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS

and the nonhomogeneous boundary conditions, reduced to the canonical form (see Paragraph 9.6.1-5): œ¥¤

œ¥¤

´

–

–

œ¥¤ – ‡

+

œ¥¤ ˆ –

´

−1

› ´

œ¥¤ – ‡

+

·

=

( ) at ‰

ˆ

= 1,

(

1 ˆ

±

, ),

!"!"!

§

ˆ

´

(9)

œ¥¤ ´

–

(2)

(1)

= ´

–

‡

(2)

=

() ‰

²

ˆ

()

¶

−1

› ´

‡

‰

²

œ¥¤

=0

œ¥¤ ´

–

(1)

·

( ) at ‰

= ˆ

(

2 ˆ

= + 1, ±

§

, ).

!"!"!

¨

ˆ

=0 ¶

Introducing a test function = ( , ) that satisfies the nonhomogeneous initial and boundary conditions (8), (9) and using the same line of reasoning as in Paragraph 9.6.1-3 for a simpler equation, we arrive at the solution of problem (1), (8), (9) in the form “

‡

( , )= ˆ

‰

2 Š

Š 

− Š

−

Ž

0(

) 



‰

’





ˆ

·

’



¾

’



2

( )

[ ]

=

Ž

” ”

1

+ 

=0 ™

’



0

‰

œ¥¤ 

(1)



½

œ¥¤

Š

( , , , ) Ž

’

=1

ˆ

Ÿ

Œ

³

œ¥¤ ²

’

–

‘

‹ ›

‰

– ‹

‹

1





1 2

ˆ

( , ) ( , , , )

‹

0

“



Œ

š Š

) ( , 0) 

º



[ ]

’

0



( , , , 0)

¢¥Ž

ˆ



‰





œ¥¤

( )

· 

0( Œ

(2)

= +1

‹

)+ 

œ¥¤

‹ ›

² •

1(

‹

1

+ ’

Š

Ž

= ” ”

2



’

,

(10)

‹ •

where the [ ] are differential operators with respect to , which are defined in relations (19), Paragraph 9.6.1-3. If the coefficients of equation (1) and those of the boundary conditions (9) are time independent, i.e., ³

Ž

¿

À¥Á

Â

îÄ



Å

œ

 

= ( ), º

º

œ

 

=

ˆ

( ), ˆ

´

´

(1)

(2)

= const,

²

then in solution (10) one should set

= const,

²

¶

¶

–

–

–

( , , , )= Ž

ˆ



‰

’

–

( , , − ),

Ž Æ

ˆ



‰

’

Ž

( , , , ) ˆ



‰

”

’

=0

”

’ ”

™

=−

( , , ).

Ž Æ

ˆ



‰

‰

9.6.3. Nonstationary Problems with Many Space Variables 9.6.3-1. Equations with the first-order partial derivative with respect to . ‰

Consider the following linear differential operator with respect to variables œ¥Ë

œ

–

œ

[ ]≡

Ç

œ –

È

‡

x, œ

›

œ¥Ë

1,

ÉÊÉÊÉ

( 1,

,

ˆ

!"!"!

,

, ) ˆ

‰

œ¥Ë ˆ  š

1

1 + ÌÊÌÊÌ 1

+

!"!"!

ˆ

1,

!"!"!

,

: ˆ

œ¥Ë š ‡ –

.

(1)

ˆ

È

The coefficients of the operator are assumed to be sufficiently smooth functions of 1, , , and (and also bounded if necessary). The coefficients of the highest derivatives 1, are assumed to be everywhere nonzero. š

ÉÊÉÊÉ

ˆ

!"!"!

ˆ

‰

š

1 . Cauchy problem ( ≥ 0, x ). The solution of the Cauchy problem for the linear nonhomogeneous parabolic differential equation with variable coefficients Í

‰

Î

Ï

š

–

– ‡

−

[ ] = (x, )

Ç

‡

x,

‰

(2)

‰ 

under the initial conditions ‡

= (x) at

‘

‰

=0

(3)

Œ

© 2002 by Chapman & Hall/CRC Page 649

650

HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS

is given by Ë

(x, ) = ‡

‰

Š



Š

Ë

(y, ) (x, y, , ) y Ð

’

0

Ñ

‰

’





+ ’

Š

(y) (x, y, , 0) y, Ð

Ñ

‘

‰



y= 





1

!"!"!



. 

(4)

Œ š

Here, = (x, y, , ) is the fundamental solution of the Cauchy problem, which satisfies for > ≥ 0 the equation Ñ

Ñ

‰

’

‰

’

–

–

−

Ñ

[ ]=0

Ç

x,

(5)

Ñ

‰

and the special initial condition



Ñ

=

” ”

= (x − y).

(6)

¼

™

The quantities y and appear in problem (5), (6) as free parameters (y ), and (x) is the -dimensional Dirac delta function. of operator (1) are independent of time , then the fundamental If the coefficients 1, , solution depends on only three arguments, (x, y, , ) = (x, y, − ). If the coefficients of operator (1) are constants, then (x, y, , ) = (x − y, − ). 

’

œ

Î

Ï

¼

œ¥Ë

š

¨

È

‰

ÉÊÉÊÉ

Ñ

Ñ

‰

’

‰

’

Ñ

Ñ

‰

‰

’

’

2 . Boundary value problems ( ≥ 0, x ). The solutions of linear boundary value problems in a spatial domain for equation (2) with initial condition (3) and homogeneous boundary conditions for x (these conditions are not written out here) are given by formula (4) in which the domain should be replaced by . Here, by we mean the Green’s function that must of integration satisfy, apart from equation (5) and the boundary condition (6), the same homogeneous boundary conditions for x as the original equation (2). For boundary value problems, the parameter y belongs to the same domain as x, i.e., y . Í

‰

Î

Ò

–

Ò

Î

Ò

Ï

Ò

Ñ

–

š

Î

Ò

Î

Ó)Ô

Ò

Reference: Mathematical Encyclopedia (1977, Vol. 1).

9.6.3-2. Equations with the second-order partial derivative with respect to . Õ

1 . Cauchy problem ( ≥ 0, x ). The solution of the Cauchy problem for the linear nonhomogeneous differential equation with variable coefficients Í

Õ

Î

Ï

š

–

–

2 Ö

−

2 Õ

[ ] = (x, )

Ç

Õ

‘ 

under the initial conditions –

= (x) at = (x) at Ö

¦

is given by (x, ) = Õ

×

Ø

×

Ù

Ú

(y, ) (x, y, , ) y Û

Ü

0

− ×

Ù

Ú

(y) ß

à

á

Ý

Ý

Õ

Ü

Þ

(x, y, , ) â

Ü

ã

ä

á

=

Þ

(8)

Ü

Þ

=0

Ü

Ý

Õ



Ö

Here,

= 0, =0 Õ

Œ

Ö

(7)

Ö

x,

y+ ×

Ù

Ú

å

(y) (x, y, , 0) y. Ý

â

Þ

(x, y, , ) is the fundamental solution of the Cauchy problem

Ý

â

Ü

2 á â

Ý

2

−

[ ] = 0,

x, æ

Ý

Ø á

ä Ý

= ç ç

Ø

= 0,

Ý ç

á

ç

ä

= ç

â ç á

= (x − y), è

where y and play the role of parameters. If the coefficients of operator (1) are independent of time , then the fundamen1, , tal solution depends on only three arguments, (x, y, , ) = (x, y, − ), and the relation (x, y, , ) =0 = − (x, y, ) holds. If the coefficients of operator (1) are constants, then (x, y, , ) = (x − y, − ). Ø

Ü

Ú

é

ì

ì

ä

ê

â

ì

Ü

â

Ý

ì

ä

Ý

ëÊëÊë

ê

ç

Ý

â

Ü

Ý

â

Ü

â

ç

Ý

â

Ü

Ý

â

Ø

Ü

© 2002 by Chapman & Hall/CRC Page 650

651

9.6. HIGHER-ORDER LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS

2 . The solution of the Cauchy problem for the more complicated linear nonhomogeneous differential equation with variable coefficients í

2 î

â

î

+ (x, )

2

á

ï

â

−

á

[ ] = (x, ) î

x, æ

Û

â

â Ø

á

á

with initial conditions (8) is expressed as (x, ) = î

â

Ø ×

Ù ×

Ú

(y, ) (x, y, , ) y Û

Ü

0

−

Ù ×

Ú

(y) ß

à

Ý

á

â

Ü

Þ

(x, y, , )

Ý

â

Ü

ã

Ü

y+ Þ

ä

=0

Ü á

Here,

Þ

Ù ×

Ú

(y) + (y, 0) (y)

ð7å

ï

ß

ñ

(x, y, , 0) y.

Ý

â

(9)

Þ

(x, y, , ) is the corresponding fundamental solution of the Cauchy problem,

Ý

â

Ü

2

+ (x, )

Ý

2

á â

ï

−

Ý

â

á

á

= 0, ä

= ç ç

Ý

Ø

á

Ý

[ ] = 0,

x, æ

â

Ý ç

ä

ç á

è

= ç

â

Ø

= (x − y).

ç

á

3 . Boundary value problems ( ≥ 0, x ). The solutions of linear boundary value problems in a spatial domain for equation (7) with initial condition (8) and homogeneous boundary conditions for x (these conditions are not written out here) are given by formula (9) in which the domain of integration should be replaced by . Here, by we mean the Green’s function that must satisfy, apart from equation (7) and the initial conditions (8), the same homogeneous boundary conditions as the original equation (7). Ø

í

â

ò

ó

ó

ò

ó

á

ô

õ

ó

Ý

9.6.4. Some Special-Type Equations 1. ö

= ( ) ÷

ù

ö

ú

+ ÷

( )+ ( ) ý

ð

ø

û

The transformation ö

ø

ö

û

ÿ

â

â

à

ö ñ

×

() ß

â

Þ

 à

×

â

+ ( ) . ÷

þ

ø

ø ö

Ü

where  ( ) = exp

ø

ú

( , ) = (
, ) exp î

ü

() â

Þ

â

, ã

=


ÿ



( )+ â

=

 á

= 

÷

2



ö

ú

( ) ( ) â

â

Þ

â

, Ü

= ×



( ) â

õ

() â

Þ

â

,


õ

á

. ÷

ú û

The transformation
= 1  ö

å

.

õ á

Ü

ö

×

, leads to the simpler constant coefficient equation ã

á

2.

÷

û

ø

ö

û

ú

ÿ

=

,

î ÿ

1−

leads to the constant coefficient equation õ

ê

á ê

=  (−1)



3. ö

ö

÷

ø



=





ú



=0

á





ö

.

õ õ á




â

õ

á

. ÷

û ö

û

The change of variable
= ln | | leads to a constant coefficient equation. ÿ

© 2002 by Chapman & Hall/CRC Page 651

652

HIGHER-ORDER PARTIAL DIFFERENTIAL EQUATIONS 

4. ö

÷

2

= ( 

+

+ ) 

û ö

ú ö

. ÷

ú û

ø

ö

The transformation

û ú

( , ) = (
, )|  î

ÿ

â

â

2 ÿ

+

¥ÿ

+ |

−1 2 , õ

=


Þ ×

 ÿ

2

ÿ

+

¥ÿ

+

leads to a constant coefficient equation. 5.

–

 ö

  

= 0, ú

= 1, 2, 

  

÷ ö

ø

Here,   is a linear differential operator of any order with respect to the space variable coefficients can depend on .

whose ÿ

ÿ

1 . General solution: í

( , )= î

ÿ

−1 õ

â

ê

ÿ

â

=0 ê

where the = ( −   ) = 0.

( , ),

ê

â

= 1:

( , ) are arbitrary functions that satisfy the original equation with

ê

ê

ÿ



â

ê

á

2 . Fundamental solution: Ø

í

( , )=

Ý

ÿ

â

õ

where

ÿ

Â

2

6. 

ö

ö

( − 1)!

1(

Ý

, ), ÿ

â

â

 

¥Á

−1

, ) is the fundamental solution of the equation with  = 1.

1(

Ý

 

âeõ

The linear differential operator 

–

2

  

= 0, ú



= 1, 2,

can involve arbitrarily many space variables. 

  

÷

ø

Here,   is a linear differential operator of any order with respect to the space variable coefficients can depend on .

whose ÿ

ÿ

1 . General solution: í

( , )= î

ÿ

−1 õ

â

= = 0. ê

ê

ê

ê

ÿ

â

=0 ê

where the ( −  )

( , ),

ê â

( , ) are arbitrary functions that satisfy the original equation with ÿ

â



= 1:

á

2 . Suppose that the Cauchy problem for the special case of the equation with  = 1 is well posed if only one initial condition is set at = 0; this means that the constant coefficient differential operator   is such that the equation with  = 1 is regular with regularity index  = 1. Then the fundamental solution of the original equation can be found by the formula Ø

Ø

í

â

( , )=

Ý

ÿ

â

õ

where  

Ý

¥Á

1( Â

ÿ

âeõ

−1

( − 1)!

Ý

1( ÿ

, ), â

, ) is the fundamental solution for  = 1.

 

â

The linear differential operator  

can involve arbitrarily many space variables.

© 2002 by Chapman & Hall/CRC Page 652

653

9.6. HIGHER-ORDER LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS

7.





  

[ ] = 0. ÷

=0

Here,  is any linear differential operator with arbitrarily many independent variables Particular solutions: ( 1, î

ÿ

where the equation ê

!"!"!

, ÿ

õ

)=

!

( 1, 

ÿ

are solutions of the equations  [ ] − # $ 

=0

 ê

# ê

= 0, and the

!"!"!

,

= 0, the

1,

!"!"!

, ÿ

õ

.

),

ÿ

=1 "

ÿ

õ

#

are roots of the characteristic

are arbitrary constants. "

© 2002 by Chapman & Hall/CRC Page 653

Supplement A

Special Functions and Their Properties Throughout Supplement A it is assumed that  is a positive integer, unless otherwise specified.

A.1. Some Symbols and Coefficients A.1.1. Factorials Definitions and some properties: 0! = 1! = 1,  ! = 1 ⋅ 2 ⋅ 3 ( − 1) , (2 )!! = 2 ⋅ 4 ⋅ 6 (2 − 2)(2 ) = 2  !, (2 + 1)!! = 1 ⋅ 3 ⋅ 5 

!! = *

= 2, 3, 

!"!"!

!"!"!

,

õ

!"!"!

(2 − 1)(2 + 1) =

!"!"!

(2 + )!! if (2 + + 1)!! if

 

= 2+ , = 2 + + 1,

2 %

õ

+1

&

' ( 

+

3 , 2)

+

= 1, 2,

0!! = 1.

A.1.2. Binomial Coefficients Definition: " , .

General case:

! , where + !( − + )! (− / ) / ( / − 1) = (−1) = + !, 

= -

" ,

,

. " 0

=

+ !"!"!

+

( / + 1) , ( 1 + 1) ' ( / − 1 + 1) '

'

= 1,

!"!"!

, ,

( / − + + 1) , !

where '

where

!"!"!

(2 ) is the gamma function.

Properties: .0

= 0 for + = −1, −2, or + > 3 , / − 1 . +1 = . −1 = . , . + . +1 = . +1 +1 , 1 +1 1 +1 0 " 0 " 0 " 0 " 0 " 0 - (23 − 1)!! (−1) , - 5 2 - = (−1) −1 4 2 = 22 (23 )!! - −1 - −1 - −1 (−1) (−1) (23 − 3)!! , 5 2 - −2 = 14 2 = 3 22 −1 3 (23 − 2)!! -2 +1 - −4 - −1 −2 2- +1 4 2 = (−1) 2 5 2- , 5 2 - +1 4 2 = 2 5 4 - +1 , "

= 1,

-

!"!"!

/ " ,

" " 5 5

14 2 5 -

= &

- +1 5 2-

22

,

- 4 2 5 -

=

22 &

-

( - −1) 4 2 . 5 -

© 2002 by Chapman & Hall/CRC Page 655

656

A. SPECIAL FUNCTIONS AND THEIR PROPERTIES

A.1.3. Pochhammer Symbol Definition and some properties ( + = 1, 2,

):

!"!"!

' (1 − / ) (/ + 3 ) = (−1) , ' (/ ) ' (1 − / − 3 ) (3 + + − 1)! , ( / )0 = 1, ( / ) - + = ( / ) - ( / + 3 ) , (3 ) = (3 − 1)! , , ' ( / − 3 ), (−1) = ( / )− - = , where / ≠ 1, , 3 ; ' (/ ) (1 − / ) - (23 )! - (23 + 1)! (1) - = 3 !, (1 6 2) - = 2−2 , (3 6 2) - = 2−2 , 3 ! 3 ! (/ )8 +( / )2 (/ ) (/ + + ), (/ + 3 )- = , (/ + 3 ) = . (/ + 7 + )- = (/ )8 , , (/ ), (/ )

( / ) - = / ( / + 1)

( / + 3 − 1) =

!"!"!

'

!"!"!

,

,

,

A.1.4. Bernoulli Numbers Definition: 9 :

2

=

−1


0

exp(−? 2 ) @ ? ,

erfc 2 = 1 − erf 2 =

Expansion of erf 2 into series in powers of 2 as 2 erf 2 = %

2 &


:

exp(−? 2 ) @ ? . ;

0: A

2 2 +1

(−1) ;

2

% &

2

=

% &

exp B −2

2C

< ;

=0

2 2 2 +1 . , 2 + , + 1)!!

,

:

(−1)

8

8 =0

2

C B 12 8 2 8 +1

+G

B |2

|−2 F

−1 CIH

, J

= 1, 2,

!"!"!

A.2.2. Exponential Integral Definition: Ei(2 ) =

:

9 K

>

@ ? ?

−

;

Ei(2 ) = Llim M

+0 N

−L 9 K

> ?

−

@ ?

+> L

:

9 K ?

@ ?PO

for 2

< 0,

for 2

> 0.

;

© 2002 by Chapman & Hall/CRC Page 656

A.3. SINE INTEGRAL

AND

657

COSINE INTEGRAL. FRESNEL INTEGRALS

Other integral representations: −: >

Ei(−2 ) = − 9

9 −: >

Ei(−2 ) =

Ei(−2 ) = −2

0

2 ;

0

>

2 ;

9 −: K ;

1

sin ? + ? cos ? 2 2+? 2 sin ? − ? cos ? @ 2 2+? 2

Expansion into series in powers of 2 as 2

ln ?

RR U

Ei(2 ) =

RR

+ ln 2 +


0,

for 2

< 0, > 0.

2

if 2

< 0,

if 2

> 0,

,

!"!"!

Ei(−2 ) =

⋅,+ ! +

=1




For small 2 , li(2 ) ≈

if 0 < 2 < 1, @ ?

2

ln(1 6

> 1. 2

.

) 2

if O

ln ?

1+ L

1: A

li(2 ) =

U

+ ln |ln 2 | +

< ;

+

=1

ln 2 . ⋅, + !

,

A.3. Sine Integral and Cosine Integral. Fresnel Integrals A.3.1. Sine Integral Definition: Si(2 ) =

: >

0

sin ? ?

@ ?

si(2 ) = − >

,

:

sin ? ;

?

= Si(2 ) −

@ ?

W

2

.

Specific values: Si(0) = 0, Properties:

Si(−2 ) = − Si(2 ),

Si( D ) = W

2

si( D ) = 0.

,

si(2 ) + si(−2 ) = − , W

lim : M −

si(2 ) = − . W

;

© 2002 by Chapman & Hall/CRC Page 657

658

A. SPECIAL FUNCTIONS AND THEIR PROPERTIES 0:

Expansion into series in powers of 2 as 2 A




cos ? ;

:

?

U

=

@ ?

:

+ ln 2 + >

0

A

+ ln 2 + 5

< ;

=1

Asymptotic expansion as 2 E F

Ci(2 ) = cos 2 where J

,Y

< −1

D

8

8 2 2

U

,

= 0.5772

!"!"!

(−1) 2 2 . 2 + (2, + )!,

,

− 1)!

(−1) (27

8 =1

= 1, 2,

:

A

@ ?

?

0:

Expansion into series in powers of 2 as 2 Ci(2 ) =

cos ? − 1

+G

C H

|−2 F

B |2

E X

+ sin 2

< −1

8

(−1) (27 )! 8 2 2 +1

8 =0

+G

B |2

|−2 X

−1 C H

,

!"!"!

A.3.3. Fresnel Integrals Definitions: Z

5

(_ ) =

:

1 2

(2 ) = [

1 2 [

sin ?

>

[

?

0

W > W

cos ?

0

^

Expansion into series in powers of _ as _

[

Z

acb

2

(_ ) = \

?

0: A _

` =0

W

(_ ) = 5

2 \ W

Asymptotic expansion as _ Z

e

f

_

acb

=

@ ?

` =0

2 \ W

@ ?

= \

>

] ^

sin ?

] ^

cos ?

0

2 Wb

>

0

2

@ ?

2

,

@ ?

.

b

(−1) _ 2 +1 , b (4 d + 3) (2b d + 1)! (−1) _ 2 . (4 d + 1) (2 d )!

:

1 cos _ sin _ − [ (_ ) − [ (_ ), 2 2 _ g 2 _ h W W 1 sin _ cos _ + [ (_ ) − [ (_ ), 5 (_ ) = 2 2 _ g 2 _ h W W 1⋅3 1⋅3⋅5⋅7 1 1⋅3⋅5 (_ ) = 1 − + − ijiji , (_ ) = − + (2_ )2 (2_ )4 2_ (2_ )3 h (_ ) =

g

ijiji

.

© 2002 by Chapman & Hall/CRC Page 658

659

A.4. GAMMA AND BETA FUNCTIONS

A.4. Gamma and Beta Functions A.4.1. Gamma Function A.4.1-1. Definition. Integral representations. The gamma function, ' ( k ), is an analytic function of the complex argument for the points k = 0, −1, −2, For Re k > 0, k

everywhere, except

!"!"!

(k ) = '

l

−1 p − q r

`

0

mon

For −(s + 1) < Re k < −s , where s = 0, 1, 2, t

(k ) = l

p −q `

0

−

u

!"!"!

. m

, w

a

v w =0

(−1) x !

−1 r

y mon

. m

A.4.1-2. Some formulas. Euler formula t

v z

Simplest properties: t

!s s

( k ) = lim

( k + 1) k

n

!"!"!

( k ≠ 0, −1, −2,

(k + s )

!"!"!

).

`

( k + 1) = k

t

t

( k ),

(s + 1) = s !,

t

t

(1) = (2) = 1.

Symmetry formulas: t

t

( k ) (− k ) = − t |

t

{

sin(

k

1 +k 2

}

k {

t |

t

, ( k ) (1 − k ) = { ) sin( { 1 { −k } = . 2 cos( k ) k

)

,

{

Multiple argument formulas: t

(2 k ) = t

t

(3 k ) = (s k

22

−1 t

[ n

33

{

(k )

−1 ~ 2

n

2

t

t |

(k )

k

t |

+

1 } , 2

k

+

{

1 } 3

) = (2 )(1− {

)~ 2

v

−1 ~ 2

s

v n

t |

 b −1 t | v

k

k

2 } , 3

+ +

=0

d s

}

.

Fractional values of the argument: t | t |

1 } = € , { 2 1 − } = −2 € 2

t |

{

,

t |

s

1 € } = { (2s − 1)!!, 2 2 v 1 2 € { − s } = (−1) . v 2 v (2s − 1)!! +

Asymptotic expansion (Stirling formula): t

(k ) = €

2

p − { n

k n

−1 ~ 2 

1+

1 −1 12 k

+

1 −2 288 k

+ ‚ (k

−3

)ƒ

(|arg | k < ). {

© 2002 by Chapman & Hall/CRC Page 659

660

A. SPECIAL FUNCTIONS AND THEIR PROPERTIES

A.4.1-3. Logarithmic derivative of the gamma function. Definition: „

Functional relations: „

t

r

(k ) =

t …

ln ( k ) r

t

(k ) . ( n k )

{

), 1 )− ,

=

k

„

1 ( k ) − „ (1 + k ) = − , „

k

„

( k ) − (1 − k ) = − cot( „

{

(† k ) − (− k „ ) =† − cot( „

1 2

„

(x

+k k

1 2

−k

+ x

− ‡

) = ln x

{

k

k {

k

‡ = w −1 „{ aˆb

1

tan( |

+ k

), k

{

d x

=0

. }

Integral representations (Re k > 0): „

(k ) = „

l

0

‰p − q `

− (1 + )− m

( k ) = ln k + l „

( k ) = −Š + l

0 1

 −1 `

„

m

m

− (1 − p

m

−1

1− 1 −m n

0

−1 r

ƒ n

r m

m

,

− q −1 p − q

)

ƒ

r n

m

,

,

is the Euler constant. where Š = − (1) = 0.5772 Values for integer argument: !"!"!

„

„

(1) = −Š ,

aˆb −1 −1 d v

( s ) = −Š +

(s = 2, 3,

!"!"!

)

=1

A.4.2. Beta Function Definition: ‹

(_ , Π) =

1

l

−1

(1 − ) 

0 m ^

−1 r

m

m

,

where Re _ > 0 and Re Œ > 0. Relationship with the gamma function: ‹

t

(_ , Π) =

t

t

(_ ) (Π) . (_ + Π)

A.5. Incomplete Gamma and Beta Functions A.5.1. Incomplete Gamma Function Definitions (integral representations): Ž

t

( , _ ) = l

( , _ ) = l

0

^ p −q

^

`

−1 r

m‘

p −q

m

t

−1 r

m 

Re  Ž > 0,

,

m

= (  ) − (  , _ ).

© 2002 by Chapman & Hall/CRC Page 660

661

A.6. BESSEL FUNCTIONS

Recurrent formulas:

(
+ 1,  ) =


(
,  ) −    − , 

(
+ 1,  ) =


(
,  ) +    − .

0:

Asymptotic expansions as  

(
,  ) =





Asymptotic expansions as  







(−1)   + ,  ! (
+  )

=0





(
,  ) = (
) −





=0



(−1)   + .  ! (
+  )

:



−1 − 

(
,  ) = (
) −  





−1 (1 −
)   (− )  =0

−1 (1 −
)   (
,  ) =   −1  −     =0 (− )

+   | |−   ,

+   |  |−   

 − 32  < arg  < 32 . 

Integral functions related to the gamma function: erf  = 

1

 1

2



,  2 ,

1   1 2 , , 2 

erfc  = 



Ei(− ) = − (0,  ).

A.5.2. Incomplete Beta Function Definition:   ( ,  ) = 

1 0

−1



(1 − ) 

−1

! , 

where Re  > 0 and Re " > 0.

A.6. Bessel Functions A.6.1. Definitions and Basic Formulas A.6.1-1. The Bessel functions of the first and the second kinds. The Bessel function of the first kind, # $ ( ), and the Bessel function of the second kind, % $ ( ) (also called the Neumann function), are solutions of the Bessel equation  2 "  &'&  +  "  & + ( 2 − ( 2 )" = 0

and are defined by the formulas # $ ( ) =

*)



) $

)

=0

(−1) ( + 2) +2 ,  , , ! ( ( + + 1)

% $ ( ) =

# $ ( ) cos  ( − # − $ ( ) . sin  (

(1)

The formula for % $ ( ) is valid for ( ≠ 0, - 1, - 2, ././. (the cases ( ≠ 0, - 1, - 2, ././. are discussed in what follows). The general solution of the Bessel equation has the form 0 $ ( ) = 1 1 # $ ( ) + 1 2 % $ ( ) and is called the cylinder function.

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A. SPECIAL FUNCTIONS AND THEIR PROPERTIES

A.6.1-2. Some formulas. 2 ( 0 $ ( ) =  [ 0 $ −1 ( ) + 0 $ +1 ( )], 1 ( !  0 $ ( ) = 2 [ 0 $ −1 ( ) − 0 $ +1 ( )] = - 2  0 $ ( ) − 0 $43 1 ( )5 , !

!

!

$ −$ 0 $ ( )] = − − 0 $ +1 ( ), !  [



6 6

1 ! 1 ! $ $

$ $ −

[ # $ ( )] =  [ − # $ ( )] = (−1)  − − # $ + ( ), # $ − ( ),  !  7  !  7



 # − ( ) = (−1) # ( ), % − ( ) = (−1) % ( ), = 0, 1, 2, ././.

[

! 

$

0 $ ( )] = 

$

0 $ −1 ( ),

A.6.1-3. The Bessel functions for ( = -  2

# 1 8 2 ( ) = 9

 

# 3 8 2 ( ) = 9

2

 

# +1 8 2 ( ) = 9

where  = 0, 1, 2, ././. : # −1 8 2 ( ) = 9

sin  , 6

1 

sin  − cos 

# −3 8 2 ( ) = 9 )

, 7

2  

2

cos  , 6

−

 

1 

cos  − sin 

) , [ 8 2]   * ) 2 (−1) ( + 2 )! , , sin   −   )  )2 2 (2 )! ( − 2 )! (2   =0 



+ cos  − # − −1 8 2 ( ) = 9

1 2,

[( −1) 8 2]

*)

 

2

=0

, (−1) ( + 2 + 1)! , , (2 ) + 1)! ( − 2 − 1)! (2 )2

) , [ 8 2]  * ) 2 (−1) ( + 2 )!    , , cos  + 2 (2 )! ( − 2 )! ) (2 )2   =0 

− sin  +



 

[( −1) 8 2]

2

=0

2

% 1 8 2 ( ) = − 9

, (−1) ( + 2 + 1)! , , (2 + 1)! ( − 2 − 1)! (2 )2

*)

% −1 8 2 ( ) = 9

cos  ,  

% +1 8 2 ( ) = (−1) +1 # − −1 8 2 ( ),

2  

, 7

) 

+1

,

) 

+1

,

sin  ,

% − −1 8 2 ( ) = (−1) # +1 8 2 ( ).

A.6.1-4. The Bessel functions for ( = -  , where  = 0, 1, 2, ././. Let ( =  be an arbitrary integer. The relations

# − ( ) = (−1) # ( ),

% − ( ) = (−1) % ( )

are valid. The function # ( ) is given by the first formula in (1) with ( =  , and % ( ) can be obtained from the second formula in (1) by proceeding to the limit (   . For nonnegative  , % ( ) ) ) ; ; can be represented in the form ) 2



% ( ) = # ( ) ln − 2  ;  ;

where (1) = −< ,

−1 

1 :)

=0

,

( − − 1)!  2  , ! 

−2

− 

1 *)



=0

 (−1)   2

) −1 = , −1 (  ) = −< + , < = 0.5772 ././. is the Euler constant, =1

,

+2 ;

,

( + 1) + ( + + 1) , , , ! ( + )! 

( ) = [ln ( )] & is the

logarithmic derivative of the gamma function.

© 2002 by Chapman & Hall/CRC Page 662

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A.6. BESSEL FUNCTIONS

A.6.1-5. Wronskians and similar formulas: >

( # $ , # −$ ) = −

2

>

sin( ( ),

 

2 sin( ( ) # $ ( ) # − $ +1 ( ) + # − $ ( ) # $ −1 ( ) = ,   > Here, the notation ( ? , @ ) = ? @  & − ?  & @ is used.

(# $ , % $ ) =

2  

,

# $ ( ) % $ +1 ( ) − # $ +1 ( ) % $ ( ) = −

2  

.

A.6.2. Integral Representations and Asymptotic Expansions A.6.2-1. Integral representations. The functions # $ and % $ can be represented in the form of definite integrals (for  > 0):  # $ ( ) =   % $ ( ) = 

0

0



A cos( sin B − ( B ) ! B − sin  (  A sin( sin B − ( B ) ! B − 

For | ( | < 12 ,  > 0,

$





0

$DC

(

0

exp(− sinh − ( ) ! ,

+

− $DC

cos  ( ) 





−  sinh C

! . 



$

21+  − sin( ) !  ,  18 2 ( 1 − ( ) ( 2 − 1) $ +1 8 2 1  2   $ $ 21+  − cos( ) ! % $ ( ) = −  .  18 2 ( 1 − ( ) ( 2 − 1) $ +1 8 2 1  2 # $ ( ) =



For ( > − 21 , # $ ( ) =

For ( = 0,  > 0, # 0 ( ) =

$ 8 2 2( + 2) $  A cos( cos ) sin2 !  18 2 ( 1 + ( ) 0     2

2 



 0

sin( cosh ) ! ,  

% 0 ( ) = −

For integer ( =  = 0, 1, 2, ././. , 1

# ( ) =   

2

A cos(

0

2

# 2 ( ) = # 2 +1 ( ) =

 

8 2



0

A

0

A

8 2

 

−  sin ) ! 

(Poisson’s formula). 

2 



cos( cosh ) ! . 

0



(Bessel’s formula), 

cos( sin ) cos(2 ) ! , 





sin( sin ) sin[(2 + 1) ] ! . 





A.6.2-2. Integrals with Bessel functions:  

0

where 

H

$  E + +1 $ 2 ( F + ( + 1) G ( ( + 1) H

 F +( +1

F +( +3

2

 , Re( F + ( ) > −1, 2 2 4 ( J , K , L ; ) is the hypergeometric series (see Section 10.9 of this supplement),

 E # $ ( ) !  =

,

, ( +1; − I

I

2 cos( (  ) G (− ( ) + $ +1  F + ( + 1 F +( +3 =− $ , ( + 1, , −I  E % $ ( )! E I I 2  ( F + ( + 1) I H 2 2 4 0 I $ 2 2 G (( ) F −( +3 $  F −( +1 − , 1−( , , −I  , Re F > |Re ( | − 1. E − +1 F − ( +1I H 2 2 4

M

© 2002 by Chapman & Hall/CRC Page 663

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A. SPECIAL FUNCTIONS AND THEIR PROPERTIES

A.6.2-3. Asymptotic expansions as | | N 2

# $ ( )= 9 I

 I

−1

W 4 − 2( R − R I 4 S TVU X

cos Q P

: O

I

X

X

(−1) ( Y , 2Z )(2 )−2 + [ (| |−2 )\ I

=0 −1

W 4 − 2Y R − R − sin Q I 4 S T UX ^ _

2

( )= ` I

W 4a − 2 Y R − R 4 S T UX

sin Q

R a P

−1

1 (4 Y X 22 Z !

2

− 1)(4 Y

2

X

I

=0

X

e

X

=0

− 32 ) b/b/b [4 Y

U

−1

X

+ 1)(2a )−2

(−1) ( Y , 2Z

f

X −1 + [ (|a |−2 −1 )\ ] ,

=0 2

U

− 1)2 ] =

− (2Z

c

( 12

Z ! c

+Y +Z ) . +Y −Z )

( 21

R a f 2 g (a ) = (−1) g (cos a + sin a ) + [ (a −2 ), R a f 2 g +1 (a ) = (−1) g +1 (cos a − sin a ) + [ (a −2 ).

A.6.2-4. Asymptotic for large Y ( Y N _

X −1 + [ (| |−2 −1 )\ ] , I U

(−1) ( Y , 2Z )(2a )−2 + [ (|a |−2 )\

For nonnegative integer d and large a , e

U

+ 1)(2 )−2

(−1) ( Y , 2Z

W 4a − 2 Y R − R + cos Q 4 S T UX

where ( Y , Z ) =

I

O

):

1 a e Q i 2 Y S 2R Y

(a ) h

_

^ _

,

(a ) h − `

2

Q i

a

2Y S

R Y

−

_

,

where a is fixed, _ f

21 j 3 1 , (2 k 3) Y 1 j 3

(Y ) h

32 j 3

^ _

(Y ) h −

c

21 j 3 1 . (2 k 3) Y 1 j 3

31 j 6

c

A.6.3. Zeros and Orthogonality Properties of Bessel Functions A.6.3-1. Zeros of Bessel functions. _

^ _

Each of the functions f (a ) and (a ) has infinitely many real zeros (for real Y ). All zeros are simple, except possibly for the point a = 0. The zeros l X of f 0 (a ), i.e., the roots of the equation f 0 (l X ) = 0, are approximately given by l X

= 2.4 + 3.13 (Z

− 1)

= 1, 2, b/b/b ),

(Z

with maximum error 0.2%. A.6.3-2. Orthogonality properties of Bessel functions. _

1 m . Let n = n X be positive roots of the Bessel function f (n ), where Y > −1 and Z = 1, 2, 3, b/b/b _ Then the set of functions f (n X o k p ) is orthogonal on the interval 0 ≤ o ≤ p with weight o : q

r 0

f

_ s n X o p

t

f

_ s n u o p

t

o v o = w

0

1 2 2p

y f _ z ( n X ){ 2 = 1 p 2 f _ 2 ( n X ) +1 2

if Z if Z

≠x , =x .

© 2002 by Chapman & Hall/CRC Page 664

665

A.6. BESSEL FUNCTIONS

_

2 m . Let n = n X be positive zeros of the Bessel function derivative f z (n ), where Y > −1 and _ Z = 1, 2, 3, / b b/b Then the set of functions f (n X o k p ) is orthogonal on the interval 0 ≤ o ≤ p with weight o : q r if Z ≠ x , }'~ 0 _ s n X o _ s n u o 2 s o v o Y _ f = | 1 2 f 1 − 2 f 2 (n X ) if Z = x . p t p t 2p 0 n X

t

_

_

3 m . Let n = n X be positive roots of the transcendental equation n f z (n ) +  f (n ) = 0, where Y > −1 _ and Z = 1, 2, 3, b/b/b Then the set of functions f (n X o k p ) is orthogonal on the interval 0 ≤ o ≤ p with weight o : q r if Z ≠ x , }~ 0 _ s n X o _ s n u o o v o = | 1 2s  2−Y 2 _2 X f f 1+ f (n ) if Z = x . p t p t 2p 2 0 n X

t

4 m . Let n = n X be positive roots of the transcendental equation _ ^ _ _ ^ _ f (€ X  ) (€ X p ) − f (€ X p ) (€ X  ) = 0 ( Y > −1, Z Then the set of functions ‚ _

_

(€ X o ) = f (€ X o )

satisfying the conditions weight o : q

ƒ

‚ _

r

‚ _

(€ X o )

^ _

_

(€ X p ) − f (€ X p )

(€ X p ) =

‚ _

‚ _

^ _

= 1, 2, 3, b/b/b ). = 1, 2, 3, b/b/b ,

( € X o ), Z

( € X  ) = 0 is orthogonal on the interval p ≤ o ≤  with }'~ 0

( € u o )o v o = |

_ _ f 2(€ X p ) − f 2(€ X  ) _ „ 2 € 2X f 2(€ X  )

2

if Z

≠x ,

if Z

=x .

5 m . Let n = n X be positive roots of the transcendental equation ^ _ ^ _ _ _ f z (€ X  ) z (€ X p ) − f z (€ X p ) z (€ X  ) = 0

( Y > −1, Z

= 1, 2, 3, b/b/b )

Then the set of functions ‚ _

^ _ ^ _ _ _ ( € X o ) = f ( € X o ) z ( € X p ) − f z ( € X p ) ( € X o ),

satisfying the conditions weight o : q r

ƒ

‚ _

(€ X o )

‚ _

‚ _

z (€ X p ) = }~ 0

( € u o )o v o = |… …

‚ _

Z

= 1, 2, 3, b/b/b ,

z ( € X  ) = 0 is orthogonal on the interval p

2

„ 2 € 2X †

s

y f _ z ( € ‡ p ){ 2 s Y 2 − 1 − y f _ z ( € ‡  ){ 2 p 2 € 2‡ t ˆ

Y 2 1 − 2 2‡  € t

≤ o ≤  with if Z

≠x ,

if ‰

=x .

A.6.4. Hankel Functions (Bessel Functions of the Third Kind) The Hankel functions of the first kind and the second kind are related to Bessel functions by Š _ (1) _ ^ _ ( ‹ ) = f ( ‹ ) + Œ ( ‹ ),

Asymptotics for ‹ N Š Š

Š _ (1) Š _ (2)

Œ 2 = −1.

0: 2Œ

(1) 0 (‹

)h

(2) 0 (‹

2Œ ) h − „ ln ‹ ,

Asymptotics for | ‹ | N

Š _ (2) _ ^ _ ( ‹ ) = f ( ‹ ) − Œ ( ‹ ),

„

ln ‹ ,

Š _ (1) (‹ ) h Š _ (2)

(‹ ) h

( Y )_ c ( ‹ k 2) Œ ( Y )_ „ ( ‹ c k 2)

−„

Œ

(Re Y > 0), (Re Y > 0).

: O

(‹ ) h `

2  ‘ 1„ 1„ D „ ‹ exp ŽŒ ‹ − 2 Y − 4

(−„ < arg ‹ < 2„ ),

(‹ ) h `

2  ‘ 1„ 1„ D „ ‹ exp  −Œ  ‹ − 2 Y − 4

(−2„ < arg ‹ < „ ).

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A. SPECIAL FUNCTIONS AND THEIR PROPERTIES

A.7. Modified Bessel Functions A.7.1. Definitions. Basic Formulas A.7.1-1. The modified Bessel functions of the first and the second kinds. _

_

The modified Bessel functions of the first kind, ’ (a ), and the second kind, “ (a ) (also called the Macdonald function), of order Y are solutions of the modified Bessel equation + a ” z − (a

a 2 ” 'z z –

_ ’

M

M M

and are defined by the formulas (a ) = •

u =0

+ Y 2 )” = 0

2

u (a k 2)2 + — , x ! ( Y + x + 1)

„

— (a ) = “

c

’ −— − ’ — , 2 sin „ Y

(see below for “ — (a ) with Y = 0, 1, 2, b/b/b ). A.7.1-2. Some formulas. The modified Bessel functions possess the properties “ − — (a ) = “

2 Y ’ — (a ) = a [ ’ — 1 ’ (a ) = [ ’ — v a — 2 v

’ − g (a ) = (−1) g ’ g (a ),

— (a );

−1 ( a

)−’ —

+1 ( a

)],

−1 ( a

)+’ —

+1 ( a

)],

2 Y “ — (a ) = −a [ “ — −1 (a ) − “ — +1 (a )], v 1 “ — (a ) = − [ “ — −1 (a ) + “ — +1 (a )]. v a 2 1 2,

A.7.1-3. Modified Bessel functions for Y = ˜ d ˜ 2

’ 1 j 2 (a ) = ` ’ 3 j 2 (a ) = ` „

2 a

− ™

’ž +1 › 2 (œ ) = Ÿ

1 2„

’ − ž −1 › 2 (œ ) = Ÿ

1 2„

© ª ©

1

„ a

d = 0, 1, 2, b/b/b

where d = 0, 1, 2, b/b/b : 2

’ −1 j 2 (a ) = `

sinh a ,

sinh a + cosh ¥ a š ,

„ a

cosh a ,

2

1

’ −3 › 2 (œ ) =  „ − cosh œ + sinh œ š , œ ™ œ a ¥ ¥ ¨ ž ž ¤*¥ (−1) (¦ + § )! (¦ + § )! ž − ¤:¥ − (−1) , ¥ œ ¢  ¡ £ =0 § ! (¦ − § )! (2œ ) ¡ £ =0 § ! (¦ − § )! (2œ ) ¥ ¥ ¨ ž ž *¤ ¥ (−1) (¦ + § )! (¦ + § )! ž − ¤:¥ + (−1) , œ  ¢¡ £ =0 § ! (¦ − § )! (2œ ) ¡ £ =0 § ! (¦ − § )! (2œ ) „

1 › 2 (œ ) = 

ž +1 › 2 (œ ) =

−

2œ ¡ £ ©

© ª

,

„

− ž −1 › 2 ( œ ) = 

„

3 › 2 (œ ) =  −

2œ

ž

¤*¥

2œ ¡ £

«

1+

1

−

œ ¬ ¡ £ ¥

(¦ + § )!

§ ! (¦ − § )! (2œ )

=0

, .

A.7.1-4. Modified Bessel functions for Y = ¦ , where ¦ = 0, 1, 2, ­/­/­ If Y = ¦ is a nonnegative integer, then ©

œ ž ž (œ ) = (−1) +1 ’ ž (œ ) ln

ž 1¤ + 2 2®

−1

(−1)

®

=0

1 ž ¤° + (−1) 2 ® ²

2® −ž

œ

(¦ − ¯

« 2¬ ž +2 ® =0

« ±2 ¬

− 1)!

¯ ! ²

(¦ + ¯

²

+ 1) + (¯ + 1) ; ¯ ! (¦ + ¯ )!

¦ = 0, 1, 2, ­/­/­ ,

where ( ³ ) is the logarithmic derivative of the gamma function; for ¦ = 0, the first sum is dropped.

© 2002 by Chapman & Hall/CRC Page 666

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A.7. MODIFIED BESSEL FUNCTIONS

A.7.1-5. Wronskians and similar formulas: ´

2

( ’/µ , ’ − µ ) = − „

2 sin(„ Y )

’/µ ( ) ’ − µ +1 ( ) − ’ − µ ( ) ’·µ −1 ( ) = − ± ± ± ± ± ´ where ( ¸ , ¹ ) = ¸ ¹ º − ¸ º ¹ . £

´

sin(„ Y ), „

,

©

( ’¶µ ,

’/µ ( )

©

1

,

© µ +1 ( ) + ± ’·µ +1 ( ) ± ±

± ±

µ )=−

1

µ ( )= ±

,

±

£

A.7.2. Integral Representations and Asymptotic Expansions A.7.2-1. Integral representations. The functions ’ µ ( ) and

©

µ ( ) can be represented in terms of definite integrals:

±

µ ±

1

µ ’/µ ( ) = „ exp(− )(1 − 2 ) −1 » 2 ¿ 1» 22µ ± ¼ (Y + 1 ) ½ −1 2 ± ± ¾ ¾ ¾ © ° ¿ µ ( )= exp(− cosh ) cosh( Y ) ½ 0 ± ± ¾ ¾ ¾ © 1 ° µ ( )= cos( sinh ) cosh( Y ) ¿ cos À 12 „ Y Á ½ 0 ± ± ¾ ¾ ¾ © 1 ° ¿ µ ( )= sin( sinh ) sinh( Y ) sin À 12 „ Y Á ½ 0 ¾ ± ± ¾ ¾ For integer Y = ¦ ,

©

’ ž ( )= „ ±

)=

0(

1 °

±

¾

¾

cos( sinh ) ¿

½ 0

±

exp( cos ) cos(¦ ) ¿

½ 0Â

±

=

¾

¾

( > 0, Y > − 12 ), ±

( > 0), ±

( > 0, −1 < Y < 1), ±

( > 0, −1 < Y < 1). ±

(¦ = 0, 1, 2, ­/­/­ ), ¾

° Ÿ

½ 0

cos( ) ¿ 2 ± +¾ 1 ¾

( > 0). ¾

±

A.7.2-2. Integrals with modified Bessel functions: £

½ 0

± Ã

where ½ 0

µ

+ +1 2 Ä +Y +1 Ä +Y +3 = µ , Re( Ä + Y ) > −1, , , Y + 1; ¼ 2 ( Ä + Y ± Ã + 1) ( Y + 1) Å « 2 2 ±4 ¬ ± ± ( Æ , Ç , È ; ) is the hypergeometric series (see Section 10.9 of this supplement),

’/µ ( ) ¿ Å © £ ± Ã

±

µ ( )¿ ±

±

=

2

µ −1 ¼ (Y )

Ä −Y +1

Ä −Y +1 ± Ã µ 2− −1 ¼ (− Y ) + Ä +Y +1

− µ +1

Å

2 «

Ä +Y +1

+ µ +1

Å

± Ã

2 «

A.7.2-3. Asymptotic expansions as Ì ’/µ ( ) = Ÿ ¡ £ 2Ë ± ©

µ ( )=  ±

Ì ±

Ë

−

±

¤

® =1

−

±

É

(−1)

1+ Í

2 ¡ £

The terms of the order of Ð (

® =1

Í

−1

Ä −Y +3

2

, 1+Y ,

,

Ä +Y +3

2

2

±4 ¬ 2

,

±4 ¬

,

Re Ä > |Re Y | − 1.

: ±

¤

1+ Í

, 1−Y ,

®

Ê

(4 Î

(4 Î

2

2

− 1)(4 Î

− 1)(4 Î

2

2

− 32 ) ­/­/­ [4 Î ® ¯ ! (8 ) ±

− 32 ) ­/­/­ [4 Î ® ¯ ! (8 )

) are omitted in the braces.

2

2

− (2¯

− (2¯

− 1)2 ] − 1)2 ]

Ï ,

Ï .

±

© 2002 by Chapman & Hall/CRC Page 667

668

A. SPECIAL FUNCTIONS AND THEIR PROPERTIES

A.8. Airy Functions A.8.1. Definition and Basic Formulas A.8.1-1. The Airy functions of the first and the second kinds. The Airy function of the first kind, Ai( ), and the Airy function of the second kind, Bi( ), are solutions of the Airy equation Ñ ± Ñ ± º'º − =0 and are defined by the formulas 1

Ai( ) = Ë

±

1

Bi( ) = ´

Wronskian:

Ô

Ë

±

±

£ £ °

1 3 3

cos À

½ 0

¾

° ½ 0 Ò

exp À

− 13 3

± ¾

¾

ÁDÓ ¿ .

Á + sin À 13 3 +

+

¾

Ai( ), Bi( ) Õ = 1 Ö Ë . ±

Á ¿ ,

+

± ¾

¾

± ¾

¾

±

A.8.1-2. Connection with the Bessel functions and the modified Bessel functions: Ai( ) = Ai(−± ) = ± Bi( ) = Ù ±

Bi(− ) = Ù ±

1 3 1 3

Ÿ ÒØ×

Ÿ ±

ÒÛÚ

1± 3 1 3

Ò× ±

ÒÛÚ ±

−1 » 3 ( ³

)−

−1 » 3 ( ³

)+

−1 » 3 ( ³

)+

−1 » 3 ( ³

)−

)Ó = Ë

1 » 3 (³

× Ú ×

1» 3 (³

)Ó ,

1» 3 (³

)Ó ,

1 » 3 (³

Ú

−1 Ù

1 3

©

1» 3 (³

³ = 23 3 » 2 , ±

),

±

)Ó .

A.8.2. Power Series and Asymptotic Expansions 0:

A.8.2-1. Power series expansions as ±

É

Ai( ) = È 1 ¸ ( ) − È 2 ¹ ( ), ±

Bi( ) = 1 ¸ ( )=1+ 3! ±

¹ ( )= ±

where È

1

=3

−2 » 3

2 + 4!

3

± 4

1 ⋅ 4± + 6! 2⋅5 + 7!

6

±

Ÿ

±

±

3 È 1 ¸ ( ) + È 2 ¹ ( )Ó ,

Ò ± ¤*¥ 1⋅4⋅7 ± 9 ° + + ­/­/­ = 9! ±

7

2⋅5⋅8 + 10!

10

+ ­/­/­ =

¥ ¥ ¥

3 À =0

¤*¥ °

¥

3 À =0

± ± ± ± ¼ −1 » 3 ¼ Ö (2 Ö 3) ≈ 0.3550 and È 2 = 3 Ö (1 Ö 3) ≈ 0.2588.

A.8.2-2. Asymptotic expansions as

1 3

3

Á

(3± § )!¥ ¥

2 3

Á

,

3 +1

(3± § + 1)!

,

.

± of É asymptotic Ê For large values of , the leading terms expansions of the Airy functions are ±

Ai(−± ) Ü Bi( ± ) Ü ÝßÞ

1 −1 » 2 −1 » 4 2Ë −1 » 2 −1 ± » 4

Ai( ) Ü

exp(− ³ ),

sin À³ +

Ë Ë

−1 » 2 ± −1 » 4

4

Á ,

³ = 23 3 » 2 , ±

exp( ³ ), Â

Ë −1 » 2 ± −1 » 4 cos À ³ + 4 Á . ± Â Reference: M. Abramowitz and I. Stegun (1964).±

Bi(−± ) Ü

© 2002 by Chapman & Hall/CRC Page 668

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A.9. DEGENERATE HYPERGEOMETRIC FUNCTIONS

TABLE A1 Special cases of the Kummer function à ( Æ , Ç ; ³ ) Æ

³ Ç

Æ

Æ

1

2

±

Æ +1

2

3 2

−¦

1 2 3 2

−¦

−¦

±

−

2 Î +1

2

± ¦ !

(2¦ )! «

±2

2¦ +2

−

±2

¦ !

( ç −1)

¼ (1+ Î ) ¼

2 ±

«

¦ +

( )

2å

á −1 ¿ ¾

¾

2 å +1 (

å å = (−1)

)

æ

±

µ ( ) ±

− 12

×

¾

¿ å

2

2

¿

Laguerre polynomials (è )

( )

×

¾

− å À¡ £ Á , ¡ £ ¦ = 0, 1, 2,± ­/­/­

±

¡ £ « ±2 ¬

½ 0

Hermite polynomials ä

± ä

−µ

−å 3 2 ¬ ¡ £ « ±2 ¬

Ë

±

2¬

(Ç )å æ å

2

±

± 1 −ž ä −å

−ã

£ ½ 0 ¡

Error function 2 2 ¿ erf = Ÿ £ exp(− )

erf

¦ ! 1 − (2¦ +1)! « 2 ¬

2

±

¦ +1

Ë

2

2

â (Æ , ) =

± Ÿ

−

Incomplete gamma function

(Æ , )

±

Ç

1 2

±

− á/â

Æ

±

Î +

sinh

¡ £

±

1 2

¡ £

1 ±

Æ

Conventional notation à

å + 12 ( ) ±

å

¿ å

−è

( )= ¡ £ ¦± ! ±

é

¿

− å À¡ £

å +è Á ,

± = Ç −1, ± ( Ç ) å = Ç ( Ç +1) ­/­/­ ( Ç + ¦ −1)

Modified Bessel functions µ ( ) ×

±

A.9. Degenerate Hypergeometric Functions A.9.1. Definitions and Basic Formulas A.9.1-1. The degenerate hypergeometric functions à ( Æ , Ç ; ) and ê ( Æ , Ç ; ). ± The degenerate hypergeometric functions à ( Æ , Ç ; ) and ê ( Æ ± , Ç ; ) are solutions of the degenerate Ñ Ñ Ñ± hypergeometric equation ± ë º'º ë + ( Ç − ) ë º − Æ = 0. í In the case Ç ≠ 0, −1, −2, −3, ­/­/­ ,± the function± à ( Æ , Ç ; )í can be represented as Kummer’s series: à (Æ , Ç ; ) = 1 + í ï

±

± í î

ì*í ° =1

(Æ ) ï , (Ç ) !

where ( Æ ) = Æ ( Æ + 1) ð/ð/ð ( Æ + − 1), ( Æ )0 = 1. Table A1 presents some special cases where à can be expressed in terms of simpler functions. î The function ê ( Æ , Ç ; ) is defined as follows: ¼ (1 − Ç ) ¼ ( Ç − 1) î î î 1− ç î ê (Æ , Ç ; ) = ¼ à (Æ , Ç ; ) + ¼ à ( Æ − Ç + 1, 2 − Ç ; ). ( Æ − Ç + 1) (Æ )

© 2002 by Chapman & Hall/CRC Page 669

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A. SPECIAL FUNCTIONS AND THEIR PROPERTIES

A.9.1-2. Kummer transformation and linear relations. Kummer transformation: ë î î à ( Æ , Ç ; ) = ñ à ( Ç − ò , Ç ; − ),

î î î ê ( ò , Ç ; ) = 1− ç ê (1 + ò − Ç , 2 − Ç ; ).

Linear relations for à : î

î

î

î

( Ç − ò ) à ( ò − 1, Ç ; ) + (2 ò − Ç + ) à ( ò , Ç ; ) − ò à ( ò + 1, Ç ; ) = 0, î î î î î Ç ( Ç − 1) à ( ò , Ç − 1; ) − Ç ( Ç − 1 + ) à ( ò , Ç ; ) + ( Ç − ò ) à ( ò , Ç + 1; ) = 0, î î î ( ò − Ç + 1) à ( ò , Ç ; ) − ò à ( ò + 1, Ç ; ) + ( Ç − 1) à ( ò , Ç − 1; ) = 0,

î î î î Ç/à ( ò , Ç ; ) − Ç/à ( ò − 1, Ç ; ) − à ( ò , Ç + 1; ) = 0, î î î î î Ç ( ò + ) à ( ò , Ç ; ) − ( Ç − ò ) à ( ò , Ç + 1; ) − ò Ç/à ( ò + 1, Ç ; ) = 0, î î î î ( ò − 1 + ) à ( ò , Ç ; ) + ( Ç − ò ) à ( ò − 1, Ç ; ) − ( Ç − 1) à ( ò , Ç − 1; ) = 0.

Linear relations for ê : î î î î ê ( ò − 1, Ç ; ) − (2 ò − Ç + ) ê ( ò , Ç ; ) + ò ( ò − Ç + 1) ê ( ò + 1, Ç ; ) = 0, î î î î î ( Ç − ò − 1) ê ( ò , Ç − 1; ) − ( Ç − 1 + ) ê ( ò , Ç ; ) + ê ( ò , Ç + 1; ) = 0, î î î ê ( ò , Ç ; ) − ò ê ( ò + 1, Ç ; ) − ê ( ò , Ç − 1; ) = 0, î î î î ( Ç − ò ) ê ( ò , Ç ; ) − ê ( ò , Ç + 1; ) + ê ( ò − 1, Ç ; ) = 0, î î î î î ( ò + ) ê ( ò , Ç ; ) + ò ( Ç − ò − 1) ê ( ò + 1, Ç ; ) − ê ( ò , Ç + 1; ) = 0, î î î î ( ò − 1 + ) ê ( ò , Ç ; ) − ê ( ò − 1, Ç ; ) + ( ò − È + 1) ê ( ò , Ç − 1; ) = 0.

A.9.1-3. Differentiation formulas and Wronskian. Differentiation formulas: ó

ò

î

ó å ó î å à (ò , ô ó å ó î å ê (ò , ô

î

ó î à ( ò , ô ; ) = à ( ò + 1, ô + 1; ), ô ó î î ó î ê ( ò , ô ; ) = − ò ê ( ò + 1, ô + 1; ),

Wronskian: ö

( à , ê ) = à ê ëº − à ëº ê = − ¼

(ò )å î à ( ò + õ , ô + õ ; ), (ô )å

î

; )= î

î

; ) = (−1) å ( ò ) å ê ( ò + õ , ô + õ ; ). ¼ (ô ) î −ç ë ñ . (ò )

A.9.1-4. Degenerate hypergeometric functions for õ = 0, 1, 2, ð/ð/ð : î ê ( ò , õ + 1; ) =

(−1) å −1 î î à ( ò , õ +1; ) ln õ ! ¼ (ò ù − õ ) ÷

ì:ù

+ ø =0

(ò ) (õ + 1)

ù úØû

û

ù û û

where õ = 0, 1, 2, ð/ð/ð (the last sum is dropped for õ = 0), derivative of the gamma function, û




(1) = − , where




î

( ò + o ) − (1 + o ) − (1 + õ + o )ü

û




(õ ) = − +

o ! ý

ù

+

å (õ − 1)! þì ù ¼ (ò )

−1 =0

(ò − õ ) (1 − õ )

ù î

ù −å

o !

,

( ÿ ) = [ln ¼ ( ÿ )] º is the logarithmic

*ì å í −1 ï −1 , =1

= 0.5772 ð/ð/ð is the Euler constant.

© 2002 by Chapman & Hall/CRC Page 670

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A.9. DEGENERATE HYPERGEOMETRIC FUNCTIONS

If ô < 0, then the formula 

î

(ò , ô ; ) =

î 1−   î ( ò − ô + 1, 2 − ô ; )

î

is valid for any . For ô ≠ 0, −1, −2, −3, ð/ð/ð , the general solution of the degenerate hypergeometric equation can be represented in the form  î î  =  1  ( ò , ô ; ) +  2 ( ò , ô ; ), and for ô = 0, −1, −2, −3, ð/ð/ð , in the form 

=

ú  î 1−  î î  1  ( ò − ô + 1, 2 − ô ; ) +  2 ( ò − ô + 1, 2 − ô ; )ü .

A.9.2. Integral Representations and Asymptotic Expansions A.9.2-1. Integral representations:  1 ¼ (ô ) î ñ   −1 (1 −  )  −  (ò , ô ; ) = ¼ (ò ) ¼ (ô − ò ) 0   1 î − −1  − −1 ó  (ò , ô ; ) = ¼ ø ñ    (1 +  ) (ò ) 0

−1

ó

(for ô > ò > 0), 

(for ò > 0,

î

> 0),

where ¼ ( ò ) is the gamma function. A.9.2-2. Integrals with degenerate hypergeometric functions: 



ô −1 î ó î î = ( ò − 1, ô − 1; ) +  ,  (ò , ô ; ) ò −1   1  î ó î î (ò , ô ; ) = ( ò − í 1, ô − 1; í ) +  í , 1−ò î í ì*å í +1 (−1) +1 (1 ï ï î − ô ) å − +1 î å î ó î ï =õ !  (ò , ô ; )  (ò − , ô − ; ) +  , í í (1 − ò ) (õ − + 1)! =1



î å  î ó î (ò , ô ; ) =õ !

ì å í +1 =1

î

A.9.2-3. Asymptotic expansion as | | 

í î ï ï î (−1) +1 å − +1  ï (ò − , ô − ; ) +  . (1 − ò ) (õ − + 1)!

: 

¼ (ô ) ì ( ô − ò ) å (1 − ò ) å î î î ñ  −    (ò , ô ; ) = ¼ (ò ) õ ! å =0 î  (ò , ô ; ) =

 

 (ô ) î (− )−   (ô − ò ) 

å =0



( , ô ;  ) =  −   å =0

where  =  ( − 

−1

(−1) å

−å

+  ,

î

> 0,

(  ) å (  − ô + 1) å (− )− å +   ,  !

(  ) å (  − ô + 1) å − å  +  ,  !

−

 < 0,

0, the hypergeometric function can be expressed in terms of a definite integral:  ( ,  ,  ;  ) =

( ) (  ) ( −  )









1 0

(

−1

(1 −  )& −(

−1

(1 − , )− '

+

,

where ( ) is the gamma function.  See M. Abramowitz and I. Stegun (1964) and H. Bateman and A. Erd e´lyi (1953, Vol. 1) for more detailed information about hypergeometric functions. # A.11. Whittaker # Functions

The Whittaker functions equation:

,.

( ) and /

,.

 ! + 0 −1 + 4  

( ) are linearly independent solutions of the Whittaker 1% 2

+1

1 4

−2

23



−2 4

 = 0.

The Whittaker functions# are expressed in terms of degenerate hypergeometric functions as # /

 5 2 1 . +1 5 2 6 −  5 2  1 , . ( ) =  ,.

( ) =  .

+1 5 2 6 −

1 2 1 2

% + 2 − , 1 + 22 ,  3 ,

% + 2 − , 1 + 22 ,  3 .

© 2002 by Chapman & Hall/CRC Page 672

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A.10. HYPERGEOMETRIC FUNCTIONS

TABLE A2 Some special cases where the hypergeometric function  (  ,  ,  ; ÿ ) can be expressed in terms of elementary functions 



ÿ



−



7 =0

−

− − 8 

# 

 

 +

1 2

1 2



2



 +

1 2

3 2



2



−

1 2

−

2



1−

1 2

−

2



 −

2 − 1 



1−

3 2

sin2 



2−

3 2

sin2 



1−

1 2

sin2 



 +1 

 +

1 2



 +

1 2

1 2

091,: 1 + 

1 : 1+

2

3 2 ' + 1: 1 + 

2+ 2' + 3

−1

+ 1 : 1+

2: 1 + 

 2



−

1

1

2

−

1 2

1

3 2



+= +2

(1 +  )1−2 ' − (1 −  )1−2 ' 2 (1 − 2  )

−2

1 1+ : 1− 3

sin[(2  − 1) (  − 1) sin(2 sin[(2  − 2) (  − 1) sin(2 cos[(2  − 1) cos 



1+ : 1− 2


1 =1

8

B >

−1 ( 

)B

−

7

>

( ).

( ) can be calculated recursively using the relations  7 7 1 2 + 1 B 0 ( ) = 1, B 1 ( ) =  , B 2 ( ) = (3 2 − 1), " " " , B +1 ( ) =   B ( ) −  B 2 +1 +1 7 7 7 = C ( ) have the form The first three functions C

−1 ( 

).

7 1 1+ 7 3 2 − 1 1 +  3  1+ , C 1 ( ) = ln − 1, C 2 ( ) = −  . ln ln 2 1− 2 1− 4 1− 2 The polynomials B ( ) have the implicit representation C

)=

0 (

 5

7

2]

[

(  ) = 2− 7 B

7

7

>

(−1) > F > F =0

2 −2

7

7

where [ G ] is the integer part of a number G .

7

 >

−2

7

> ,

A.12.2. Zeros of Legendre Polynomials and the Generating Function All zeros of B ( ) are real and lie on the interval −1 <  < +1; the functions B ( ) form an orthogonal system on the7 interval −1 ≤  ≤ +1, with 7 H +1 0 if M ≠ N , 2 B I (J ) B (J ) K J = L > if M = N . −1 2M + 1 The generating function is O

S

1 1 − 2 PQJ + P

2

= R I

I (J ) P =0

I

(| P | < 1).

T

A.12.3. Associated Legendre Functions The associated Legendre functions I > (J ) of order N T

I > (J ) = (1 − J

2

)> U

2

KT >

K J > T 0 I (J ) =

I (J ),

are defined by the formulas

M = 1, 2, 3, " " " ,

= 0, 1, 2, " " " N

It is assumed by definition that I (J ). T The functions I > (J ) form Tan orthogonal system on the interval −1 ≤ J ≤ +1, with T

H

+1 −1

#

W!X 0 2 (M + N )! I > (J ) > (J ) K J = V T T 2M + 1 (M − N )!

The functions I > (J ) (with N (1 − J 2 )−1 , that is, T H

+1 −1

%

if M ≠ , %

if M = .

≠ 0) are orthogonal on the interval −1 ≤ J ≤ +1 with weight #

W!X 0 I > (J ) > (J ) K J = V (M + N )! T (1 −T J 2 ) N (M − N )!

%

if M ≠ , %

if M = .

© 2002 by Chapman & Hall/CRC Page 674

675

A.14. MATHIEU FUNCTIONS

A.13. Parabolic Cylinder Functions A.13.1. Definitions. Basic Formulas The Weber parabolic cylinder function Y Z ( [ ) is a solution of the linear differential equation: \ ]^! ] + _ − 1 [ 4

2

+ ` + 12 a \ = 0,

where the parameter ` and the variable [ can assume arbitrary real or complex values. Another linearly independent solution of this equation is the function Y − Z −1 (b,[ ); if ` is noninteger, then Y Z (− [ ) can also be taken as a linearly independent solution. The parabolic cylinder functions can be expressed in terms of degenerate hypergeometric functions as Y Z ( [ ) = 21 U

2

exp _ − 14 [ 2 a c

_d d

1 2

_ 1 2

−

a Z

Z _ − 2 , 12 , 12 [

a e

2

2

−1 a +2 U

d

For nonnegative integer ` = M , we have I Y I ( [ ) = 2− U

2

1

2

exp _ − 41 [ 2 a g I _ 2−1 U 2 [ a ,

I g I ( [ ) = (−1) exp _ [

2

_ −2a _ d _ −Z [ 2a e

1 2

Z

− 2 , 32 , 12 [ 2 Qa f .

M = 0, 1, 2, h h h ;

I

K a K [ I exp _ − [

2

a ,

where g I ( [ ) is the Hermitean polynomial of order M .

A.13.2. Integral Representations and Asymptotic Expansions Integral representations: Y Z ([ ) = i Y Z ([ ) =

2 j k exp _

d

1 2 4[ a

Z

0

I

=0

exp _ − 21 2 a cos _ [

l 0

l

− Z −1

R l

exp _ − [ l

−

l

− 12 k ` a K

1 2 2 a

l

K

l

l

for

Re ` > −1,

for

Re ` < 0.

: n I

S

exp _ − 41 [ 2 a c o

Z R

H

1 exp _ − 41 [ 2 a (− ` )

Asymptotic expansion as | [ | m Y Z ([ ) = [

H

Z

(−2) _ − 2 a I _ M !

1 2

−

Z

2

a I

1

+p [ 2I

_ | [ |−2 o

−2

a f

for | arg [ |


ce2 I

2I

S

+1

=R =0 v

>

2

=0 v

2 I +1 2 +1

>

S

>

se2 I

S

+1

=R >

>

=0 x

=0 x

r

cos(2N +1)J r

v 2I 2

se2 I = R

+1 +1 sin(2 N

r r

+1)J r

x

v

>

v

2I 4

2x I 2 +2

>

x −r

S

=0

v

2I 2

>

)2

I

= ( ô 2 I −4) 22 ; I = ( ô 2 I −4N x 2) 22 > 2I N ≥2 x 2 −2 ,

S

=0

v

R

(

( R =0

>

>

>

x

=0

> S

2 I +1 2 2 +1 ) = 1

( R

>

I 2 I +1 = ( ô 2 I +1 −1− r ) 12 +1; 3 2x I +1 x 2 2 I +1 2 +3 = [ ô 2 I +1 −(2 N +1) ] 2 +1 > > 2 I +1 − r 2 −1 , N ≥ 1 x x

(

>

>

x

S

)2 + R

2 if M = 0 =w 1 if M ≥ 1

>

v

sin 2N J ,

r

2I 0

(

I 2 I +1 = ( q 2 I +1 −1− r ) 12 +1; 3 2v I +1 2 2 I +1 v 2 +3 = [ q 2 I +1 −(2 N +1) ] 2 +1 > > 2 I +1 − r 2 −1 , N ≥ 1 v

se0 = 0 2I 2 >

I

= q 2 I 20 ; I I r = (v q 2 I −4) 22v −2 r 20 ; 2 v 2I v v r 2I ) 2 2 +2 = ( q 2 I −4 N > > 2I v − r 2 −2 , N ≥ 2v r

2I 4

cos 2N J >

2I 2

Normalization conditions

x

2I 2

>

)2 = 1

2 I +1 2 2 +1 )

>

=1

A.14.1-2. Properties of the Mathieu functions. The Mathieu functions possess the following properties: I

ce2 I (J , − r ) = (−1) ce2 I y se2 | (z , − r ) = (−1) |

−1

k

2

se2 | y

−z , r { , k

2

ce2 |

− z , r { , se2 |

+1 ( z +1 ( z

, − r ) = (−1) | se2 | , − r ) = (−1) | ce2 |

k y

+1 +1

k y

2 2

−z , r { , −z , r { .

Selecting sufficiently large number N and omitting the term with the maximum number in the recurrence relations (indicated in Table A3), we can obtain approximate relations for eigenvalues q | (or ô | ) with respect to parameter r . Then, equating the determinant of the corresponding homogeneous linear system of equations for coefficients | (or | ) to zero, we obtain an algebraic > > equation for finding q | ( r ) (or ô | ( r )). v x For fixed real r ≠ 0, eigenvalues q | and ô | are all real and different, while if

r >0

then q

0

0, the functional differential equation (10) has solutions

  6  ( ) =  exp 405  7 ,

  ( ) = exp *  8

  + ( )9 - ,

(11)

where  is an arbitrary constant. Solution (11) for  makes the expression in square brackets in (10) vanish, which allows separation of variables. 2 3 . There is a more general solution of the functional differential equation (10) for  > 0:

  6  6  ( ) =  1 exp 4  7 +  2 exp 4 −  7 , !;:  ! :   −1 < 2 2  ()=  3 + 8  1  2 8 9  , where  1 ,  2 , and  constant, namely,

3

= = 8

  + ( )9 ,



are arbitrary constants. The function  =  ( ) makes each of the terms in (10) that depend on

 &2  >  = const,

(    )2 −  

2



= const .

It is this circumstance that makes it possible to separate the variables.  Note that the function  =  ( ) satisfies the Bernoulli equation   =  + ( ) − 4  

1  2

3.

3 3 . There is another solution of the functional differential equation (10) for  < 0:

  6  6  ( ) =  1 sin 4 −  7 +  2 cos 4 − 7 , !;: ! :   −1 < 2 2  ()= 9 , *  3 + 2  (  12 +  22 ) 8 

= = 8

  + ( )9 ,

where  1 ,  2 , and  3 are arbitrary constants. The function  =  ( ) makes both terms in (10) that depend on  Note that the function  =  ( ) is determined by the Bernoulli equation   =  + ( ) −  (  12 +  22 )  3 .



constant.

© 2002 by Chapman & Hall/CRC Page 683

684

B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES Example 5. Consider the third-order equation

   .

 2     2 +  

We look for additive separable solutions



 2  3  3  . 2 =?   3 +@  . 3 . 

(12)

.

= + ( ) + / ( ).

(13)

Substituting (13) into (12) yields

/ 1 + 2&  +  +   / #1&  1 = ? + 2&&2  + @ / #1& &1# 1 .

(14)

This expression cannot be rewritten as the sum of two functions depending on different arguments. It is not difficult to see that the functional differential equation (14) is satisfied if

/ 1 = 

1

A B

+  = 

1

A B

. .  / ( ) =  1 +  2, + ( ) =    . + ( ) =  1 +  2, / ( ) = 

3

exp( 

3

exp(  

1

 1

> ? )+ .

4

> @ )+

 4

(case 1),

.

(case 2),

where  1 ,  2 ,  3 , and  4 are arbitrary constants. In both cases, two terms of the four in (14) vanish, which makes it possible to separate the variables. In addition, equation (12) has a more complicated solution of the form (13):



! =

1

−C

" 

+

@ED  + 

!#" 1

2

.

− )?D

+

3,

where  1 ,  2 ,  3 , and D are arbitrary constants. The mechanism of separation of variables is different here: both nonlinear terms on the left-hand side in (14) contain terms which cannot be rewritten in additive form but are equal in magnitude and have unlike signs. In adding, the two terms cancel, thus resulting in separation of variables:

/ 1 + 2&  +  +   / 1#& 1

= 

1 2

2D 3

= −

1 2

2D 3

/ 1 + 2&  +  +   / 1#& 1 = − 

1?

( )D )3

!

! "

1 −C

! " −C

"

1 −C 

"

 " 

+

( )D ! )" 3

−

1?

+

2 @ED

2 @ED

3

! "

1

3

1

!

−C

"

 .

= ? + 2&&2  + @ / 1#&&1# 1

Example 6. Consider the second-order equation with a cubic nonlinearity (1 +



2

)

  2     2    2 +  . 2  −2   

2

−2



    . 

2

=



(1 −



2

).

(15)

We seek an exact solution of this equation in the product form





.

= + ( )/ ( ).

(16)

Substituting (16) into (15) yields (1 + +

/ )(/ + 2&  + + / 1#& 1 ) − 2 + / F&/ 2 ( +   )2 + + 2 (/ 1 )2 G =  + / (1 − + 2 / 2 ).

2 2

(17)

This expression cannot be rewritten as the sum of two functions with different arguments. Nevertheless, equation (15) has  . solutions of the form (16). One can make sure by direct check that the functions + = + ( ) and / = / ( ) satisfying the nonlinear ordinary differential equations ( +   )2 = H +

4

+I +

2

4

+ (  − I )/

( / 1 ) =  /

2

+ , 2

+H ,

(18)

where H , I , and  are arbitrary constants, reduce equation (17) to an identity; to verify this, one should use the relations + 2&  = 2 H + 3 + I + and / 1#& 1 = 2  / 3 + (  − I )/ that follow from (18).

J KML NPORQ S

By the variable change T = 4 arctan sinusoidal source, U T =  sin T .



equation (15) can be reduced to a nonlinear heat equation with a

The examples considered above illustrate some specific features of separable solutions to nonlinear equations. Sections B.2 and B.3 outline fairly general methods for constructing similar and more complicated solutions to nonlinear partial differential equations.

© 2002 by Chapman & Hall/CRC Page 684

B.2. METHODS OF GENERALIZED SEPARATION OF VARIABLES

685

B.2. Methods of Generalized Separation of Variables B.2.1. Structure of Generalized Separable Solutions B.2.1-1. General form of solutions. The classes of nonlinear equations considered. To simplify the presentation, we confine ourselves to the case of mathematical physics equations with two independent variables
,  and a dependent variable  (one of the independent variables can play the role of time). Linear separable equations of mathematical physics admit exact solutions in the form  (
,  ) =  1 (
) 1 ( ) +  2 (
) 2 ( ) + V)V)V +  W (
) W ( ),

(1)

where the  X =  X (
) X ( ) are particular solutions; the functions  X (
), as well as the functions  X ( ), with different numbers Y are not related to one another. Also having exact solutions of the form (1) are many nonlinear partial differential equations with quadratic or power nonlinearities

1 (
)Z 1 ( ) [ 1 [ ] + 2 (
)Z 2 ( ) [ 2 [ ] + V)V)V + \ (
)Z \ ( ) [ \ [ ] = 0,

(2)

where the [ X [ ] are differential forms that are the products of nonnegative integer powers of the function  and its partial derivatives   ,    ,   ,    ,     ,   , etc. We will refer to solutions (1) of nonlinear equations (2) as generalized separable solutions. Unlike linear equations, in nonlinear equations the functions  X (
) with different subscripts Y are usually related to one another [and to the functions  ] ( )]. Subsections B.1.2 and B.1.3 give examples of exact solutions (1) to nonlinear equations (2) for some simple cases with ^ = 1 or ^ = 2 (for  1 =  2 = 1). _ `#a b cPd e

general form

If the f (
) and Z f ( ) in (2) are all constant, then one can seek solutions in the more  (
,  ) =

g

W h = j 1
+ j 2 ,

 \ (h ) \ (i ),

\ =1

i = k 1
+ k 2 ,

where j 1 , j 2 , k 1 , and k 2 are constants. Some solutions of this sort are discussed in Subsections B.7.1 and B.8.1. B.2.1-2. General form of functional differential equations. In general, on substituting expression (1) into the differential equation (2), one arrives at a functional differential equation l l l p p 1 (m

)n

1(o

)+

2 (l m

)n

2(o

) + V)V)V +

(m ) n

(o ) = 0

(3)

for the  X (
) and  X ( ). Thel functionals ] (m ) and n ] ( o ) depend only on
and  , respectively, l ] (m

) ≡ ] qr
, 

1 ,  1s ,  1sts ,

u)u)u ,  W ,  sW ,  Wsts v ,

n ] ( o ) ≡ n ] qr ,  1 ,  1s ,  1sts , u)u)u ,  W ,  W s ,  W sts v .

(4)

Here, for simplicity, the formulas are written out for the case of a second-order equation (2); for higher-order equations, the right-hand sides of relations (4) will contain higher-order derivatives of  X and  ] .

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B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES

B.2.2. Solution of Functional Differential Equations by Differentiation p x

p

B.2.2-1. Description of the method. 1 w . Assume that n l 0. We dividel equation (3) by n results in a similar equation but with fewer terms: y

l y

ly

) + 2 (m ) n y ] (m ) = ] (m ),

1 (m

)n

1( o

l

y

differentiate with respect to  . This l and p p y

py

) + V)V)V + −1 (m ) n −1 ( o ) = 0, y n ] ( o ) = [ n ] ( o ) z n ( o )] s .

2( o

l

We continue the above procedure until we obtain a separable two-term equation {

l

{ { l ) n 1 ( o )x + 2 (m

1 (m

)n

{

Three cases must be considered. { { { Nondegenerate case: l | 1 (m )| + l | 2 (m )| 0 and | n equivalent to the ordinary differential equations {

) = 0.

1(o

)| + | n

{

(5) x

2(o

0. Then equation (5) is

)|

{ { | n 1 ( o ) − n 2 ( o ) = 0, { { is an arbitrary constant. The equations 2 = 0 and n 1 = 0 correspond to the limit case 1 (m

)+|

{

2(o

where | | =} . l Two degenerate cases: {

{

l

) = 0,

2 (m

l

) ≡ 0,

1 (m

n 1 ( o ) ≡ 0,

{

)≡0

~ 

n 2(o ) ≡ 0

~ 

{

2 (m

l{

n 1,2 ( o ) are any; { 1,2 ( m ) are any.

2 w . The solutions of the two-term equation (5) should be substituted into the original functional differential equation (3) to “remove” redundant constants of integration [these arise because p p equation (5) is obtained from (3) by differentiation]. 3 w . The case n ≡ 0 should be treated separately (since we divided the equation by n at the first stage). Likewise, we have to study all other cases where the functionals by which the intermediate functional differential equations were divided vanish. _ `#a b cPd

€ e

_ `#a b cPd

 e

The functional differential equation (3) can happen to have no solutions.

l At p x each subsequent stage, the number of l p terms in the functional differential equation can be reduced by differentiation with respect to either ‚ or ƒ . For example, we can assume at the first stage that 0. On dividing equation (3) by and differentiating with respect to ƒ , we again obtain a similar equation that has fewer terms.

B.2.2-2. Examples of constructing exact generalized separable solutions. Below we consider specific examples illustrating the application of the above method to constructing exact generalized separable solutions of nonlinear equations. Example 1. The two-dimensional stationary equations of motion of a viscous incompressible fluid are reduced to a single fourth-order nonlinear equation for the stream function (see equation 1 in Subsection B.7.1), specifically,

„ … „ †

„ „ … … )− „ ‡ „ ‡ (U

„ … … )=ˆ U U , „ † (U

We seek exact separable solutions of equation (6) in the form

…

Substituting (7) into (6) yields Differentiating (8) with respect to

‡

‡

U

…

„ 2…

= „ ‡

†

= ‰ ( ) + Š ( ).

Š Œ‹ ‰ 2‹&‹&2‹  − ‰  ‹ Š Œ#‹&‹&Œ#‹ Œ = ˆ ‰ 2 &‹ ‹&2‹&‹ 2 + ˆ Š Œ#‹&‹&Œ#‹&‹ Œ#Œ . † and , we obtain Š #Œ&‹ ‹ Œ ‰ 2&‹ ‹&2‹&‹ 2 − ‰ 2&‹ ‹  Š Œ#&‹ ‹&Œ#‹&‹ Œ#Œ = 0.

„ 2…

2

+ „ †

2

.

(6) (7) (8) (9)

Nondegenerate case. If ‰ 2‹&‹  Ž 0 and Š Œ#‹&‹ Œ Ž 0, we separate the variables in (9) to obtain the ordinary differential equations

‰ 2 ‹&‹&2‹&‹ 2 =  ‰ 2‹&‹  , Š Œ#‹&‹&Œ#‹&‹ Œ#Œ =  Š #Œ‹&‹ Œ ,

(10) (11)

which have different solutions depending on the value of the integration constant  .

© 2002 by Chapman & Hall/CRC Page 686

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B.2. METHODS OF GENERALIZED SEPARATION OF VARIABLES 1  . Solutions of equations (10) and (11) for 

= 0:

‡

‰ ( )=‘ † Š ( )=’

+‘

1

+’

1

‡

2

+‘

†

2

3

+’

3

‡

2

†

+‘

2

4

+’

4

‡

3

†

3

,

(12)

,

where ‘ “ and ’ “ are arbitrary constants ( ” = 1, 2, 3, 4). On substituting (12) into (8), we evaluate the integration constants. Three cases are possible: ‘ 4 = ’ 4 = 0, ‘ • , ’ • are any numbers (– = 1, 2, 3); ‘ “ = 0, ’ “ are any numbers ( ” = 1, 2, 3, 4);

’ “ = 0,

‘ “ are any numbers

( ” = 1, 2, 3, 4).

The first two sets of constants determine two simple solutions (7) of equation (6):

… … where 

—˜—E— , 

1,

5

=

1

=

1

‡

2

†

+

2

3

+

2

2

> 0:

‡

+

†

2

†

3

+

2

3

†

†

+

4

+

4,

+

5,

are arbitrary constants.

2  . Solutions of equations (10) and (11) for 

=™

 › , Œ Œ † + ’ 4 š −› . 1 + ’ 2 + ’ 3š ›

‡

‰ ( )=‘ † Š ( )=’ Substituting (13) into (8), dividing by ™

‘

3 (ˆ

™ −’

2 )š

 ›

3,

+‘

1

‡

2

+‘



3 š#›

+‘

−

4š

(13)

+‘

and collecting terms, we obtain 4 (ˆ

™ +’

−

2 )š

 ›

+’

3 (ˆ

™ +‘

2 )š

Œ ›

+’

4 (ˆ

™ −‘

−

2 )š

›

Œ

= 0.

Equating the coefficients of the exponentials to zero, we find

‘

3

‘

3

‘

3

=‘ =’ =’

4 3 4

= ’ 3 = 0, ‘ 2 = ˆ ™ = 0, ‘ 2 = ˆ ™ , ’ 2 = − ˆ ™ = 0, ‘ 2 = − ˆ ™ , ’ 2 = − ˆ ™

(case 1), (case 2), (case 3).

(The other constants are arbitrary.) These sets of constants determine three solutions (7) of equation (6):

… … … where 

1,



2,



3,

1š

−

›

=

1š

−

›

1š

−

Œ

= =

+



+ˆ ™

 ›

2

† ‡ ‡

−ˆ ™

+

3

+

2š

+

2 š#›

‡

+ˆ ™ −

›

Œ

,

Œ

−ˆ ™ −ˆ ™

† †

+ +

3, 3,

and ™ are arbitrary constants.

3  . Solution of equations (10) and (11) for 

2

= −™

‡

‰ ( )=‘ † Š ( )=’

1 1

+‘ +’

< 0: 2 2

‡

+‘ +’

†

‡

cos( ™ ) + ‘ † 3 cos( ™ ) + ’ 3

‡

sin( ™ ), † 4 sin( ™ ). 4

(14)

Substituting (14) into (8) does not yield new real solutions.

†

‡

Degenerate cases. If ‰ 2‹&‹  ≡ 0 or Š Œ#‹&‹ Œ ≡ 0, equation (9) becomes an identity for any Š = Š ( ) or ‰ = ‰ ( ), respectively. ‡ ‡ These cases should be treated separately from the nondegenerate case. For example, if ‰ 2‹&‹  ≡ 0, we have ‰ ( ) = ‘ +’ , where ‘ and ’ are arbitrary numbers. Substituting this ‰ into (8), we arrive at the equation − ‘ Š Œ#‹&‹&Œ#‹ Œ = ˆ Š Œ#‹&‹&Œ#‹&‹ Œ#Œ . Its general † † œ † 2 † solution is given by Š ( ) =  1 exp(− ‘ ˆ ) +  2 +  3 +  4 . Thus, we obtain another solution (7) of equation (6):

…

=

1š

−

›

Œ

+

2

†

2

+

3

†

+

4

+ˆ ™

‡

(‘

=ˆ ™ , ’

= 0).

Example 2. Consider the second-order nonlinear parabolic equation

„ … „ … „ … … 2 „  =ž „ ‡ 2 +Ÿ   „ ‡ ¡

2

+¢ .

(15)

We look for exact separable solutions of equation (15) in the form

…





‡

= £ ( ) + ¤ ( ) ¥ ( ).

(16)

Substituting (15) into (16) and collecting terms yields

£ ‹¦ − ¢ + ¤ ¦R‹ ¥ = ž £ ¤ ¥#2‹&‹  + ¤

2

§ ž)¥2¥#2‹&‹  + Ÿ ( ¥#‹ )2 ¨ .

(17)

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B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES 

2

On dividing this relation by ¤

‡

and differentiating with respect to and , we obtain

œ

2

( ¤ ¦‹ ¤

œ

) ‹¦R¥#‹ = ž ( £

¤ )‹¦¥#2‹&‹&2‹  .

Separating the variables, we arrive at the ordinary differential equations

¥#2‹&‹&2‹  = © #¥ ‹ , œ 2 œ ) ‹¦ = ž © ( £ ¤ ) ‹¦ ,

(18)

( ¤ ¦‹ ¤ where ©

is an arbitrary constant. The general solution of equation (18) is given by

‘

1

¥ = ª ‘ ‘

where ‘

(19)

‘

1,

and ‘

2,

3

‡

2

+ ‘

‘

+‡ 1š › 1 sin( ™

‡

2

+‘  −

if © if © if ©

3

+ ‘ 3‡ ) + ‘ 2 cos( ™ ) + ‘ 2š

›

3

= 0, = ™ 2 > 0, = − ™ 2 < 0,

(20)

are arbitrary constants. Integrating (19) yields

’

¤ = 

+

 £ ( ) is any

,

1

1 ¤ ¦‹

£ =’ ¤ +

ž ©

,

¤

if ©

= 0,

 ¤ ( ) is any if ©

≠ 0,

(21)

where ’ is an arbitrary constant. On substituting solutions (20) and (21) into (17), one can “remove” the redundant constants and define the functions £ and ¤ . Below we summarize the results. 1  . Solution for ž ≠ − Ÿ and ž ≠ −2 Ÿ :

… where 

1,



2,

=

¢ (ž + 2Ÿ )  ( + 2( ž + Ÿ )

and 

3

1)

+

2(



+

1)

−

‡

«

«

+2 ¬

( +  3 )2  2( ž + 2 Ÿ )( + 

−

1)

(corresponds to ©

= 0),

are arbitrary constants.

2  . Solution for Ÿ = − ž :

…

=



where the function ¤

1

ž)™

2

¤ ¦‹ + ¤ (‘ ¤



1 š#›

+‘

2š

−

›



(corresponds to ©

)

=™

2

> 0),

= ¤ ( ) is determined from the autonomous ordinary differential equation

­

2

¦'‹&‹¦ = ž)¢E™

+ 4ž

2 4

™ ‘

1‘

2®

2š

whose solution can be found in implicit form. In the special case ‘

1

¤ =š ® ,

, = 0 or ‘

2

= 0, we have ¤

=

1

exp ¯

2

< 0).

1 2

ž)¢E™

2 2

+

2

±°

.

3  . Solution for Ÿ = − ž :

… where the function ¤

=−



1

ž)™

2

¤ ¦‹ + ¤ [‘ ¤

1

sin( ™

‡

)+‘

2

cos( ™

‡

)]

(corresponds to ©

= −™

= ¤ ( ) is determined from the autonomous ordinary differential equation

­

¦'‹&‹¦ = − ž)¢E™

2

+ ž 2™

4

(‘

2 1

+‘

2 2® 2 )š

,

¤ =š ® ,

whose solution can be found in implicit form.

B.2.3. Solution of Functional Differential Equations by Splitting B.2.3-1. Preliminary remarks. Description of the method. As one reduces the number of terms in the functional differential equation (3) by differentiation, redundant constants of integration arise. These constants must be “removed” at the final stage. Furthermore, the resulting equation can be of a higher-order than the original equation. To avoid these difficulties, it is convenient to reduce the solution of the functional differential equation to the solution of a linear functional equation of a standard form and solution of a system of ordinary differential equations. Thus, the original problem splits into two simpler problems. Below we outline the basic stages of the splitting method.

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B.2. METHODS OF GENERALIZED SEPARATION OF VARIABLES

The case of even number of terms in equation (3), ² = 2 ³ . 1 ´ . At the first stage, we treat equation (3) as a purely functional equation that depends on two variables µ and ¶ , where · 1 (µ ), ¸)¸)¸ , · ¹ (µ ), º ¹ +1 ( ¶ ), ¸)¸)¸ , º 2¹ ( ¶ ) are unknown quantities and the functions · ¹ +1 (µ ), ¸)¸)¸ , · 2¹ (µ ), º 1 ( ¶ ), ¸)¸)¸ , º ¹ ( ¶ ) are assumed to be known. It can be shown (by induction and differentiation) that the functional equation (3) has a solution depending on ³ 2 arbitrary constants · » (µ

) = ¼ » 1 · ¹ +1 (µ ) + ¼ » 2 · ¹ +2 (µ ) + ½)½)½ + ¼ »0¹P·

2¹

(µ )

º ¹ + » ( ¶ ) = − ¼ 1 »º 1 ( ¶ ) − ¼ 2 »º 2 ( ¶ ) − ½)½)½ − ¼ ¹±»º ¹ ( ¶ )

(¾ = 1, ¸)¸)¸ , ³ ), (¾ = 1, ¸)¸)¸ , ³ ),

(22)

where the ¼ »À¿ are arbitrary constants. Note that there are also “degenerate” solutions depending on fewer arbitrary constants (see Item 2 ´ in Paragraph B.2.3-2). 2 ´ . At the second stage, we substitute the · » (µ ) and º ¿ ( ¶ ) of (4) into (22). This results in an overdetermined system of ordinary differential equations for the unknown functions Á  (ƒ ) and Ã Ä (Å ). The case of odd number of terms in equation (3), ² = 2 ³ − 1. 1 ´ . If the number or terms is odd ( ² = 2 ³ − 1), the functional equation (3) has two different solutions with ³ ( ³ − 1) arbitrary constants. One of them can be obtained from formulas (22) by setting · 2¹ ≡ 0 and discarding the last term with º 2¹ . The other solution can be obtained from the first one by renaming · » (µ ) Æ º » ( ¶ ). 2 ´ . Further analysis for each solution should be performed following the same scheme as in the case of even number of terms in (3). B.2.3-2. Solutions of simple functional equations and their application. Below we give solutions of two simple functional equations of the form (3) that will be used subsequently for solving specific nonlinear partial differential equations. 1 ´ . The functional equation

· 1º 1 + · 2º 2 + · 3º 3 = 0

(23)

where the · » are all functions of the same argument and the º » are all functions of another argument, has two solutions: · 1 = Ç 1 · 3 , · 2 = Ç 2 · 3 , º 3 = −Ç 1 º 1 − Ç 2 º 2 , (24) º 1 = Ç 1 º 3 , º 2 = Ç 2 º 3 , · 3 = −Ç 1 · 1 − Ç 2 · 2 , where Ç

1

and Ç

2

are arbitrary constants.

2 ´ . The functional equation · 1 º 1 + · 2 º 2 + · 3 º 3 + · 4 º 4 = 0,

(25)

where the · » are all functions of the same argument and the º » are all functions of another argument, has a solution · 1 = Ç 1 · 3 + Ç 2 · 4, · 2 = Ç 3 · 3 + Ç 4 · 4, (26a) º 3 = −Ç 1 º 1 − Ç 3 º 2 , º 4 = −Ç 2 º 1 − Ç 4 º 2 depending on four arbitrary constants Ç È [see solution (22) with ³ = 2, ¼ and ¼ 22 = Ç 4 ]. Equation (25) has also two “degenerate” solutions

11

=Ç

1,

¼ 12 = Ç 2 , ¼ 21 = Ç 3 ,

· 1 = Ç 1 · 4,

· 2 = Ç 2 · 4,

· 3 = Ç 3 · 4,

º 4 = −Ç 1 º 1 − Ç 2 º 2 − Ç 3 º 3 ,

º 1 = Ç 1 º 4,

º 2 = Ç 2 º 4,

º 3 = Ç 3 º 4,

· 4 = −Ç 1 · 1 − Ç 2 · 2 − Ç 3 · 3

(26b)

involving three arbitrary constants.

© 2002 by Chapman & Hall/CRC Page 689

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B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES Example 3. Consider the nonlinear hyperbolic equation

É 2Ê É Ê É   É  2 = ž É Ë   Ê É Ë ¡ + Ì ( )Ê + Í ( ), 

(27)



where Ì ( ) and Í ( ) are arbitrary functions. We look for generalized separable solutions with the form

Ê (Ë ,  ) = £ (Ë )¤ ( ) + Î ( ).

(28)

Substituting (28) into (27) and collecting terms yield

ž ¤

2

( £ £ ÏÐ ) ÏÐ + ž ¤ Î £ Ï&Ð2Ï Ð + ( Ì ¤ − ¤ ¦'Ï&Ï ¦ ) £ + Ì Î + Í − Î Ï&¦'Ï ¦ = 0.

This equation can be represented as a functional equation (25) in which

Ñ Ò

1 1

Ñ

= ( £ £ ÏÐ ) ÏÐ , 2

= ž ¤

Ò

,

Ñ

2

= £ Ï&Ð2Ï Ð ,

2

= ž ¤ Î ,

Ò

Ñ

3

=£ ,

3

= Ì ¤ − ¤ ¦'Ï&Ï ¦ ,

Ò

4

= 1,

4

= Ì Î + Í − Î Ï&¦'Ï ¦ .

(29)

On substituting (29) into (26a), we obtain the following overdetermined system of ordinary differential equations for the Ë  functions £ = £ ( ), ¤ = ¤ ( ), and Î = Î ( ): ( £ £ ÏÐ ) ÏÐ = ‘

Ì ¤ − ¤ ¦'Ï&Ï ¦ = − ‘

1£

+‘ 1 ž ¤

£ Ï&Ð2Ï Ð = ‘

2, 2

−‘

3 ž ¤

3£

+‘

4,

Î , Ì Î + Í − Î Ï&¦'Ï ¦ = − ‘

2 ž ¤

2

4

= 2’

−‘

4 ž ¤

(30)

Î .

The first two equations in (30) are consistent only if

‘ where ’

0,

’

1,

and ’

2

1

= 6’

2,

‘

=’

2

2 1

− 4’

0 ’ 2,

‘

3

= 0,

‘

2,

(31)

are arbitrary constants, and the solution is given by

Ë £ ( )=’

2

Ë

2

+’

1

Ë

+’

0.

(32)



On substituting the expressions (31) into the last two equations in (30), we obtain the following system of equations for ¤ ( )  and Î ( ):  ¤ ¦'Ï&Ï ¦ = 6 ž ’ 2 ¤ 2 + Ì ( )¤ , (33)   Î Ï&¦'Ï ¦ = [2 ž ’ 2 ¤ + Ì ( )]Î + ž ( ’ 12 − 4 ’ 0 ’ 2 )¤ 2 + Í ( ), Relations (28), (32) and system (33) determine a generalized separable solution of equation (27). The first equation  in (33) can be solved independently; it is linear if ’ 2 = 0 and is integrable in quadrature for Ì ( ) = const. The second equation in (33) is linear in Î (for ¤ known). Equation (27) does not have other solutions with the form (28) if Ì and Í are arbitrary function and £ Ó 0, ¤ Ó 0, and Î Ó 0.

Ô ÕMÖ ×PØRÙ Ú

It can be shown that equation (27) has a more general solution with the form

Ë Ë Ê (Ë , Û ) = £ 1 (Ë )¤ 1 ( ) + £ 2 (Ë )¤ 2 ( ) + ¤ 3 ( ), £ 1( ) = 2,  where the functions ¤ Ü = ¤ Ü ( ) are determined by the ordinary differential equations  ¤ 1Ï&Ï = 6 ž ¤ 12 + Ì ( )¤ 1 ,  ¤ 2Ï&Ï = [6 ž ¤ 1 + Ì ( )]¤ 2 ,   ¤ 3Ï&Ï = [2 ž ¤ 1 + Ì ( )]¤ 3 + ž ¤ 22 + Í ( ). 

Ë Ë £ 2( ) = ,

(The prime denotes the derivative with respect to .) The second equation in (35) has a particular solution ¤ its general solution can be represented as (see Polyanin and Zaitsev, 1995)

¤

2

=Ý

1¤ 1

+Ý

2¤ 1

The solution obtained in Example 3 corresponds to the special case Ý

Þ ¤ 2

(34)

(35)

2

=¤

1.

Hence,

ß2à 2 . 1

= 0.

Example 4. Consider the nonlinear equation

É Ê

É 2Ê É Ë É

2

+ á É Ë â

à

−Ê

É 3Ê É 2Ê É Ë 2 =ã É Ë 3,

which arises in hydrodynamics (see equations B.6.2.1, Item 3 ä and B.7.2.1, Item 2 ä ). We look for exact solutions of the form Ê = £ ( ) ¥ (Ë ) + ¤ ( ).

à

à

(36)

(37)

© 2002 by Chapman & Hall/CRC Page 690

691

B.2. METHODS OF GENERALIZED SEPARATION OF VARIABLES Substituting (37) into (36) yields

§ ( ¥#ÐÏ )2 − ¥2¥#Ð2Ï&Ï Ð ¨ − ã £ ¥#Ð2Ï&Ï&Ð2Ï Ð = 0.

2

£ ϦR¥#ÐÏ − £ ¤ ¥#Ð2Ï&Ï Ð + £

This functional differential equation can be reduced to the functional equation (25) by setting

Ñ Ò

1

= £ Ϧ ,

1

= ¥ ÐÏ ,

Ñ Ò

Ñ

2

=£ ¤ ,

2

= − ¥ Ð2 Ï&Ï Ð ,

Ò

2

Ñ

3

=£

3

= ( ¥ ÐÏ )2 − ¥2¥ Ð2 Ï&Ï Ð ,

,

Ò

4

=ã £ ,

4

= − ¥ Ð2 Ï&Ï&Ð2Ï Ð .

On substituting these expressions into (26a), we obtain the system of equations

£ Ϧ = å

1£

2

+å

2ã

£ ,

2

( ¥#ÐÏ ) − ¥2¥#Ð2 Ï&Ï Ð = − å

£ ¤ =å

1 ¥#ÐÏ

Ï&Ï Ð 3 ¥#Ð2

+å

¥#Ð2Ï&Ï&Ð2Ï Ð = å

,

2

3£

+å

2 ¥#ÐÏ

4ã

£ , Ï&Ï Ð 4 ¥#Ð2

−å

(38)

.

It can be shown that the last two equations in (38) are consistent only if the function ¥ and its derivative are linearly dependent, ¥#ÐÏ = æ 1 ¥ + æ 2 . (39) The six constants æ

1,

æ

2,

å

1,

å

2,

å

3,

and å

4

must satisfy the three conditions

æ

1 (å

1

æ

2 (å

1

+æ +æ

2 2

2 1

+å

exp( æ

1

æ Integrating (39) yields

¥ = ª æ æ

3 2

Ë

+æ

−å −å

3 æ 1)

4æ 1

Ë

)−

3 æ 1)

= 0, = 0,

−å

= 0.

æ

2

æ

1

3

2

(40)

if æ

1

≠ 0,

if æ

1

= 0,

(41)

where æ 3 is an arbitrary constant. The first two equations in (38) lead to the following expressions for £ and ¤ :

å 2ã Ý exp(− å 2ã ) − å £ = èçè à 1 − å + Ý 1 èèê à é

1

if å

2

≠ 0,

if å

2

= 0,

¤ =å

3£

+å

4ã

,

(42)

where Ý is an arbitrary constant. Formulas (41), (42) and relations (40) allow us to find the following solutions of equation (36) with the form (37):

Ê =

Ë

Ê = Ý

à

+Ý +Ý 1š

™

1 2 −

›

Ð

+Ý

+Ý +1 2

Ê = Ý 1 š à − › ( Ð +ë Ê = ã í + Ý 1š 1 + Ý 2 š −ì

3

+㠙

if

å

2

=æ

if

å

2

= 0, æ

1

=å

3

1

=å

ì ¦ ) + ã ( ™ + í ) if å − Ð › + ã ( ™ − í ) if å ë ¦ ›

1

= 0, æ

= −å

2

1

= −å

=æ

2

3æ 1

−æ

4,

æ

= 0, å 2,

å

2

1;

2

= −å

2

=æ

=æ

1 2 1

2 1

−å

+å

+å

3å

4;

4 æ 1; 4 æ 1,

where Ý 1 , Ý 2 , Ý 3 , í , and ™ are arbitrary constants (these can be expressed in terms of the å î and æ î ). The analysis of the second degenerate solution (26b) of the functional equation (25) leads to the following two more general solutions of the differential equation (36):

Ë

Ê = à

+Ý

Ê = £ ( )š − › à where £ ( ) and ¤ ( ) are arbitrary functions, and Ý

à

à

1

+ ¤ ( ),

1

Ð

à

−

£ Ϧ ( ) à +㠙 , ™ £ () à

and ™ are arbitrary constants.

B.2.4. Simplified Scheme for Constructing Exact Solutions of Equations with Quadratic Nonlinearities B.2.4-1. Description of the simplified scheme. To construct exact solutions of equations (2) with quadratic or power nonlinearities that do not depend explicitly on ï (all ð » constant), it is reasonable to use the following simplified approach. As

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B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES

before, we seek solutions in the form of finite sums (1). We assume that the system of coordinate functions { Á » (ï )} is governed by linear differential equations with constant coefficients. The most common solutions of such equations are of the form » Á » (ï ) = ï , Á » (ï ) = ñ ò ó±ô , Á » (ï ) = sin( õ »±ï ), Á » (ï ) = cos(ö »±ï ). (43) Finite chains of these functions (in various combinations) can be used to search for separable solutions (1), where the quantities ÷ » , õ » , and ö » are regarded as free parameters. The other system of functions {à » (Å )} is determined by solving the nonlinear equations resulting from substituting (1) into the equation under consideration. This simplified approach lacks the generality of the methods outlined in Subsections B.2.2 and B.2.3. However, specifying one of the systems of coordinate functions, { Á » (ï )}, simplifies the procedure of finding exact solutions substantially. The drawback of this approach is that some solutions of the form (1) can be overlooked. It is significant that the overwhelming majority of generalized separable solutions known to date, for partial differential equations with quadratic nonlinearities, are determined by coordinate functions (36) (usually with ø = 2). B.2.4-2. Examples of constructing exact solutions of higher-order equations. Below we consider specific examples that illustrate the application of the above simplified scheme to constructing generalized separable solutions of higher-order nonlinear equations. Example 5. The equations of laminar boundary layer on a flat plate are reduced to a single third-order nonlinear equation for the stream function (see Schlichting 1981, Loitsyanskiy 1996):

É Ê É

É Ê

É 2Ê É Ë É

Û

É 3Ê É 2Ê É 2 =ã É 3. Û Û

− É Ë

Û

(44)

We look for generalized separable solutions with the form Ê (Ë , Û ) = Ë ù (Û ) + ú (Û ), (45) Ë Ë Ë which corresponds to the simplest set of functions û 1 ( ) = , û 2 ( ) = 1 with – = 2 in formula (1). On substituting (45) into (44) and collecting terms, we obtain

Ë

ù

ù ù

ù

ù

ù

[( Ï )2 − Ï&Ï − ã Ï&Ï&Ï ] + [ Ï ú Ï − ú Ï&Ï − ã ú Ï&Ï&Ï ] = 0. Ë (The prime denotes the derivative with respect to Û .) To meet this equation for any , one should equate both expressions in ù ù square brackets to zero. This results in a system of ordinary differential equations for = ( Û ) and ú = ú ( Û ):

ù

( Ï )2 −

ù

ù ù ù

Ï&Ï − ã

ù

Ï&Ï&Ï = 0,

Ï'ú#Ï − ú#Ï&Ï − ã ú#Ï&Ï&Ï = 0. For example, this system has an exact solution ù Ý 2 Ý 3 6ã = , ú = + + Ý 4, Û +Ý 1 Û + Ý 1 ( Û + Ý 1 )2 where Ý 1 , Ý 2 , Ý 3 , and Ý 4 are arbitrary constants. Other generalized separable solutions of equation (44) can be found in Polyanin (2001b, 2001c) and Subsection B.6.1. Example 6. Consider the – th-order nonlinear equation

É 2Ê É ü Ê Ë (46) É 2 =Ì ( )É ü , Û Û Û Û Ë Ë where Ì ( ) is an arbitrary function. In the special case – = 3 with Ì ( ) = ã = const, this equation coincides with the boundary É Ê É

É 2Ê É Ë É

É Ê

− É Ë

layer equation (44). We look for generalized separable solutions of the form

Ê (Ë , Û ) = û (Ë ) š;› ý + ú (Ë ), (47) ù which correspond to the set of functions 1 ( Û ) = š › ý , 2 ( Û ) = 1 in (1). On substituting (47) into (46) and rearranging terms, we obtain ü Ë ™ 2 š › ý û [ ú#ÐÏ + ™ −2 Ì ( )] = 0. ù

This equation is met if

ü Ë ú ( ) = −™

Ë Ë Ë Ì ( ) + Ý , û ( ) is any, (48) ß where Ý is an arbitrary constant. (The other case û = 0 and ú is any is of little interest.) Formulas (47) and (48) define an exact solution of equation (46),

−2

Þ

Ê (Ë , Û ) = û (Ë ) š › ý − ™ ü

Ë Ë Ì ( ) +Ý , (49) ß Ë which involves an arbitrary function û ( ) andË two arbitrary constants Ý and ™ . Note that solution (49) with – = 3 and Ì ( ) = const was obtained by Ignatovich (1993) by a more complicated approach. −2

Þ

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B.3. METHODS OF FUNCTIONAL SEPARATION OF VARIABLES

B.3. Methods of Functional Separation of Variables B.3.1. Structure of Functional Separable Solutions B.3.1-1. Functional separable solutions. Suppose a nonlinear equation for þ = þ (ï , Å ) is obtained from a separable linear mathematical physics equation for ÿ = ÿ (ï , Å ) by a nonlinear change of variable þ = ( ÿ ). Then, obviously, the former has exact solutions of the form þ (ï , Å ) =

( ÿ ),

ÿ =

where






È =1

Á È (ï )Ã È (Å ).

(1)

It is noteworthy that many nonlinear partial differential equations that are not reduced to linear equation have exact solutions of the form (1) as well. We will call such solutions functional separable solutions. In general, the functions Á È (ï ), Ã È (Å ), and ( ÿ ) in (1) are not known in advance and are to be identified.  

 



 

 



In functional separation of variables, searching for solutions in the forms þ = (ï ) +  ( )  and þ = RÁ (ï ) ( )  leads to equivalent results, because the two forms are functionally equivalent. Indeed, we have RÁ (ï ) ( )  = 1 RÁ 1 (ï ) +  1 ( )  , where 1 ( ÿ ) = ( ñ  ), Á 1 (ï ) = ln Á (ï ), and  1 ( ) = ln  ( ).

RÁ

In constructing functional separable solutions with the form þ it is assumed that Á  const and   const.

=

RÁ

(ï ) +  ( )  ,

B.3.1-2. Various modifications. Below we give three more general modifications of solution structure (1): þ (ï ,  ) =

( ÿ ),

ÿ =






 =1

Á  ( )  ( ),

þ (ï ,  ) =  1 (ï ) ( ÿ ) +  2 (ï ),

ÿ =

þ (ï ,  ) =  1 ( ) ( ÿ ) +  2 ( ),

ÿ =







=  1ï +  2 , 

=  1ï +  2 ;

(2)




 =1


 =1

Á  (ï )  ( );

(3)

Á  (ï )  ( ).

(4)

These can also be used to find exact solutions of nonlinear mathematical physics equations. The solution structures of (1)–(4) cover all most common types of solutions—traveling wave, self-similar, and additively and multiplicatively separable solutions (as well as many invariant solutions). In general, the functions Á  ( ),   ( ), Á  (ï ),   ( ), ( ÿ ),   (ï ), and   ( ) are not known in advance and are to be determined in the analysis. Note that Miller and Rubel (1993) studied functional separable solutions of a different form (for a stationary heat equation with a nonlinear source); see also Clarkson and Kruskal (1989) and Burde (1994).

B.3.2. Special Functional Separable Solutions To simplify the analysis, some of the functions in (1) can be specified a priori and the other functions will be defined in the analysis. We call such solutions special functional separable solutions.

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B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES

B.3.2-1. Solutions of the form (1) with ÿ linear in one of the independent variables. Consider functional separable solutions of the form (1) in the special case where the composite argument ÿ is linear in one of the independent variables (e.g., in ï ). We substitute (1) into the equation under study and eliminate ï using the expression of ÿ to obtain a functional differential equation with two arguments. In many cases, this equation can be solved by the methods outlined in Section B.2. Example 1. Consider the nonstationary heat equation with a nonlinear source

   

 2  =   2 +  ( ).

(5)

We look for functional separable solutions of the special form     ù  = (  ),  = û ( ) + ( ). (6)   ù   The functions (  ), û ( ), ( ), and  ( ) are to be determined.  On substituting (6) into (5) and on dividing by ! , we have  #  " !$! ù "  ( ) û + =û 2  +  . (7) ! !   We express from (6) in terms of  and substitute into (7) to obtain a functional differential equation with two variables and  , " ù  #  !$! " ù " û  ( )  +û 2  +  = 0, − + û − ! ! û û which can be treated as the functional equation (25) in Section B.2 where " ù " ù " û % % % 2 = − + û , = − , 1 2 3 = û , û û  # !$! & & & , 1 = 1, 2 =  , 3 =  !

%

4

= 1,

4

=

&

  ( ) .  !

Substituting these expressions into relations (26a) of Section B.2 yields the system of ordinary differential equations " ù " ù " û − + û = ' 1 û 2 + ' 2, − = ' 3 û 2 + ' 4, û û  #  !$!  ( ) = −' 1 − ' 3  , = −' 2 − ' 4  ,   ! !

(8)

where ' 1 , ' 2 , ' 3 , and ' 4 are arbitrary constants. The solution of system (8) is given by



û ()=( ) *

−2 , 4

1+

ù 

 ( ) = − û ( ) /0'



( ) = *



 ( ) = −*

3

1 3 ('

1

1

" −

+'

−1 . 2

3

'

4

-

  2 ( )3 +'

exp 5 − 12 ' 4

'

3

2

2 ) exp 5

−' − 12

,

2

3



 +* 2 () 1

1 76

37 + *

'

2

3

−'

24

,

(9)

4, 1 76

,



where * 1 , * 2 , * 3 , and * 4 are arbitrary constants. The dependence  =  ( ) is defined by the last two relations in parametric form (  is considered the parameter). In the special case ' 3 = * 4 = 0, ' 1 = −1, and * 3 = 1, the source function can be represented in explicit form as     ( ) = − ( ' 4 ln + ' 2 ). (10) If '

3

≠ 0 in (9), the source function is expressed in terms of elementary functions and the inverse of the error function.

Example 2. Consider the more general equation

   

     2  =8 ( )  2 +9 ( ) 

  + : ( ) ( ).

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B.3. METHODS OF FUNCTIONAL SEPARATION OF VARIABLES

We look for solutions in the form (6). In this case, only the first two equations in system (8) will change, and the functions   (  ) and  ( ) will be given by (9). Example 3. The nonlinear heat equation    

 =  

    / ? + ( * 2 + * 3 ) + > @ . Example 4. Likewise, we can treat the A th-order nonlinear equation    B     =   B +  ( ).

B  # As before, we look for solutions in the form (6). In this case, the quantities 2 2 and !$! in (8) must be replaced by 2  (B ) and ! , respectively. In particular, for ' 3 = 0, apart from equations with logarithmic nonlinearities of the form (10), we obtain other equations. Example 5. For the A th-order nonlinear equation    B    =   B

   + ( )   ,

    the search for exact solutions of the form (6) leads to the following system of equations for 2 ( ), = ( ), (  ), and  ( ): " " " B B = 2 −= + 2 = ' 1 2 + ' 22 , − = ' 32 + ' 42 , 2 2  (! B )  = −' 1 − ' 3  ,  ( ) = −' 2 − ' 4  ,  ! where ' 1 , ' 2 , ' 3 , and ' 4 are arbitrary constants.  In the case A = 3, we assume ' 3 = 0 and ' 1 > 0 to find in particular that  ( ) = − '

 arcsin( C ).  Example 6. In addition, searching for solutions of equation (5) with  quadratically dependent on ,      = (  ),  = 2 ( ) 2 + = ( ), 2

−'

4

also makes sense here. Indeed, on substituting (11) into (5), we arrive at an equation that contains terms with   not contain terms linear in . Eliminating 2 from the resulting equation with the aid of (11), we obtain "  #  #  " " !$! !$! =  ( ) 2 −= + 2 + 2 2 − − 42 =  +  = 0.  + 42   ! ! ! 2 2

(11)



2

and does

To solve this functional differential equation with two arguments, we apply the splitting method outlined in Subsection B.2.3. It can be shown that, for equations (5), this equation has a solution with a logarithmic nonlinearity of the form (10).

B.3.2-2. Solution by reduction to equations with quadratic nonlinearities. In some cases, solutions of the form (1) can be searched for in two stages. First, one looks for a transformation that would reduce the original equation to an equation with a quadratic (or power) nonlinearity. Then the methods outlined in Section B.2 are used to find solutions of the resulting equation. Sometimes, quadratically nonlinear equations can be obtained using the substitutions D D D

(E ) = E F ( E ) = G ln E

(for equations with power nonlinearities), (for equations with exponential nonlinearities),

(E ) = H F 

(for equations with logarithmic nonlinearities),

where G is a constant to be determined. This approach is equivalent to specifying the form of the function I ( E ) in (1) a priori.

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696

B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES Example 6. The nonlinear heat equation with a logarithmic source  2          = 8   2 + J ( ) ln + K ( )

can be reduced by the change of variable



!



=+

to the quadratically nonlinear equation

 2       =8   2 +8 )  

2

-

  + J ( )  + K ( ),

which admits separable solutions with the form



where 2 1 ( ) = equations.



2,

2

2(



 =2 

1(



)=

1(





)+ 2

2(



)=

2(



)+=

3(



),

) = , and the functions = > ( ) are determined by an appropriate system of ordinary differential

B.3.3. Differentiation Method B.3.3-1. Basic ideas of the method. Reduction to a standard equation. In general, the substitution of expression (1) into the nonlinear partial differential equation under study leads to a functional differential equation with three arguments—two arguments are usual, L and  , and the third is composite, E . In many cases, the resulting equation can be reduced by differentiation to a standard functional differential equation with two arguments (either L or  is eliminated). Two solve the two-argument equation, one can use the methods outlined in Section B.2. B.3.3-2. Examples of constructing functional separable solutions. Below we consider specific examples illustrating the application of the differentiation method for constructing functional separable solutions of nonlinear equations. Example 7. Consider the nonlinear heat equation      =  

   /MJ ( )   4 .

(12)

   = 2 ( ) + = ( ).

(13)

We look for exact solutions with the form

 

=



(  ),

On substituting (13) into (12) and dividing by ! , we obtain the functional differential equation " #  = = 2 NON J ( ) + ( 2 N )2 P (  ), where

Differentiating (14) with respect to



 # P (  ) = J ( )  !$! + J ! (  ), ! 

=



(  ).

yields ## #   3P P 2 NONON J ( ) + 2 N 2 NON [ J ! ( ) + 2 (  )] + ( 2 N ) ! = 0.

(14) (15)

(16)

This functional differential equation with two variables can be treated as the functional equation (23) of Section B.2. This three-term functional equation has two different solutions. Accordingly, we consider two cases. Case 1. The solutions of the functional differential equation (16) are determined from the system of ordinary differential equations J ! + 2 P = 2' 1 J , P ! = ' 2 J , (17) ## # 3 2 NONON + 2 ' 1 2 N 2 NON + ' 2 ( 2 N ) = 0, where ' 1 and ' 2 are arbitrary constants. The first two equations (17) are linear and independent of the third equation. Their general solution is given by , 1 ! (U > ! +U −> ! ) if ' 21 > 2 ' 2 , 1+ 2+ S + ! P = ' J − 1J ! , , 1 J = QR + C = V | ' 21 − 2 ' 2 |. (U 1 + U 2  ) if ' 21 = 2 ' 2 , 1 2 ,+ 1 ! [ U 1 sin( C  ) + U 2 cos( C  )] if ' 2 < 2 ' 2 , 1 RT

(18)

© 2002 by Chapman & Hall/CRC Page 696

B.3. METHODS OF FUNCTIONAL SEPARATION OF VARIABLES Substituting P

of (18) into (15) yields a differential equation for



=*

1

1 +

,

1

!



=



| J (  )|−3 . 2 7 3  +*

697

(  ). On integrating this equation, we obtain 2,

(19)

where  * 1 and * 2 are arbitrary constants. The expression of J in (18) together with expression (19) define the function J = J ( ) in parametric form. Without full analysis, we will study the case ' 2 = 0 ( C = ' 1 ) and ' 1 ≠ 0 in more detail. It follows from (18) and (19) that ! !  J ( ) = U 1 + 2, 1 + U 2 , P = ' 1 U 2 , (  ) = * 3 ( U 1 + U 2 + −2 , 1 )−1 . 2 + * 2 ( * 1 = ' 1 U 2 * 3 ). (20) Eliminating  yields

 J ( )=

U

*

2 3

2 2* 3

−U

1



2

.

(21)

# The last equation in (17) with ' 2 = 0 has the first integral 2 NON + ' 1 ( 2 N )2 = const. The corresponding general solution is given by  1 1 2 ln / W for  4 2 ( )=− 1 > 0 and 2 > 0; 2 W W 2' 1 1 sinh 50' 1 X 2 + 36 W W W  1 1 2 ln / − W for  (22) 2 ( )=− 4 1 > 0 and 2 < 0; 2 W W 2' 1 1 cos 5 ' 1 X − 2 + 3 6 W W W  1 1 2 ln / − W for  4 2 ( )=− 1 < 0 and 2 > 0; 2 W W 2' 1 1 cosh 50' 1 X 2 + 36 W W W where 1 , 2 , and 3 are constants of integration. In all three cases, the following relations hold: W W W # ( 2 N ) = 1 + −2 , 1 Y + 2 , 2 NON = − ' 1 1 + −2 , 1 Y . (23) W W W We substitute (20) and (23) into the original functional differential equation (14). With reference to the expression of   in (13), we obtain the following equation for = = = ( ): " = = −' 1 U 1 1 + 2, 1 Z + ' 1 U 2 2 . W W Its general solution is given by  1 U 2 2 W = ()= ln , (24)  2 2' 1 4 exp(−2 ' 1 U 2 2 ) + U 1 1 W W W where 4 is an arbitrary constant.   W Formulas (13), (20) for , (22), and (24) define three solutions of the nonlinear equation (12) with J ( ) of the form (21) [recall that these solutions correspond to the special case ' 2 = 0 in (18) and (19)]. Case 2. The solutions of the functional differential equation (16) are determined from the system of ordinary differential equations ## # 3 3 2 NONON = ' 1 ( 2 N ) , 2 N 2 NON = ' 2 ( 2 N ) , (25) P P ' 1 J + ' 2 ( J ! + 2 ) + ! = 0. The first two equations in (25) are consistent in the two cases

  2 ( ) = U 1 + U 2, (26)   1 ' 1 = 2 ' 22 [ \ ln | U 1 + U 2 |. 2 ( )=− ' 2     The first solution in (26) eventually leads to the  traveling solution = ( U 1 + U 2 ) of equation (12) and the second   2^wave  solution to the self-similar solution of the form = ] ( ). In both cases, the function J ( ) in (12) is arbitrary. '

1

='

2

=0

[ \

A more detailed analysis of functional separable solutions (13) of equation (12) can be found in the reference cited below. _a` Reference: P. W. Doyle and P. J. Vassiliou (1998). Example 8. One can look for more complicated functional separable solutions of equation (12) with the form      = (  ),  = 2 (b ) + = ( ), b = + 8 ( 8 = const).   We substitute this into (12), divide the resulting functional differential equation by ! , and differentiate with respect to to obtain # ## #   − 8 2 cdc + 2 cdcdc J ( ) + 2 c 2 cdc [ J ! ( ) + 2 P (  )] + ( 2 c )3 P ! = 0, where the function P = P (  ) is defined by (15). This functional differential equation with two variables b and  can be treated as the functional equation (25) of Section B.2. The solution of (25) is given by relations (26), thus representing a system of ordinary differential equations for J , P , and 2 .

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B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES Example 9. Consider the nonlinear Klein–Gordon equation  2  2    2 −   2 =  ( ).

(27)

We look for functional separable solutions in additive form:     = (  ),  = 2 ( ) + = ( ).

(28)

Substituting (28) into (27) yields

=

"0# "

# " N N + e ( = )2 − ( 2 N )2 f K (  ) = g (  ), −2 O

(29)

where

 #    K (  ) = !$! ^ ! , g (  ) =  5 (  ) 6 ^ ! . "   On differentiating (29) first with respect to and then with respect to and on dividing by = 2 N , we have # "0# " " # # 2( = − 2 NON ) K ! + e ( = )2 − ( 2 N )2 f K !$! = g !$! . "0# " # Eliminating = − 2 NON from this equation with the aid of (29), we obtain # # " e ( = )2 − ( 2 N )2 f ( K !$! − 2K7K ! ) = g !$! − 2K ! g . This relation holds in the following cases: # # K !$! − 2K7K ! = 0, g !$! − 2K ! g = 0 " ( = )2 = ' = + U , ( 2 N )2 = − ' 2 + U where ' , U , and *

(case 1),

# # g $! ! − 2K ! g = ( '  + * )(K !$! − 2K7K ! )

−* ,

(case 2 h ),

(30)

(31)

(32)

are arbitrary constants. We consider both cases.

Case 1. The first two equations in (32) enable one to determine K (  ) and g (  ). Integrating the first equation once yields K ! = K 2 + const. Further, the following cases are possible:

K =C , K = −1 ^ (  + * 1 ), K = − C tanh( C  + * 1 ), K = − C coth( C  + * 1 ), K = C tan( C  + * 1 ),

(33 8 ) (33 9 ) (33 : ) (33 3 ) (33 + )

where * 1 and C are arbitrary constants. The second equation in (32) has the particular solution g = K (  ). Hence, its general solution in expressed by (e.g., see Polyanin and Zaitsev 1995) 37 g = * 2 K ( ) + * 3 K ( ) 1 , (34) K 2 ( ) where * 2 and * 3 are  arbitrary constants.  The functions (  ) and  ( ) are found from (30) as

 and U

( ) = U

1

i (  ) 37 + U 1

2,

  ( )= U

1g

(  ) i (  ),

i (  ) = exp / 1

where

K (  ) 37 4 ,

(35)

1 and U 2 are arbitrary constants ( is defined parametrically). Let us dwell on the case (33b). According to (34),

g =' where '

1

= −*

3

^ 3 and '

2

= −*

2



1 (

+*

1)

2

+

' 2  +*

1

,

(36)

are any numbers. Substituting (33b) and (36) into (35) yields

=U

1

ln |  + *

1|

+U

2,



='

1 U 1 (

+*

1)

+

' 2U ( + *

Eliminating  , we arrive at the explicit form of the right-hand side of equation (27):   −U  ( ) = ' 1 U 1 + j + ' 2 U 1 + −2 j , where k = U 1 For simplicity, we set *



1

= 0, U

= 1, and U 2 = 0 and denote '  (  ) = ln |  |,  ( ) = 8 + @ + 9 + −2 @ , 1

1

= 8 and '

2

1 1)

2

2

.

.

(37)

= 9 . Thus, we have

K (  ) = −1 ^  ,

g (  ) = 8l

2

+9 ^  .

(38)

* In case 2, equation (31) can be represented as the functional equation considered in Paragraph B.3.5-1.

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B.3. METHODS OF FUNCTIONAL SEPARATION OF VARIABLES

TABLE B1 "0"on n n n n Nonlinear Klein–Gordon equations m − m NON =  ( ) admitting functional separable solutions of the form = (  ),   = 2 ( ) + = (p ). Notation: ' , * 1 , and * 2 are arbitrary constants; q = 1 for  > 0 and q = −1 for  < 0

n Right-hand side  ( )

No

8

1

ln

n

8 + @ +9 +

2

3

n

8 sin

4

8 sinh

5

8 sinh

n

n

n

+ 2 9 ) sinh

n +

−2 @

ln tan

n

ln tanh

n

n

( )

n 4 n 4

+ 2 sin

4 arctan +

4-

n

+ 2 sinh

arctan + @ . 2 + cosh

2n 2-

Equations for = (p ) and 2 ( r ) (=

!

"

2

) =* 2

1+

(2 N ) = * 2+ " ( = )2 = 2 87=

ln |  |

n n

+ 9 ) sin + 9 ) sinh

+9

Solution

!

 2 ln s coth s s s 2 s s s s  2 ln s tan s s 2 ss s s s

−2 Z

+ 87= −

−2 Y

−8 2 +

3

+' =

2

+*

1 8 2 1 8 2 1=

+9 +' , +' +*

2,

( 2 N )2 = −2 8 2 3 + ' 2 2 − * 1 2 + * 2 + 9 " ( = )2 = * 1 + 2 Z + * 2 + −2 Z + 9d= + 8 + ' , ( 2 N )2 = − * 2 + 2 Y − * 1 + −2 Y − 9 2 + ' " ( = )2 = * 1 + 2 Z + * 2 + −2 Z − q 9d= + 8 + ' , ( 2 N )2 = * 2 + 2 Y + * 1 + −2 Y + q 9 2 + ' " ( = )2 = * 1 sin 2 = + * 2 cos 2 = + q 9d= + 8 + ' , ( 2 N )2 = − *

1 sin 2 2

+*

2 cos 2 2

−q 9 2 +'

It remains to determine = ( p ) and 2 (r ). We substitute (38) into the functional differential equation (29). Taking into account (28), we find # # "0# " " "0# " [ = = − ( = )2 − 87= 3 − 9 ] − [ 2 NON 2 − ( 2 N )2 + 8 2 3 ] + (= − 3 87= 2 ) 2 − = ( 2 NON + 3 8 2 2 ) = 0. (39) Differentiating (39) with respect to p and r yields the separable equation* ## "0##"0" " " (= − 6 87= = ) 2 N − ( 2 NONON + 6 8 2 2 N ) = = 0, whose solution is determined by the ordinary differential equations "0##"0" " " = − 6 87= = = ' = , ## 2 NONON + 6 8 2 2 N = ' 2 N , where ' is the separation constant. Each equation can be integrated twice, thus resulting in " ( = )2 = 2 87= 3 + ' = 2 + * 1 = + * 2 , (40) ( 2 N )2 = −2 8 2 3 + ' 2 2 + * 3 2 + * 4 , where * 1 , * 2 , * 3 , and * 4 are arbitrary constants. Eliminating the derivatives from (39) with the aid of (40), we find that the arbitrary constants are related by * 3 = − * 1 and * 4 = * 2 + 9 . So, the functions = (p ) and 2 ( r ) are determined by the first-order nonlinear autonomous equations " ( = )2 = 2 87= 3 + ' = 2 + * 1 = + * 2 , ( 2 N )2 = −2 8 2 3 + ' 2 2 − * 1 2 + * 2 + 9 . The solutions of these equations are expressed in terms of elliptic functions. For the other cases in (33), the analysis is performed in a similar way. Table B1 presents the final results for the cases (33a)–(33e). Case 2. Integrating the third and fourth equations in (32) yields

= = ( X U p + 1, if ' = 0; 2 =( X U −* p + 2 W W (41) U U −* 1 1 2 2 = = (' p + 1 ) − , 2 =− (' r + 2 ) + if ' ≠ 0; W W 4' ' 4' ' n where 1 and 2 are arbitrary constants. In n both n cases, the function  ( ) in equation (27) is arbitrary. Then first n row in (41) W W corresponds to the traveling wave solution = ( C r + t p ). The second row leads to a solution of the form = ( r 2 − p 2 ). _a` References: A. M. Grundland and E. Infeld (1992), J. Miller and L. A. Rubel (1993), R. Z. Zhdanov (1994), V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachev, and A. A. Rodionov (1994). Example 10. The nonlinear stationary heat (diffusion) equation n n n m 2 m 2 + =v ( ) m r 2 m u 2 is analyzed just as the nonlinear Klein–Gordon equation considered in n Example 9. The final results are listed n n n n in Table B2; the traveling wave solutions = ( C r + t p ) and solutions of the form = ( r 2 + u 2 ), existing for any v ( ), are omitted. _a` References: A. M. Grundland and E. Infeld (1992), J. Miller and L. A. Rubel (1993), R. Z. Zhdanov (1994), V. K. Andreev, O. V. Kaptsov, V. V. Pukhnachev, and A. A. Rodionov (1994). * To solve equation (39), one can use the solution of equation (25) in Section B.2 [see (26a)].

© 2002 by Chapman & Hall/CRC Page 699

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B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES TABLE B2 n n n n n Nonlinear equations m NON + mlww = v ( ) admitting functional separable solutions of the form = ( x ), x = 2 ( r ) + = (u ). Notation: y , * 1 , and * 2 are arbitrary constants; q = 1 for x > 0, q = −1 for x < 0

n Right-hand side z ( )

No

8

1

4

5

ln

n

8 + @ +9 +

2

3

n

8 sin 8 sinh 8 sinh

n

n

n

n

+ 2 9 ) sinh

n

−2 @

ln | x |

n + 2 sin

4

ln tanh

4 arctan + {

4-

n

n

(x )

+ {

ln tan

n

n

Equations for 2 (r ) and = ( u ) 2

n

+ 9 ) sin + 9 ) sinh

+9

Solution

n 4

+ 2 sinh

arctan + @ . 2 + cosh

2n 2-

x 2 ln s coth s s s 2 s s s s x 2 ln s tan s s 2 ss s s s

( 2 |N ) = *

1+

( = w | )2 = *

2+

−2 Y

+ 8 2 − 12 8 + 9 + y ,

−2 Z

+ 87= − 12 8 − y

( 2 | N )2 = 2 8 2

3

2

( = w| ) = 2 87=

3

2

2Y

+*

2Z

+*

1+

+*

2+

( 2 |N ) = *

1+

( = w | )2 = *

2+

( 2 | N )2 = *

1+

2

( = w| ) = − *

+y 2

2

+*

12

+*

2,

−y =

2

+*

1=

−*

2−9

−2 Y

+9 2 +8 +y ,

2Y 2+

2Z

2+

−2 Z

−*

−2 Y 1+

−2 Z

+ 9d= − y −q 9 2 +8 +y , − q 9d= − y

2

1 sin 2 2

+*

2 cos 2 2

+q 9 2 +8 +y ,

2

1 sin 2 =

−*

2 cos 2 =

+ q 9d= − y

( 2 |N ) = * ( = w| ) = *

B.3.4. Splitting Method. Reduction to a Functional Equation with Two Variables B.3.4-1. Splitting method. Reduction to a standard functional equation. The general procedure for constructing functional separable solutions, which is based on the splitting method, involves several stages outlined below. 1 } . Substitute expression (1) into the nonlinear partial differential equation under study. This results in a functional differential equation with three arguments—the first two are usual, L and ~ , and the third is composite, E . 2 } . Reduce the functional differential equation to a purely functional equation with three arguments L , ~ , and E with the aid of elementary differential substitutions (by selecting and renaming terms with derivatives). 3 } . Reduce the three-argument functional differential equation by the differentiation method to the standard functional equation with two arguments (either L or ~ is eliminated) considered in Section B.2. 4 } . Construct the solution of the two-argument functional equation using the formulas given in Subsection B.2.3. 5 } . Solve the (overdetermined) system formed by the solution of Item 4 } and the differential substitutions of Item 2 } . 6 } . Substitute the solution of Item 5 } into the original functional differential equation of Item 1 } to establish the relations for the constants of integration and determine all unknown quantities. 7 } . Consider all degenerate cases possibly arising due to violation of assumptions adopted in the previous analysis. The splitting method reduces solving the three-argument functional differential equation to (i) solving a purely functional equation with three arguments (by reducing it to a standard functional equation with two arguments) and (ii) solving a system of ordinary differential equations. Thus, the initial problem splits into several simpler problems. Examples of constructing functional separable solutions by the splitting method are given in Subsection B.3.5.

© 2002 by Chapman & Hall/CRC Page 700

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701

B.3.4-2. Three-argument functional equations of special form. The substitution of expression (1) with = 2 into nonlinear partial differential equation often leads to functional differential equations of the form


1 (

)

, )+

1 (

+



where






( ) and 





2 (

)

2 (

, )+ 

+1 ( 



,  ) +  + +2 ( 




( ) 



( ,  )

,  ) +  +  ( ,  ) = 0,

(42)

( ,  ) are functionals dependent on the variables  and  ,  , respectively,

( ) ≡




 , ,  ,    ), 



( ,  ) ≡ 



 ,  ,   ,     ,  ,   ,      .

(43)

(These expressions correspond to a second-order equation.) It is reasonable to solve equation (42) by the splitting method. To this end, we treat (42) at the first stage as a purely functional equation, thus disregarding (43). Assuming that  1  0, we divide (42) by  1 and differentiate with respect to  to obtain a similar equation but with fewer terms:

   (2) ( )  (2) ( ,  ) +  (2)+1 ( ,  ) +  +  (2) (44) ( ,  ) = 0, 2 (  )  2 (  ,  ) +  + 



 =       1  +         where  (2)   equation independent of  explicitly: 











( +1) +1 ( 

1

. We continue this procedure until we arrive at an

,  ) +  +  ( 







+1)

( ,  ) = 0,

(45)

  where  ( +1) =    ( )   ( )  +      ( )   ( )  .    Relation (45) can be regarded as an equation with two independent variables  and  . If  ( +1) ( ,  ) =   ( )   (  ) for all  =  + 1,  , , then equation (45) can be solved using the results of Section B.2.

B.3.5. Some Functional Equations and Their Solutions. Exact Solutions of Heat and Wave Equations In this subsection, we discuss several types of three-argument functional equations that arise most frequently in functional separation of variables in nonlinear equations of mathematical physics. The results are used to construct exact solutions for some classes of nonlinear heat and wave equations. B.3.5-1. The functional equation ( ) + ! ( ) =  (  ), where  = ( ) +  ( ). Here, one of the two functions ( ) and ( ) is prescribed and the other is assumed unknown, also one of the functions ! ( ) and  ( ) is prescribed and the other is unknown, and the function  (  ) is assumed unknown.* Differentiating the equation with respect to  and  yields     = 0. Consequently, the solution is given by ( ) = " ( ) + # , ! ( ) = "  ( ) − # + $ ,  (  ) = "  + $ , (46) where " , # , and $ are arbitrary constants. B.3.5-2. The functional equation (% ) + ! ( ) + & ( )  (  ) +  (  ) = 0, where  = ( ) +  (% ). Differentiating the equation with respect to  yields the two-argument equation !  + &  

+ &    +    = 0.

(47)

* In similar equations with a composite argument, it is assumed that ' ( ( ) ) const and * (+ ) ) const.

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B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES

Such equations were discussed in Section B.2. Hence, the following relations hold [see formulas (25) and (26a) in Section B.2]: !  = " 1 &  + " 2  , &  = " 3 &  + " 4  , (48)   = − " 1 − " 3  ,   = − "

−"

2

,

4

where " 1 , " 2 , " 3 , and " 4 are arbitrary constants. By integrating system (48) and substituting the resulting solutions into the original functional equation, one obtains the results given below. Case 1. If " 3 = 0 in (48), the corresponding solution of the functional equation is given by = − 12 " ! = & ="

1"

1 2"

1"

2

4 2

4

+ ("

+ ("

1# 1#

+"

1

+"

1

+"

2

+#

2 )

−#

3 )

4#

−#

2

1#

−#

3

4,

2,

+ # 1, = −" 1  + # 3 ,



= 12 " 



(49)

4

1"

2

4

− ("

+"

2

+#

3 )

4#

4,



where the " and # are arbitrary constants and = ( ) and  =  (% ) are arbitrary functions. Case 2. If " 3 ≠ 0 in (48), the corresponding solution of the functional equation is = −# " " 1,

=#

3,



= 

-

1

,

3

& =#

"

-

3

3

1 + / "

1 − " 3



4 3

"

−

3 −3

. + / "

3

"

−-

4#

"

−-

3,

1#

! =



1#

3

,

3



−

1"

4

"

"

0

3

1"

"

4 3

 −# +#

0

2

−#

4

−

"

1"

"

2 3

4

,

2,

(50)

,

1

"

2

"

−

2

, 1"

"

+ /

"

4 3

−"

2

0

 +#

4,



where the " and # are arbitrary constants and = ( ) and  =  (% ) are arbitrary functions. Case 3. In addition, the functional equation has the two degenerate solutions: ="

1

+#

1,

! ="

where = ( ),  =  (% ), and  constants; and ="

1

+#

1,

! ="

+#

1

& ="

2,

2,



= −"

1

=  (  ) are arbitrary functions, " +"

1

2&

+#

2,



= −"

2,



−" 1,

"

= −"

−#

2 2,

# 1

1,

1

−#

and #

−#

1

−#

2, 2

(51a)

are arbitrary 2,

(51b)

where = ( ),  =  (% ), and & = & ( ) are arbitrary functions, " 1 , " 2 , # 1 , and # 2 are arbitrary constants. The degenerate solutions (51a) and (51b) can be obtained directly from the original equation or its consequence (47) using formulas (26b) in Section B.2. Example 11. Consider the nonstationary heat equation with a nonlinear source

2 3 2 23 3 2 4 = 2 2 + 5 ( ). (

(52)

We look for exact solutions of the form

3

=

3

( 6 ),

4 6 = ' (( ) + * ( ).

(53)

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B.3. METHODS OF FUNCTIONAL SEPARATION OF VARIABLES

3 7 8 yields the functional differential equation 3 7:7 3 8=8 7 7:7 7 5 ( ( 6 )) * 9 = ' ; 0,

3,

D

2 3,

D

2 3,

D

2

=

1 2

D

2

−2 − 3 P 1 , 3

H I

( +F

1T

Q +F

2

for D

1

=

1 2

D

2 3,

D

2

=

1 2

D

3 D 4,

E

2

= 0, D

3D

4

< 0,

( +F

1T

Q +F

2

for D

1

=

1 2

D

2 3,

D

2

=

1 2

D

3 D 4,

E

2

= 0, D

3D

4

< 0,

Q

Q

where F 1 and F 2 are arbitrary constants. The right-hand sides of equation (52) corresponding to these solutions are represented in parametric form. 7 Case 3. Traveling wave solutions of the nonlinear heat equation (52) and solutions of the linear equation (52) with 5 U = const correspond to the degenerate solutions of the functional equation (51). Example 12. Likewise, one can analyze the more general equation

2 3 2 23 2 3 3 + 5 ( ). (56) 2 4 = V (( ) 2 2 + W (( ) 2 ( ( It arises in convective heat/mass exchange problems ( V = const and W = const), problems of heat transfer in inhomogeneous 7 media ( W = V ; ≠ const), and spatial heat transfer problems with axial or central symmetry ( V = const and W = const B ( ). Searching for exact solutions of equation (56) in the form (53) leads to the functional equation B.3.5-2 in which

> 4

7

7:7

7

7

3 7:7

3 7

> 3

3 7

( ) = −* 9 , ? ( ( ) = V (( ) ' ; 0, (6) where the functions F (Y ) and c (Y ) are determined by solving the following system of first-order ordinary differential equations with variable coefficients: ] 2 F :H = _LM r'F + c 2) + ` F + , (7) : c H = 2 _LM F c + ` c − ; _?c . (8) One can express F in terms of c from (8) and then substitute into (7). As a result, one obtains a ] second-order nonlinear equation for c ; if M , ` , = const, this equation is autonomous and, hence, admits reduction of order. npo Reference: V. F. Zaitsev, A. D. Polyanin (1996).

15. N

2

=

O W

a

N

O

N

N

2

2

+ S (W ) Q N

O

P

N

+

(W )

= S

P

R

X

N

+ O

O N

(W )

b S

2 O

N

W

a

N

O

N

N

+ V (W ).

G

),

2

2

=

O

O

P

The equation has exact solutions of the form (G , Y ) = F (Y ) + c (Y ) exp( d where d is a root of the quadratic equation d 2 + _Ld + K = 0. 16.

+ U (W )

2

+ S ( )Q P

P

N

+U ( )N O

+ O

P

X

N

P

Exact solution in additive form:

R

N

(G , Y ) = F (G ) + Z

+ V ( ) + z (W ).

= O

P

P : [ {

+[

:

|

{

[

−

:~} {

(Y )  Y ,

where Z is an arbitrary constant and the function € ( ) is determined by the following second-order ordinary differential equation with variable coefficients: ] 2 ; € ƒ‚I‚ ƒ + „ (  )( € ‚ƒ ) + ` (  ) € ‚ƒ + _*€ + (  ) = 0. 17. …

† …

W

=

2

2 a

Exact solution:

… …

†

2

+ S (W ) Q

‡

…

+ [U 1 (W ) + U 0 (W )] …

…

† ‡

X

+ V (W ) †

R

‡

…

†

+ z (W )

2 ‡

+ ˆ (W ) + s(W ). ‡

‡

( , Y ) = € (Y ) 2 + c (Y ) + f (Y ), where the functions € (Y ), c (Y ), and f (Y ) are determined by solving the following system of first-order ordinary differential equations with variable coefficients: ] } 2 € :‚ = 4 „ € + (2` 1 + ) € + , (1) ] c : ‚ = (4 „ € + ` 1 + ) c + 2` 0 € + ‰ , (2) ] 2 Š f ‚ = f + 2‹ € + „ c + ` 0c + Œ . (3) Equation (1) for € (Y ) is a Riccati equation; it can be reduced to a second-order linear equation. For solutions of Riccati equations, see Kamke (1977) and Polyanin and Zaitsev (1995). Whenever a solution of equation (1) is known, the solutions of equations (2) and (3) can be obtained successively (the equations are linear in c and f ).

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B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES

18. …

=

† W

2 a

…

…

† …

+

2

S

,

Q ‡

…

‡

+ †

…

‡

X Exact solution in additive form:

+ U (W ).

 †

R Š

( , Y ) = € ( ) + Z

[ {

Š

+[

|

{

− [

Š {

(Y )  Y , `

where Z is an arbitrary constant and the function € ( ) is described by the second-order ordinary differential equation ‹ € ƒ‚I‚ ƒ + „ (  , € ‚ƒ ) + _*€ = 0. X

B.5.1-3. Equations of the form 19. …

=

† W …

(W ) S

…

Q

‡



Exact solution:

…

…

+ U (W ) †

 ‡

‡

…

R

ƒ 2 Ž

ln †

X

‡

2

= „ ( , Y ) Ž

Ž Š Ž

+`

,Y , ,

r'

Ž ƒ Ž v

.

. †

2

( , Y ) = exp w € (Y )

+ c ( Y )y ,

where the functions € (Y ) and c (Y ) are determined by solving the following system of first-order ordinary differential equations with variable coefficients (the arguments of „ and ` are not specified): 2

€

Š‚

= 4„

c

Š‚

= 2(‘ + 1) „

€

+`

, €

+` €

. c

Integrating successively yields (Y ) = €

[ g

where h and x 20. …

Q

W

„

[ g

,

 Y

(Y ) = c

x

[ g

+ 2(‘ + 1) [

| „ g

€

[

− g

 Y

, i

=

| `

 Y

,

R

are arbitrary constants. 2

= S (W ) …

†

−1 |

−4 h

…

† …

Exact solution:

+

2

j ‡

(W ) V

(W ) + U

X

‡

k

‡

…

+ s(W )

…

†

ln †

+[ †

2 ‡

z

(W ) + ˆ (W )] . †

‡

2

( , Y ) = exp w € (Y )

+ c ( Y )y ,

where the functions € (Y ) and c (Y ) are determined by solving the following system of first-order ordinary differential equations with variable coefficients: € c

}

= 4 „ € 2 + (2` + Œ ) € + , ] = Œ*c + 2( „ + ) € + ‰ .

Š‚ Š‚

(1) (2)

The Riccati equation (1) for the function € (Y ) can be reduced to a second-order linear equation. For solutions of the Riccati equation, see Kamke (1977) and Polyanin and Zaitsev (1995). On solving (1), one can determine the solution of the linear equation (2) for c (Y ). 21. …

† …

W

= S (W )

Exact solution:

…

…

Q ‡

’

“ ”

…

+ U (W )

…

† ‡

R

X

†

ln †

+ V (W ) .

( , Y ) = exp wp€ (Y ) [

†

−•

ƒ

+ c ( Y )y ,

where the functions € (Y ) and c (Y ) are determined by the ordinary differential equations €

Š‚ c

Š‚

= d 2 „ (Y ) € 2 + ` (Y ) € , ] = ` (Y )c + (Y ).

© 2002 by Chapman & Hall/CRC Page 714

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B.5. SECOND-ORDER NONLINEAR EQUATIONS

Integrating yields €

c

where h and x 22. …

W

(Y ) =

(Y ) j i

x

2

−d h

−1 |

] |

(Y ) + i (Y ) i

(Y ) i (Y )  „

(Y ) (Y )

i

,

Y k

(Y ) = exp j i

| `

(Y ) 

Y k

,

,

 Y

are arbitrary constants.

=

†

(Y ) =

…

( )…

j–S

+a

†

k

…

‡

…

‡

…

Exact solution:

ln †

. †

X

‡

Š

( , Y ) = exp w h

+ € (  )y ,

[ —

where h is an arbitrary constant and the function € ( ) is determined by the ordinary differential equation ( „ € ‚ƒ ) ‚ƒ + „ ( € ‚ƒ )2 + ‹ € = 0. 23. …

=

† W …

…

( )…

j–S

†

+a k

…

‡

‡

…

ln †

+ [U ( ) + V (W )] . †

‡

†

‡

Exact solution in multiplicative form: X Š

( , Y ) = exp jtZ

Š

+[

[ —

|

− [

—

Š ]

(Y ) 

—

Y3k €

( ),

where Z is an arbitrary constant and the function € ( ) is determined by the ordinary differential equation ( „ € ‚ƒ ) ‚ƒ + ‹ € ln € + ` ( ) € = 0. 24. …

2

= S ( )…

†

†

2

‡

W …

…

+U ( )…

+a †

‡

‡

…

ln †

+ [ V ( ) + s(W )] . †

‡

†

‡

Exact solution in multiplicative form: X Š

( , Y ) = exp j'Z

[ —

Š

+[

|

—

[

−

Š —

Œ

(Y ) 

Y3k

€

( ),

where Z is an arbitrary constant and the function € ( ) is determined by the ordinary differential equation ] „ (  ) € ƒ‚I‚ ƒ + ` (  ) € ‚ƒ + ‹ € ln € + ( ) € = 0. 25. …

† …

W

2

= S ( )…

†

‡

…

2

‡

+U ˜

‡

Exact solution in additive form: ›

, …

+ †

…

‡

š †

+ V (W ).

™

( , œ ) = € ( ) + 

Š

+ž

ž —

Š

| ž

—

−

Š Ÿ —

(œ )  œ ,

where  is an arbitrary constant and the function € ( ) is determined by the following second-order ordinary differential equation: „

( ) €

ƒ‚I‚ ƒ

+ ¡ ( , €

‚ƒ

)+‹ €

= 0.

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B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES › ›

B.5.1-4. Equations of the form 26. …

2

=

† W

š

…

†

…

+¤ ( )

2

†

†

‡

27.

‡

± °

š

°

± °

+¤ ( )

2

±

²

+© − 

Ž ƒ Ž

1 ‹

ª

. £

²

(¬ − ­ ) ® (­ ) ¯ ­ , «

0

are arbitrary constants.

0 2

=

°

,œ , ,

¢t

+¥ .



( , œ ) = ( ¦? + § )œ + ¨ where ¨ , © , and ¬

+„

Ž ƒ 2 Ž ‹

›

…

Exact solution:

+

‡

2

=

Ž Š Ž

«

+ ´ ( ). ±

²

³

1 µ . Exact solution: › (¬ , œ ) = where ¨ and ©

(œ )( ¨ ¶

¡

+ © ) + ¶ (œ ) ¬

ª

¶

(œ ) ¯ œ , (œ )

(œ ) = exp · ¶

(œ ) ¯ ®

ª

œ3¸

,

are arbitrary constants.

2 µ . Exact solution: ›

2

(¬ , œ ) = ¹ (œ )(¬ (œ ) = ¹

+ © ) + ¹ (œ ) ¬

−1

(œ ) ·t − 2 º ¶

+¨ (œ ) ¯

¶ ª

,

œ3¸

¡ ª

¹

(œ ) ¯ œ , (œ )

(œ ) = exp · ¶

® ª

(œ ) ¯

,

œ3¸

where ¨ , © , and  are arbitrary constants. 28. °

= ±

°

2 š

°

±

±

²

°

Exact solution:

+¤ ( )

2

³

°

+´ ( ) ±

±

³

°

²

› ³

+ » ( ). ±

²

(¬ , œ ) = ¹ (œ ) ¼ (¬ ) + ½ (œ ), where the functions ¹ (œ ), ½ (œ ), and

(¬ ) are described by ordinary differential equations ¼

¹ ½ º

Š¾

=

Š¾

=

¼



2 ¹

¿

+ ¡ (œ ) ¹ ,

+ ® (¬ ) ¼

¾I¾

Ÿ

+ ¡ (œ )À?½ + (œ ), ¹

= ¾

, 

« « « where  is an arbitrary constant. Integrating successively yields

(œ ) = ¹

(œ ) = ¨

2¹

(¬ ) = ©

1

½

¼

where ¨

(œ ) ·Â¨ Á

1, ¨

2, ©

1,

−1

−

1

ª

(œ ) + ¹ (œ ) ¯ ¬ ª

¶

and ©

(¬ ) 2

Á Ÿ

ª

+©

¹

2

(œ ) ¯

œ3¸

(œ ) (œ )

¯ œ

+

 º

ª

, Á

(œ ) = exp · ª

¡

(œ ) ¯

œ3¸

,

, · ª

¶

(¬ ) ¯ ¬

¸ ¶

¯ ¬

(¬ )

, ¶

(¬ ) = exp ·

1 º

ª

®

(¬ ) ¯ ¬

¸

,

are arbitrary constants.

© 2002 by Chapman & Hall/CRC Page 716

717

B.5. SECOND-ORDER NONLINEAR EQUATIONS

29.

2

=

°

± °

š

°

±

2

±

²

°

2

+ ¤ ( )Ã °

²

³

Exact solution:

± › °

+´ ( )°

+» ( ) ±

²

³

²

™

°

+ s( ). ±

²

³

2

(¬ , œ ) = ¹ (œ )¬

+ ½ (œ )¬ + Ä (œ ),

where the functions ¹ (œ ), ½ (œ ), Ä (œ ) are determined by solving the following system of first-order Ÿ ordinary differential equations with variable coefficients (the arguments of ® , ¡ , , and Å are not specified): Š¾ ½ Ä

= (2 º

Š¾

Ÿ

2

= 2(2 ® + º ) ¹ = (4 ® ¹ + 2 º

Š¾ ¹

¹

+ ¹ , Ÿ + ) ½ + 2¡

Ÿ

+ )Ä + ® ¹

½

2

+¡

,

(1) (2)

+Å .

(3)

¹

½

Equation (1) for ¹ = ¹ (œ ) is a Bernoulli equation; it is easy to integrate. After that, one can successively construct the solutions of equations (2) and (3); each equation is linear in the unknown function.

ÆpÇ

30.

Reference: V. F. Zaitsev, A. D. Polyanin (1996).

°

=

°

± ²

2

2 š

°

±

± °

1 µ . Exact solution:

2

+

³

È

Ã

°

+ ±

°

³

2 ¥

+¤ ( )

±

²

+ ´ ( ). ±

²

™

›

(¬ , œ ) = ¹ (œ ) + ½ (œ ) exp(É Ê

¬

),

= Ê

Ã

−§ +¦ º

1Ë 2

,

(1)

™

where the functions ¹ (œ ) and ½ (œ ) are determined by solving the following first-order ordinary differential equations with variable coefficients (the arguments of ® and ¡ are not specified): Š¾ ¹

Š¾ ½

2

= §L¹ = (º

+® 2

Ê

¹

+¡ , + 2 §L¹ + ® )½ .

(2) (3)

¹

Equation (2) for ¹ = ¹ (œ ) is a Riccati equation; it can be reduced to a second-order linear equation. The books by Kamke (1977) and Zaitsev and Polyanin (1995) present many solutions of equation (2) for various ® and ¡ . Given a solution of equation (2), the solution of equation (3) for ½ = ½ (œ ) is evaluated by ½

(œ ) =

exp · 

(º ª

2 Ê

¹

+ 2 §L¹ + ® ) ¯

œ3¸

,

º

−§ +¦

(4)

where  is an arbitrary constant. 2 µ . Exact solution ( ¨ › is an arbitrary constant): (¬ , œ ) = ¹ (œ ) + ½ (œ ) cosh( Ê ¬

+ ¨ ), Ê

= Ã

1Ë 2

,

(5)

™

where the functions ¹ (œ ) and ½ (œ ) are determined by solving the following first-order ordinary differential equations with variable coefficients (the arguments of ® and ¡ are not specified): Š¾ ¹ ½

Š¾

= §L¹ = (º Ê

2

− ¦LÊ 2 ½ 2 + ® ¹ + ¡ , ¹ + 2 §L¹ + ® )½ .

2

(6) (7)

One can express ¹ from (7) in terms of ½ and substitute the resulting ¹ into (6). As a result, one arrives at a second-order nonlinear equation for ½ (if ® , ¡ = const, this equation is autonomous and, hence, admits reduction of order).

© 2002 by Chapman & Hall/CRC Page 717

718

B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES

3 µ . Exact solution ( ¨ › is an arbitrary constant): (¬ , œ ) = ¹ (œ ) + ½ (œ ) sinh( Ê where the functions ¹ (œ ) and differential equations

+ ¨ ), ¬

= Ê

Ã

1Ë 2

º

−§ +¦

, ™

(œ ) are determined by solving the system of first-order ordinary ½

=

¹

Š¾ ½

Š¾

2

§L¹

= (º

2 Ê

2

+ ¦LÊ

2 ½

+®

+¡ , ¹

+ 2 §L¹ + ® )½ . ¹

4 µ . Exact solution ( ¨ ›is an arbitrary constant): (¬ , œ ) = ¹ (œ ) + ½ (œ ) cos( Ê where the functions ¹ (œ ) and differential equations

+ ¨ ), ¬

= Ê

1Ë 2 §

Ã

,

+¦ º

(8)

™

(œ ) are determined by solving the system of first-order ordinary ½

=

¹

Š¾ ½

Š¾

2

§L¹

= (− º

2

+ ¦LÊ 2 Ê

2 ½

+®

+¡ , ¹

(9)

+ 2 §L¹ + ® )½ . ¹

(10)

One can express ¹ from (10) in terms of ½ and substitute the resulting ¹ into (9). Thus, one arrives at a second-order nonlinear equation for ½ (if ® , ¡ = const, this equation is autonomous and, hence, admits reduction of order).

ÆpÇ

Reference: V. F. Zaitsev, A. D. Polyanin (1996).

31.

=

°

± °

2

2 š

°

±

±

²

°

2

+¤ ( )Ã °

+ [´ 1 ( ) + ´ 0 ( )] ° ±

²

²

³

›³

°

³

+» ( ) ±

³

²

°

™

The equation has an exact solution of the form 2

(¬ , œ ) = ¹ (œ )¬

+Ì ±

2( ²

)

2 ³

+Ì

1( ²

) +Ì ³

0( ²

).

³

+ ½ (œ )¬ + Ä (œ ),

where the functions ¹ (œ ), ½ (œ ), and Ä (œ ) are determined by a system of first-order ordinary differential equations with variable coefficients (the system › is not specified here). › B.5.1-5. Equations of the form 32.

=

°

± °

š

°

Ã

²

°

Ñ

±

º

Ï

³

°

³

™

( )Í

¿ ®

1–«

²

The change of variable Ð =

Í Í

+¤ ( )

›

° ±

=

Í ÎŠ Í

±

. Ï

Ò Ò

Ð œ

=

°

± ²

Ö

°

Ã

°

°

±

=

³

°

The change of variable Ð =

Ô

²

³ Ø

Ò Ð Õ

Ð

2 ¬

º

+

2

²

2 Ð

Ò Ã

Ó

+´ ( ) ±

=

º Ð

2 Ò

¬

£

.

Ô

+Ó ®

(Õ ),

+ ½ (Õ )¬ + Ä (Õ ). 1–

±

. Ï

¬

Ò Ð

2

+ Ó

º Ã

which admits solutions with the form Ð = ¹ (Õ )¬

ÆpÇ

Í Î Í

leads to an equation of the form B.5.1.29,

Ò Ò

,œ , ,

«

Ò

2 Ò

º Ð

+× ( )

Ñ ±

Ï

¢'¬

«

which admits solutions with the form Ð = ¹ (Õ )¬ °

+® À

leads to an equation of the form B.5.1.29, Ò

33.

Î Í

2 Ð

Ò ¬

2

Ô

+Ó ®

(Õ )Ð + Ó Ù

(Õ ),

+ ½ (Õ )¬ + Ä (Õ ).

Reference: V. F. Zaitsev, A. D. Polyanin (1996).

© 2002 by Chapman & Hall/CRC Page 718

719

B.5. SECOND-ORDER NONLINEAR EQUATIONS

34.

=

°

± °

Ö

°

Ã

° ±

²

°

±

³

+ Ô

Ï °

1+ È

³

For Ú = 0, see equation B.5.1.33. The change of variable Ð = Ø Ò Ò

Õ

=

Ò

2 Ð

Ò

º Ð

²

+´ ( ) ±

²

1– ±

. Ï

leads to an equation of the form B.5.1.30,

Ò Ð

+× ( ) Ï

±

Ñ

+

2 ¬

º

2 Ð

Ò Ã

Ó

¬

+Ú

Ô

Ó Ð

2

+Ó

(Õ )Ð + Ó ®

Ù

(Õ ),

which admits solutions with the forms (¬ , Õ ) = ¹ (Õ ) + ½ (Õ ) exp(É Ð

Ê

Ð

(¬ , Õ ) = ¹ (Õ ) + ½ (Õ ) cosh( Ê (¬ , Õ ) = ¹ (Õ ) + ½ (Õ ) sinh( Ê Ð

(¬ , Õ ) = ¹ (Õ ) + ½ (Õ ) cos( Ê

Ð

), ¬

+ Û ), + Û ), ¬

¬ ¬

+ Û ),

where the functions ¹ (Õ ) and ½ (Õ ) are determined by a system of first-order ordinary differential equations; the parameter Ê is a root of a quadratic equation and Û is an arbitrary constant.

ÆpÇ

Reference: V. F. Zaitsev, A. D. Polyanin (1996).

35.

=

°

± °

Ö

°

²

Ã

°

Ü

Ý Þ

°

+ × ( ) + ´ ( )Ü

±

³

°

The change of variable Ð =

Ô

²

³ ß à Î

Ò Ò

Ð

= Õ

2 Ò

º Ð

Ð

= ±

°

Ö

°

²

Ã

°

Ü

°

+ × ( ) + (È

±

Ý Þ

³

°

The change of variable Ð =

Ô

³

³ ß à Î

+Ê

2 ¬

which admits solutions with the form Ð = ¹ (Õ )¬ °

.

Ý Þ

leads to an equation of the form B.5.1.27, Ò

36.

–

²

2

®

Ù

(Õ ),

+ ½ (Õ )¬ + Ä (Õ ).

+ á )Ü ³

(Õ )Ð + Ê

–

Ý Þ

.

leads to an equation of the form B.5.1.26, Ò

Ò Ð Ò

= Õ

2 Ò

º Ð

Ð

2 ¬

+Ê

(¬ )Ð + Ê ( Ú?¬ + § ), ®

which admits solutions with the form Ð = Ê ( Ú?¬ + § )Õ + ¹ (¬ ). 37. °

=

°

± ²

Ö °

°

Ã

Ü

Ý Þ

³

° °

±

+ Ô

È Ü

+ × ( ) + ´ ( )Ü

Ý Þ

²

²

–

Ý Þ

.

³

For Ú = 0, see equation B.5.1.35. The change of variable Ð = ß à Î

leads to an equation of the form B.5.1.30,

Ò

Ò Ð Ò

= Õ

º Ð

2 Ò ¬

Ð

2

+ Ú?Ð

2

+Ê

(Õ (Õ (Õ (Õ

)+½ )+½ )+½ )+½

(Õ (Õ (Õ (Õ

) exp(É â ¬ ), ) cosh(â ¬ + Û ), ) sinh(â ¬ + Û ), ) cos(â ¬ + Û ),

®

(Õ )Ð + Ê Ù

(Õ ),

which admits solutions with the forms Ð Ð Ð Ð

(¬ (¬ (¬ (¬

,Õ ,Õ ,Õ ,Õ

)= )= )= )= ¹ ¹ ¹ ¹

where the functions ¹ (Õ ) and ½ (Õ ) are determined by a system of first-order ordinary differential equations; the parameter â is a root of a quadratic equation and Û is an arbitrary constant.

ÆpÇ

Reference: V. F. Zaitsev, A. D. Polyanin (1996).

© 2002 by Chapman & Hall/CRC Page 719

720

B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES

B.5.1-6. Equations of the form 38.

=

°

±

2

( ) ×

³

Ö

° ² ± Exact solution:

+

°

± °

+Ù

Î2 Í

ã

,Õ ,Ø , ¬

.

Í Î Í «

«

.

2

È

2

= ® (¬ , Õ , Ø ) Í

Í ÎŠ Í

ä

³

1

(¬ , Õ ) = Ø

(¬ )Õ + ½ (¬ ) − Ú À ,

¿ ¹

º

where the functions ¹ (¬ ) and ½ (¬ ) are determined by the ordinary differential equations (¬ ) ¹ ®

2

−¹

¾I¾

= 0,

(¬ )½ ®

−¹

¾I¾

= 0. ½

The first equation can be treated « independently. The second equation has a particular solution « « « ½ ( ¬ ) = ¹ ( ¬ ), and hence, its general solution is given by (¬ ) = ½

where Û 39.

and Û

1

±

°

Ã

°

²

°

±

²

°

Exact solution:

±

³

+´ ( ) Ô

Ï °

²

where the functions equations

= ¹ (¬ ) and ¹

¶

where æ and ç

ÆpÇ

2 (¬ ¹

. Ï

)

,

= ½

1Ë

+ ½ ( Õ )À

,

+ 2) ®

Ó

2 ¹

,

= 2®

å¾ ½

¹

+Ó ½

Ù

.

Ñ Ñ

− ¶

2

= ½ (¬ ) are determined by the first-order ordinary differential ½

Ñ

,

Ñ

(Õ )¬

¿ ¹

Ñ

Integrating yields 1

1– ±

(¬ , Õ ) =

2(Ó

=

å¾ ¹

=

ª

³ Ø

¹

¯ ¬

(¬ )

2¹

are arbitrary constants.

2

=× ( )

°

(¬ ) + Û

1¹ Û

+2 Ã

+Ó æ

Ù

+2 ¶

ª

¯ Õ

, Ô

= ¶

− ç

2(Ó

+ 2) Ó ª

®

¯ Õ

,

are arbitrary constants.

Reference: V. F. Zaitsev, A. D. Polyanin (1996).

40.

=

°

± °

°

( )Ü

·–×

è

° Þ

±

. ¸

³

²

°

³

°

³

Exact solution in additive form: (¬ , Õ ) = − Ø

1 é

ln(

é Õ

+Û )+ é

1

é

æ

ln ·–ê

®

ë

− ë ( )

ë ì

+ç í

,

where æ , ç , and Û are arbitrary constants. B.5.1-7. Equations with three independent variables. 41. î

=

î

ï ð

î

×

( ,ó )î ï

í

î

ñ

ñ

ò

î

+

( ,ó )î

î

ñ

ï

î

òô

Exact solution in multiplicative form:

ó

ln

õ ï

. ï

î å

ë

( , ö , Õ ) = exp ã Ø

+ í

ñ

ó

æ

ë

) ¼ ( , ö ),

ß ÷ ë

where æ is an arbitrary constant and the function ¼ ( , ö ) satisfies the stationary equation ø ø ë ò

®

ë

( ,ö )

ø

ø ø

¼ ë

í

+

Ù ø ö

ë

( ,ö )

ø ø

¼ ö

í

+ù ¼

ln ¼ = 0.

ò

© 2002 by Chapman & Hall/CRC Page 720

721

B.5. SECOND-ORDER NONLINEAR EQUATIONS


42.



= 








( , ) 



+ 






( , ) 





+ ( ) 




ln




.

Incomplete separable exact solution (the solution is separable in the space coordinates and but not in time  ):  ( , ,  ) =  ( ,  ) ( ,  ). The functions  ( ,  ) and  ( ,  ) are determined from the one-dimensional nonlinear parabolic differential equations      





= 



  



( ,  ) 

+  ( )  ln  +  ( )  , 





 



=



( ,  ) 

+  ( ) ln  −  ( ) , 

where  ( ) is an arbitrary function.

B.5.2. Hyperbolic Equations B.5.2-1. Equations of the form 1.

2






2

=

2






2




+ !

ln






2



=

2











2



+

2

 





, , ,



  



.




+ [ ( ) + ( )] .

Exact solution in multiplicative form: 

( ,  ) =  ( ) ( ),

where the functions  ( ) and  ( ) are determined by the second-order ordinary differential equations  #" " − %'& ln  $   "#" + %'& ln   

+ ( ) + 

(

= 0,

+ ( ) − 

()

= 0,

where  is an arbitrary constant. 2.

2


2

=

2





2

+ !

*

2 

+, +

Exact solution in additive form: 




 

+ ( ).

( ,  ) =  ( ) + - (. ), .

= +/ ,

where / is an arbitrary constant and the functions  =  ( ) and - = - (. ) are determined by solving the second-order ordinary differential equations 

2

(  − / )-

23 "#" 2

 #" " $

− 01 − ( ) = 0,

(1)

+ 0)- = 0.

(2)

+&

- 2" 

2

The general solution of equation (1) is given by 

( ) =

 1

cosh( 4  ) + 



( ) =

 1

cos( 4  ) + 

2

2

sinh( 4  ) +

sin( 4  ) +

1 4

1 4



 5 0

5 0 



(6 ) sinh[ 4 ( − 6 )] 7 (6 ) sin[ 4 ( − 6 )] 7

where  1 and  2 are arbitrary constants. Equation (2) can be solved with the change of variable 8 (- ) = linear equation.

6

6  - 2"

2 

2

if 0

=

if 0

= −4

4

> 0, 2

< 0,

, which leads to a first-order

© 2002 by Chapman & Hall/CRC Page 721

722 3.

B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES 2


2

=

2





2

+ !

2

* 




+,

+

Exact solution in additive form: Here,

( ) = 

( ) = 



+ ( ). 

( ,  ) =  ( ) +  ( ).

cosh( 4  ) + 

 1

cos( 4  ) + 

 1

2

sinh( 4  ) if

sin( 4  )

2

= 0

if

4

2

= −4 0

> 0, 2

< 0,

where  1 and  2 are arbitrary constants, and the function  ( ) is determined by the ordinary   differential equation 2   "#" + &  " + 01 + ( ) = 0.  

4.

2


2

=

2





2

+ !

* 

2

+, +

Exact solution: 

 





+ 

9




2



+ ( )






+ ( ).

( ,  ) =  ( ) +  ( ) exp( /

),

where / is a root of the quadratic equation &1/ 2 + 0/ + 4 = 0 and the functions  ( ) and  ( ) are determined by the following system of second-order ordinary differential equations: 

 #" " $  "#" $ 



2

= 4  + ( )  +  ( ), = % ( 0/ + 2 4 )  + ( ) +  /



2

(1) (2)

.

()

In the special case ( ) = const and ( ) = const, equation (1) is autonomous and has particular solutions of the form  = const and, hence, can be integrated in quadrature. Equation (2) is linear in  , and consequently, with  = const, its general solution is expressed in terms of exponentials or sine and cosine. 5.

2


2

=

2




2



+ ( )*






2 

Exact solution:

+ 



+ ( )






+  ( ).

( ,  ) =  ( )

2

+  ( ) + : ( ),

(1)

 where the functions  ( ),  ( ), and : ( ) are determined by solving the following  system of secondorder ordinary differential equations with variable coefficients (the arguments of , , and  are not specified): 

 #" " $  "#" $  : "#" $



2

= 4   +  , = (4   + ) ,  = : +  2 +  + 2 

(2) (3) (4)

.

Equation (2) has the trivial particular solution  ( ) ≡ 0; the corresponding solution (1) is linear in the coordinate . Equation (3) has a particular solution  =  ¯ ( ), where  ¯ ( ) is any nontrivial particular solution of equation (2). Hence, the general solution of equation (3) is given by 

( ) =

 1

¯ ( ) + 

2  ¯ (

)

7  5

where  1 and  2 are arbitrary constants. If the functions (  = const) is a particular solution of equation (2). Equation (4) linear in : = : ( ).



 ¯ 2 (

)

and

, 

 ; 

proportional, then 

= − 41

© 2002 by Chapman & Hall/CRC Page 722

723

B.5. SECOND-ORDER NONLINEAR EQUATIONS

6.

2


2

2


=








2



2



+ ( )* 





+ ( ) +  ( ).

+

Exact solution in additive form: 

1 2
0, 2

< 0,

are arbitrary constants.

2


=

2





2




+ ( )*

1 ? . Exact solution: 



2 

+

+



( ) !











+  ( ).

@

( ,  ) =  ( ) +  ( ) exp

A

−& ,

< 0, &



where the functions  ( ) and  ( ) are determined by solving second-order ordinary differential equations with variable coefficients (the arguments of , , and  are not specified): 





2

= &   +  +  , = (2 &  + −  & ) .

 #" " $ "#" $

2 ? . Exact solution of a more general form: 

 the  following



( ,  ) =  ( ) +  ( ) % < exp

 A

−& 

+ = exp − A

−& 

(

, &

< 0,

where < and = are arbitrary constants and the functions  ( ) and  ( ) are determined by the following   system of second-order ordinary differential equations with variable coefficients (the arguments of , , and  are not specified): 

 #" " $  "#" $

3 ? . Exact solution: 

= & (  = (2 &





2

2

+ 4 < =  ) + + −  & ) .

+ , 



( ,  ) =  ( ) +  ( ) cos

A

&

+ 

, &

> 0,

(9)

© 2002 by Chapman & Hall/CRC Page 723

724

B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES

where  is an arbitrary constant and the functions  ( ) and  ( ) are determined by the following system of second-order ordinary differential equations with variable coefficients: 

=

 #" " $  "#" $

9.

2


=

2

2






+

2

*

,

&





2



= 2&






2

+  ) + + −



&

+ ,



 

.



+ ( ). +

Exact solution in additive form: 

( ,  ) =

1 2< 

2

+=

+ + 

 

( − 6 ) (6 ) 7

5 0

Here, < , = , and  are arbitrary constants and the function second-order nonlinear ordinary differential equation 

10.

2


2

2


=





2

+




*



, 

+ +

Exact solution in additive form: 



!

,




+  ( ).

( ) is determined by solving the



+

  "#"  



6

− < = 0.

"

 



+ ( ).

( ,  ) =  ( ) +  ( ).

Here, the functions  ( ) and  ( ) are determined by the second-order ordinary differential equations 

+

"#"  

 

− &) − ( ) = 0,

  #" " $

,

"  

+ &> = 0.

The general solution of the first equation is given by

where  11.



( ) =

 1

cosh( 4  ) + 



( ) =

 1

cos( 4  ) + 

1

and 

2

2


2

=

2

2

sinh( 4  ) +

1 4

1

sin( 4  ) +



(6 ) sinh[ 4 ( − 6 )] 7

5 0 

(6 ) sin[ 4 ( − 6 )] 7

5 0 4

 

6

6

2

if &

=

if &

= −4

4

> 0, 2

< 0,

are arbitrary constants. 2


2

+




 *



,


1




+

.

Exact solution in multiplicative form: 

( ,  ) =

B C  

( ),

where / is an arbitrary constant and the function  ( ) is determined by the second-order linear ordinary differential equation   #" " $

=

%D /

2

+ ( , / )(1 .

© 2002 by Chapman & Hall/CRC Page 724

725

B.5. SECOND-ORDER NONLINEAR EQUATIONS 2

B.5.2-2. Equations of the form 12.

2




2





2

= ( + E )F










= ( ) 

2



 

2

 




+ ( ),

2



+

2

, , ,



   

.

> 0.

Exact solution for G ≠ 2: 



=

( 8 ),

= 8

I ( 2(2− I )

(2 − G )2 ( +  )2 − ( + H )2− I

1 4 %





,

where  is an arbitrary constant and the function = ( 8 ) is determined by the generalized Emden–Fowler equation    4 4(1− I ) I J> "#" J − 8 ( ) = 0.  (1) 2   G



A number of exact solutions to equation (1) for some specific = ( ) can be found in Polyanin and Zaitsev (1995). In the special case G = 1, the general solution of equation (1) is given by  5

where  13.

1

and 

2 

2


=

2

( ) K

=

7

F 

+

2

Exact solution for G ≠ 2:




!





–1 F








=

 

+
0.



(2 − G )2 ( +  )2 −

2"

( ) = 0,

For < ≠ 1, the change of variable Emden–Fowler equation 

where =

.

J> "#" J


J " = 0. "#" J + %D (1 − G ) + ( )(  (2 − G ) 8   The change of variable N ( ) = 8 J" leads to a first-order separable equation. Integrating this equation yields the general solution in implicit form: 

 5

where 

1

and 

 G

2

7 

−2 ( )+ K

1

= 

1 ln | 8 | +  (2 − G )

 

2, K

( )= 5



( )7



,

are arbitrary constants.

© 2002 by Chapman & Hall/CRC Page 725

726

B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES 2


15.



2

2


=

F 

+

2

  F





–1



( )




+ ( ). 

Exact solution for G ≠  2: @ 1L 2 = ( 8 ), 8 = % 4  (2 − G )2 ( +  ) 2 − 4 4 2− I ( , 4 = 1, where  is an arbitrary constant and the function = ( 8 ) is determined by the ordinary differential equation       2 1 1 J> J" − "#" J + %  (1 − G ) + ( )( ( ) = 0.  (2 − G ) 8  4 (2 − G )2 2


16.



2

2


=

O P Q






2

+

! O P Q




+ ( ), 

> 0.

@ Exact solution for /  ≠ 0:  1L 2 = ( 8 ), 8 = % 4 4 B − C  −   4 / 2 ( +  )2 ( , 4 = 1, where  is an arbitrary constant and the function = ( 8 ) is determined by the ordinary differential equation     2(  / − & ) 1 1 J> J" + "#" J + ( ) = 0. (1) 2

For & =

 /

 /



 1 5

where 

1

and 

For & ≠ equation

8

 4 /

, the solution of equation (1) is given by

1 2 /

2

−

( )

 4 / 2K

@

−1 L 2  

2

=

7

are arbitrary constants.

, the change of variable 

+

23 "#" 2

= .

8

+ 8

2R −S S C C

 .

 

2, K

( )= 5



( )7



,

brings (1) to the generalized Emden–Fowler

4( S − R ) C 2R −S C

 

( ) = 0,

(2 & −  / )2 whose solvable cases are presented in Polyanin and Zaitsev (1995). 17.

2


2

=

2
O P Q 






2

+

4




( )

O P Q

. 

Exact solution for /  ≠ 0:  @ − 2 2 1L 2 = ( 8 ), 8 = % 4 4 B C  −   4 /  ( + < ) ( , 4 = 1. Here, < is an arbitrary constant and the function = ( 8 ) is determined by the ordinary differential equation      2 1 J> "#" J + 1− ( ) J" = 0,  8



 /

which by the change of variable N ( ) = 8 J" is reduced to a separable first-order equation. Integrating this equation yields the general solution in implicit form:      7 1   = ln | 8 | +  2 , ( )= ( )7 , 5 5 2 ( )− / +  1  / K K where  1 and  2 are arbitrary constants. 18.

2


2

=

2
O P Q 



2

+




( )

O P Q







+ ( ). 

Exact solution for /  ≠ 0:  @ − 2 2 1L 2 = ( 8 ), 8 = % 4 4 B C  −   4 /  ( +  ) ( , 4 = 1, where  is an arbitrary constant and the function = ( 8 ) is determined by the ordinary differential equation        2 1 1 J> "#" J + 1− ( ) J " + ( ) = 0. 2 8

 /

 4 /

© 2002 by Chapman & Hall/CRC Page 726

727

B.5. SECOND-ORDER NONLINEAR EQUATIONS

B.5.2-3. Other equations. 2


19.



 

= ( )

2





* 

1 ? . Exact solutions:  ( ,  ) = (  

where 

( ,  ) = ( 

1,  2,  3,  4,

.



+

1

+

1

+

2 )(  3

+

+

2 )

 5 S

4)

1L 2

, 

( − 6 )( 

+

16

2)

2

(6 ) 7

+ 6

3

+

4,

and  are arbitrary constants.

2 ? . Exact solution: 

where the functions  =  ( ), differential equations

( ,  ) =  ( ) 2 +  ( ) + : ( ), =  ( ), and : = : ( ) are determined by the system of ordinary





= 6  ( )  2 , = 6  ( )   ,

 #" " $  "#" $ : "#" $



= 2 ( )  : + ( ) 2 . 3 ? . Exact solution in multiplicative form:  ( ,  ) = T ( ) U ( ), where the functions T = T ( ) and U = U ( ) are determined by the ordinary differential equations  (  is an arbitrary constant) T "#" =  ( ) T 2 , $ ( U U " )" =  U . 



1 6

The latter equation is autonomous and has a particular solution U = in quadrature. 2


20.



 

= ( )

2

* 







Exact solution: 

where the functions  =  ( ), differential equations



+ +

2( )



 2



+





= 2 ( ) 

+  2 ( ), +  1 ( ), + ( )

:

B.5.3. Elliptic Equations

1.



2

+

2


2

=




ln






and, hence, is integrable

).



2

= 6  ( )  = 6  ( ) 

 #" " $  "#" $ : "#" $

2


0(

2

( ,  ) =  ( ) 2 +  ( ) + : ( ), =  ( ), and : = : ( ) are determined by the system of ordinary



B.5.3-1. Equations of the form



1( ) +







2

  

+&

2







2

 V 

=

2





2

+





0 (

, , , 

  

).

, 

  V



,

 &

> 0.




+ [ ( ) + ( )] .

Exact solution in multiplicative form: 

( , ) =  ( ) ( ), where the functions  ( ) and  ( ) are determined by the ordinary differential equations  "#" − %D ln  +  ( ) +  ( = 0, 

where  is an arbitrary constant.

  "#" V1V

−

%D

ln  + ( ) − 

(1

= 0,

© 2002 by Chapman & Hall/CRC Page 727

728 2.

B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES 2


2

+

2


 

= ( )

2




ln






+[








( ) + ( )] .

Exact solution in multiplicative form: 

( , ) = B − S V  ( ),  the ordinary differential equation where the function  ( ) is determined by 2  "#" = ( )  ln  + % ( ) −  ( . 3.

2


2

+

2


2

 




 

= ( )*



+ ( )



+

Exact solution:



2






+  ( ).



( , ) =  ( ) 2 +  ( ) + : ( ). (1)  Here, the functions  ( ),  ( ), and : ( ) are determined by the  following second-order ordinary  differential equations with variable coefficients (the arguments of , , and  are not specified): 2  "#" = 4   +  , (2)    "#" = (4  + )  , (3)     2 : "#" = : +  +  − 2  . (4)   If a solution  =  ( ) of the nonlinear equation (2) is found, then the functions  =  ( ) and : = : ( ) can be determined successively from equations (3) and (4), which are linear in  and : . By comparing equations (2) and (3), one can see that equation (3) has a particular solution  =  ( ). Hence, the general solution of (3) is given by (see Polyanin and Zaitsev 1995) 

( ) =

( ) + 

 1

7

( )

2

,  W 0. )  ;  ( )  ≡ 0, to which there is a corresponding 1 and are proportional, then  = − 4

 2 (   Note that equation (2) has the trivial particular solution 5

solution (1) linear in the coordinate . If the functions (  = const) is a particular solution of equation (2). 4.

2


2

+

2




2






= ( )*

1 ? . Exact solution:



2

+



+

!



( )




2



+ ( )






+  ( ).

@ 

( , ) =  ( ) +  ( ) exp A − & , & < 0,   where the functions  ( ) and  ( ) are determined by the following second-order ordinary differential   equations with variable coefficients (the arguments of , , and  are not specified): 2  "#" = &   +  +  ,    "#" = (2 &  + +  & ) .     2 ? . Exact solution of a more general form:  ( , ) =  ( ) +  ( ) % < exp A − & + = exp − A − & ( , & < 0,   where the functions  ( ) and  ( ) are determined by the following system of second-order ordinary    differential equations with variable coefficients: 2 2  "#" = &   + 4 < =  ) +  +  , 

3 ? . Exact solution:



  "#"  

= 2&

 

+

+ 

&>

.



( , ) =  ( ) +  ( ) cos A & +  , & > 0,  where  is an arbitrary constant and the functions  ( ) and  ( ) are determined by the following   system of second-order ordinary differential equations with variable coefficients: 2 2  "#" = &   +  )+  + ,

X'Y



  "#"  

= 2&

 

+ 

+

&>

.

Reference: V. F. Zaitsev, A. D. Polyanin (1996).

© 2002 by Chapman & Hall/CRC Page 728

729

B.5. SECOND-ORDER NONLINEAR EQUATIONS 2


5.



2


+

2

! 



=

2




1*

,

 

+ +

Exact solution in additive form:




2*

, 

+ +


9

.



( , ) =  ( ) +  ( ). Here, the functions  ( ) and  ( ) are determined by solving the second-order ordinary differential   equations   "#" −  1  ,  " −4  = ,   #" " V1V

&>

where  is an arbitrary constant.

−



B.5.3-2. Equations of the form 6.




 *



F 

+ +

1 ? . For G ≠ 2 and [  




=



(. ),

 V

= .

and 

2



2G 2  1 + ( )  & (2 − G )4 K are arbitrary constants. 1− M

 V




(2 − [ )2

%'&

2− I

+  (2 − G )2

2− \ (

1L 2

.

+






b

−d

V (

1L 2

,

(. ) is determined by the ordinary differential equation 

9.

= .

ln

 &>b




4 ( ). 2 (2 − G )2

+ [

1(



)+

2(






)] .

( , ) =  ( ) ( ),

where  =  ( ) and  =  ( ) are determined by the ordinary differential equations 

[  ( )  [ ( )

] " = [  ln  +  1 ( ) +  ]  ,  ] " = [  ln  +  2 ( ) −  ] ,

"  " V

V

where  is an arbitrary constant. B.5.3-3. Other equations with two independent variables. 10.

2


2

+ 

f

gih

( j )k

+ l ( j )m

k n

= 0.

n o

1 q . Exact solution:r (s , t ) = ( u

p

+ v )t − w s

x

(s − y )( u y

+ v )2 z ( y ) { y

+|

1s

+|

2,

0

where u , v , |

1, | 2,

and s

2 q . Exact solution:

0

x

are arbitrary constants. r

where the functions } = } (s ), equations

(s , t ) = } (s )t

2

+ ~ (s )t +  (s ),

= ~ (s ), and  =  (s ) are determined by the ordinary differential ~

} €#€ ~ x €#€ x  €#x € x

+ 6z + 6z + 2z

2 }

} ~ } 

= 0, = 0, + 2}  + z ~

2

(1) (2) (3)

= 0.

x x

The nonlinear equation (1) can be treated independently. For z ≡ const, its solution can be 2 expressed in terms of elliptic integrals. For z = ‚ ƒ „ , a particular solution of (1) is } = − 6„ … ƒ − „ . x x Equations (2) and (3) can be solved successively (these are linear in the unknowns). Because ~ = } ( s ) is a particular solution of equation (2), the general solution of (2) is given by (see Polyanin and Zaitsev 1995) ~

where |

1

and |

2

(s ) =

| 1}

(s ) + |

2}

(s ) w

{ s } 2 (s

)

,

are arbitrary constants.

© 2002 by Chapman & Hall/CRC Page 730

731

B.5. SECOND-ORDER NONLINEAR EQUATIONS 2

11.

† j ‡

n

k

j 2 n

2

+

For Š ≠ 2 and ‹

ˆ

k

n

o ‰ r

+

2

n ro

–1

9 j ‡

k n

+s j

n

–1

k n

o ‰

= h (k ).

n o

≠ 2, there is an exact solution of the form r

r

1 2

2 2− Ž = (Œ ), Œ = g' (2 − ‹ ) s + ‚ (2 − Š )2 t 2−  m . r r r = (Œ ) is determined by the ordinary differential equation

Here, the function

v

+

‘3 €#€ ‘ u

= z ( ),

‘€ Œ

where

”'•

= 14 (2 − Š )(2 − ‹ ) ’D‚  (3Š

v

− 4Š − 4‹ ‹

+ 4) + 2 1“ (2 − ‹ ) + 2 ‚ e (2 − Š ) m .

Reference: V. F. Zaitsev and A. D. Polyanin (1996). 2—

12.

† – ‡

n

– 2 n

2—

+

For Š ≠ 2 and ‹

ˆ

n

o ‰ r

n ro

† ™ š ›

œ

– 2 œ

–1 ˜

9 – ‡

+s –

n

+

(Œ ), 2—

ˆ ™  ž

œ

2

œ rŸ

= Œ

+

(2 − ‹ )2 s

’

¡

r

r

— 9 ™ š ›

œ

r

+ s™ –

œ

where

‘3 €#€ ‘

2— – 2 œ

+

2 2 1 4‚    ¡ ,

= u

œ

n o

+ ‚ (2 − Š )2 t

2— œ

2— ˆ ™  ž

œ

2

œ rŸ

– 2 œ

2—

+

m

1 2

.

œ Ÿ

+

˜

¦ ™ š ›

v

+

—

(— ) œ

r

= z ( ),

‘€ Œ

1 4  ¡

= v

– œ

= (Œ ), † – ‡

2− 

= ˜ (— ).

œ

(3 ‚

   ¡

+ s™

 ž

− 2 1“ ˜

ˆ ™ š ž

œ

2

+

rœ Ÿ

¦ – ‡

Œ

–1

— œ

r

œ

=

¢  ¡

+ s™ –

− 2‚ ¡ —

(— ) œ

e) 

).

= § (— ).

œ Ÿ

≠ 0, there is an exact solution of the form ¡

= l (— ).

1 2

u

† ™ š ›

o ‰

—  ž

—

(— ) n

2 −£ = (Œ ), Œ = ¢  ¡ ƒ + ‚   2 ƒ −d ¤ ¥ . x r r r (Œ ) is determined by the ordinary differential equation

=

Here, the function

14.

–1 ˜

2− Ž

≠ 0, there is an exact solution of the form

For  

For  

—

(— ) n

≠ 2, there is an exact solution of the form

2—

13.

+

2

=

15.

(2 − Š )2 (2 − ‹ )2 ,

1 4‚ 

= u

2 −£ ƒ — š ž

+‚ x

 

2 −d ¤ ¥ 1  2 ƒ .

= ˜ (— ).

œ œ Ÿ

For   ≠ 0 and Š ≠ 2, there is an exact solution of the form r

Here, the function

r

1 2

2 2− Ž = (Œ ), Œ = ’    s + ‚ (2 − Š )2 ƒ −£ ¤¨ . r r r (Œ ) is determined by the ordinary differential equation

=

‘3 €#€ ‘ u

where ”'•

16.

u

=

2 1 4 ‚    (2

− Š )2 , v

1 4 

=

+

v Œ

‘€

(2 − Š ) ’D‚

= z ( ),   

(4 − 3Š ) + 2 1“  

− 2 ‚ e (2 − Š )¨ .

Reference: V. F. Zaitsev and A. D. Polyanin (1996). 2— † – ‡

œ œ

– 2

+

2— ˆ ™ š ž

œ rœ Ÿ

2

+

¦ – ‡

–1 ˜

—

(— ) œ

r

œ

–

+ s™

š ž

˜

For   ≠ 0 and Š ≠ 2, there is an exact solution of the form =

(Œ ), Œ

=

’   

2 2− Ž s

—

(— ) œ

= § (— ).

œ Ÿ

+ ‚ (2 − Š )2 ƒ

−£ ¤¨ 1  2

.

© 2002 by Chapman & Hall/CRC Page 731

732

B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES

B.5.3-4. Equations with three independent variables. —

17. œ

–

† – ‡

œ

©

For Š ≠ 2, ‹

œ –

ª

œ

—

+ œ

ˆ

Ÿ ‰

©

œ Ÿ

—

+

œ ª

œ Ÿ

œ

œ

© ¬ « ­

œ «

= ˜ (— ). ª

œ «

≠ 2, and ® ≠ r r 2, there is an exact solution of the form = r

(Œ ), Œ

2

2− Ž s

=4¯

+

(2 − Š )2 ‚

2−  t

(2 − ‹ )2 

2− ±

+

°

²

(2 − ® )2 ³

,

by the ordinary differential equation where the functionr (Π) is determined r r

”'•

u

+

‘3 €#€ ‘

= z ( ),

‘€ Œ

=2 u

1 1 + 2−Š 2−‹ ©

1 2−®

+

− 1. ª

Reference: A. D. Polyanin and A. I. Zhurov (1998). —

18. œ

–

† ™ ´ ›

œ

©

œ – œ

—

+ ª

œ

ˆ ™  ž ©

œ Ÿ

œ ª

œ Ÿ

—

+ œ

™ µ ¶

© ¬

œ «

œ

= ˜ (— ). ª

œ «

there is an exact solution of the form For · ≠ 0, ¡ ≠ 0, and ¸ ≠ 0, r r = (Œ ), r

2 Œ

ƒ

=4 ©

− „

−d ¤ ƒ

+

x ‚ · 2

 ¡ 2

− ¹1º ƒ

+

²

,

¸ 2 ª

differential equation where the function (Œ ) is determined by r the ordinary r r −

‘3 €#€ ‘ —

19. œ

–

† – ‡

œ

©

For Š ≠ 2, ‹

œ –

ª

œ

—

+ œ

ˆ ©

œ Ÿ

Ÿ ‰

œ

= z ( ).

‘€

Œ

—

+ ª

œ Ÿ

1

œ

™ µ ¶

© ¬

œ «

œ

= ˜ (— ). ª

œ «

≠ 2, andr ¸ ≠ r 0, there is an exact solution of the form = (Œ ), r

2 Œ

2− Ž s

=4¯

+

(2 − Š )2 ‚

2−  t

(2 − ‹ )2

+



− ¹1º ƒ ²

¸ 2 ³

,

where the function (r Π) is determined by the ordinary differential equation r r

—

20. œ œ

–

©

† – ‡

œ œ

–

ª

‘3 €#€ ‘

+

+ œ

u

‘€ Œ

= z ( ), —

œ Ÿ

©

ˆ ™  ž

œ ª

œ Ÿ

=2 u

©

1 1 + 2−Š 2−‹ —

+ œ

™ µ ¶

© ¬

œ «

œ ª

œ «

ª

− 1.

= ˜ (— ).

is an exact solution of the form For Š ≠ 2, ¡ ≠ 0, and ¸ ≠r 0, there r r

= (Œ ), Œ

2

=4¯ ‚

s

2− Ž

(2 − Š )2

+

ƒ

−d ¤ 2  ¡

+

− ¹1º ƒ ²

¸ 2 ³

,

where the function (Œ ) is determined differential equation r by the ordinary r r ‘3 €#€ ‘ ”'•

+

Š

2−Š

1 Œ

‘€

= z ( ).

Reference: A. D. Polyanin and A. I. Zhurov (1998).

© 2002 by Chapman & Hall/CRC Page 732

733

B.5. SECOND-ORDER NONLINEAR EQUATIONS

B.5.4. Equations Containing Mixed Derivatives B.5.4-1. Monge–Amp`ere equations. 2

2—

1.

œ ©

–

œ

2—

= ª

œ Ÿ

œ

2— œ

– 2 œ

œ Ÿ

1 q . Exact solutions: r (s , t ) = | r

1t

(s , t ) = where |

2

+|

1 +| s

2 | 2

+

2s t

2

| 2t

1 ©

+ ˜ ( – ).

2

1, | 2, | 3, | 4, | 5,

4|

+|

2 s

1

and |

2|

1

2 | 3

+

3t

1

− 4|

w

0

1

−

2ª

(s − y ) z (y ) { x

2|

w

2

+|

3t

− y )(y + |

x (s

0

+| y

1)

z

+|

4s

(y ) { y

5,

+|

4t

+|

5s

+|

3t

+|

4,

+|

6,

are arbitrary constants.

6

2 q . Exact solutions for z (s ) > r 0: (s , t ) = » t

w

z ¼

(s ) {

+ } (s ) + | s

1t

,

y

+|

where } (s ) is an arbitrary function. 2. 1q .

2

2— ©

œ œ

–

2—

= ª

œ

– 2

œ Ÿ œ Exact solution: r

(s , t ) =

where |

1 , ¾¾¾

,|

5,

Ÿ

œ Ÿ

2

−t

1 2|

+ ˜ (– ) .

2

| 1t

(s ) = ½

2— œ

w z

w

1

(s ) { ½

(s ) {

+ s

+| s

1 2|

w

1

2

(s − y ) ½ x

…

(y ) {

2s

5,

and ‚ are arbitrary constants. r

2 q . Exact solution:

(s , t ) = } (s )t

2

+ ~ (s )t +  (s ),

4

+

where (s ) = }



| 1s

1 +|

1 (s ) = 2

x w …

2

,

(s ) = ~

(s ) + |

[~ ¿ € (y )]2 (s − y ) } (y )

| 3}

+|

{ y

5s

+|

(s ) }

2|

1

z w

1 (s ) { s − 3 [ } (s )] 2| 1

3t

z w

(s ) { s , [ } (s )]2

6,

3 q . Exact solutions cubic in t : r r

(s , t ) = (s , t ) =

where |

1 , ¾¾¾

,|

5,

3

| 1t

t

−

(|

1s

1 6|

1

3

+|

2

)2

x w …

−

(s − y ) z (y ) { 1 6

x w …

y

+|

2s

+|

(s − y )( |

1y

+|

2)

2z

+|

(y ) { y

4,

+|

3s

+|

4t

+|

5,

and ‚ are arbitrary constants.

4 q . See the solution of equation B.5.4.5 in Item 2 q with “

= 1.

© 2002 by Chapman & Hall/CRC Page 733

734

B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES 2

2—

3.

œ ©

–

œ

ª

œ Ÿ

2—

= œ

2— œ

– 2 œ

2

2

+ ˜ (– )

.

œ Ÿ

Ÿ

r solution quadratic in t : 1 q . Exact

2

(s , t ) = } (s )t

+

¯À| 1 w

2 }

(s ) {

+| s

1 | 2

+

2³ t

2 1 w

3

(s − y ) } x

…

(y ) {

+| y

+|

3s

4.

The function } = } (s ) is determined by the ordinary differential equation

r

(s , t ) =

1 , ¾¾¾

,|

5,

(|

1 12 |

−

+|

1s

…

2)

−

3

1 12

1z 2

w

x …

y

+|

2s

+|

(s − y )( |

1y

+|

2)

2

2— œ ©

–

œ

2—

= ª

œ Ÿ

œ

2— œ

– 2 œ

Exact solution:

2

2

5.

œ ©

–

œ

œ Ÿ

2—

= ª

œ

œ

œ

2

r

(s , t ) = where |

1 , ¾¾¾

+2

| 1t Ä

t Ä

1s

,|

5,

+2

+|

~

y

+|

+|

3s

+|

4t

5,

2

=

=  (s ) are determined by the system of ordinary

(s ), and ~

−

2)Ä

+1

w

x …



+ ~ (s )t +  (s ),

)2 −

1z 2 2 } € x ~ € − 12  1 x 1 x2 2 (~ € ) − 2  x

} } €#€

= 2( }

} ~ x €#€ x

=

}  x€#€ x

=

+ (– )

| 1

= 2. “

(s , t ) = } (s )t

˜

1

−

( “ + 1)( “ + 2) (|

(y ) {

4,

Ÿ

€

(s ), (s ), (s ).

.

Ÿ Ã

œ Ÿ

1 q . r Exact solutions: (s , t ) =

3z

+|

+ § ( – ) + Á ( – ).

Ÿ

r

x x

2—

– 2

2

+ ˜ (– )

œ Ÿ

where the functions } = } (s ), differential equations

2—

3t

and ‚ are arbitrary constants.

3 q . See the solution of equation B.5.4.5 in Item 2 q with 4.

(s ).

x

(s − y ) z (y ) { x

w

1

4 t

(s , t ) = where |

4

| 1t

€

x x

r 2 q . Exact solutions quartic in t :

)2 −

= 2( }

} } €#€

(s − y ) z (y ) {

1 ( “ + 1)( “ + 2)

+|

2s

(s − y )( | x

w

+| y

…

1y

3t

+|

+|

2)Ä

4, +1 z

(y ) { y

+|

3s

+|

4t

+|

5,

and ‚ are arbitrary constants. r

2 q . Exact solution:

Ä

(s , t ) = } (s )t

+2 2

.

The function } = } (s ) is determined by the ordinary differential equation ( “ + 2) } “

6.

2

2— ©

Exact

œ

ª œ Ÿ r solution: œ

–

=

2— œ œ

– 2

2— œ

2

(s , t ) = } (s )t Ä

+ (– )

+2

x x 2 +2

˜

œ Ÿ

−

− ( “ + 2)2 ( }

} €#€

Ÿ

Ã

)2 + 4 z (s ) = 0. € x

+ § (– )

1 ( “ + 1)( “ + 2)

Ÿ Ã

x w …

.

(s − y ) }



(y ) (y )

{ y

+|

1s

+|

2t

+|

3,

© 2002 by Chapman & Hall/CRC Page 734

735

B.5. SECOND-ORDER NONLINEAR EQUATIONS

where } = } (s ) is determined by the ordinary differential equation ( “ + 1)( “ + 2) } } €#€ − ( “ + 2)2 ( } € )2 + z (s ) = 0. 2

2—

7.

œ ©

–

œ

2—

= ª

œ Ÿ

œ

2— œ

– 2

2

r

œ

r

(s , t ) =

| 1 w

(s , t ) =

| 1ƒ

1, | 2, | 3, | 4, ‚

(s − y ) z (y ) { x

…

+ ¤ „

£ x

−

2

2— œ ©

–

œ

2—

= ª

œ Ÿ

œ

2—

x w

| 1· 2

x x

œ

– 2

r œ

Exact solution:

…

| 1· 2

−£

(s − y ) ƒ

2

¿ z

ƒ „

¤

(y ) {

+| y

+|

3t 2s

+| +|

4, 3t

+|

4,

2

2— –

œ

œ

2—

– 2 œ

Exact solution:

(s , t ) = where |

.

´ ž

œ Ÿ

2—

= ª

œ Ÿ r

+ § (– )™

´ ž

1

(y ) { y + | 1s + | 2 t + | 3, (y ) … = } (s ) is determined by the ordinary differential equation 2 −2 z } } €#€ − ( } € ) + · (s ) = 0.

where the function }

œ

x

2

+ ˜ (– )™

(s , t ) = } (s ) ƒ

©

1

1

−

2s

(s , t ) = } (s ) exp ¢ 12 · t ¥ , = } (s ) is determined by the ordinary differential equation 2 −2 z } } €#€ − ( } € ) + 4 · (s ) = 0.

where }

9.

+| y

, and   are arbitrary constants. r

2 q . Exact solution:

8.

x

.

´ ž

œ Ÿ

1 q . Exact solutions:

where |

x x

+ ˜ (– )™

−

x

w

· 2

x x

œ

2

…



}

Ÿ

(s − y ) z (y ) { x

(s − y )

x

+ ˜ ( – )§ ( ).

œ Ÿ

| 1 w

1, | 2, | 3, | 4, ‚

¤

„

− y

1 | 1

¤ w Å

( t − Œ ) ( Œ ) { Œ

+|

2s

+|

3t

+|

4,

, and  are arbitrary constants.

B.5.4-2. Other equations with quadratic nonlinearities. 10.

— œ

œ œ Ÿ

œ

–

r

2—

–

œ Ÿ

2— —

œ œ

–

œ

= ˜ ( – ).

2

r

r

œ Ÿ

1 q . Suppose (s , t ) is a solution of the equation. Then the functions ¥ ¢ s , » t + } (s ) + | , 1 =» where } (s ) is an arbitrary function and | is an arbitrary constant, are also solutions of the equation. 2 q . Exact solutions:r r

(s , t ) = » t

(s , t ) =

| 1t

z

2w ¯

2

(s ) { s

+|

1 2 1³

+ } (s ),

1 2 ¯} (s ) − 2 w z (s ) { s ³ + | 4| 1 and | 2 are arbitrary constants. r

+ } (s )t +

where } (s ) is an arbitrary function and |

1

3 q . Exact solutions in implicit form:

2,

{

= » É + } (Ç ), 2 ½ (Ç ) + ~ (È ) where } (Ç ) and ~ (È ) are arbitrary functions and ½ (Ç ) = Ê Ë (Ç ) Ì Ç . w

Æ

© 2002 by Chapman & Hall/CRC Page 735

736 11.

B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES 2— — œ œ œ Ÿ

–

œ

Ÿ

œ Ÿ

2— —

+˜ ( )œ

– œ

Exact solution:

œ

= § ( )—

2

+ Á ( ) – + s( ).

Ÿ

œ Ÿ

Ÿ

Ÿ

= Í (É )Ç + Î (É ), È

where the functions Í (É ) and Î (É ) are determined by the system of ordinary differential equations Ë Í Í ¤1 Ï#Ï ¤

+ (Í

Ë Í Î 1 ¤ Ï#Ï ¤

+Í

)2 = РϤ

=Ð

Ϥ Î ¤ Ï

+Â , Í

+Ñ . Î

B.5.5. General Form Equations B.5.5-1. Equations of the form 1.

Ö ×

= Ù

2

,

Ú Û

Ö

Ö Ø

× Û 2 Ü Ö

Exact solution:

=

Ò Ó¿ Ò

½

,Ô ,È ,

¢ Ç

,

Ò Ó Ò Õ

Ò

2

Ò Õ

Ó2

¥

.

. Ý

(Þ , Ô ) = ( ß

+ à )Ô + Í (Þ ), Þ

where ß and à are arbitrary constants and the function Í (Þ ) is determined by the ordinary differential equation á â Þ , Í Ï#Ï =ß Þ +à . Õ Õ ã

2.

Ö ×

= Ù

Ú

Ö ×

Ö Ø

Ö

Exact solution:

2

, Û

Ö

.

× Û 2 Ü Ö

Ý

(Þ , Ô ) = ß

Ô

+ à + Í (ä Þ

+ å Ô ),

where ß , à , ä , and å are arbitrary constants and the function Í ( æ ) is determined by the ordinary differential equation á

â

,

ä Í Ïç

3.

Ö ×

=

Ù

Ö Ø

×

Ø

2

1

,

Ú

Ö Ö

Í ç> Ï#Ï ç ã

−å

− ß = 0,

Í Ïç

= æ

ä Þ

+å Ô.

.

× Û 2 Ü

×

2 ä

Exact solution in multiplicative form: Ý

(Þ , Ô ) = ( ß Ý

+à

è é Õ

(Þ , Ô ) = [ ß cos( å Þ

è

−

é Õ

)ê

1 (Ô

ê

1 (Ô

) = exp ëì

ê

2 (Ô

) = exp ëì

),

) + à sin( å Þ

)] ê

2 (Ô

),

á

(Ô , å 2 ) í á

Ôïî

(Ô , − å 2 ) í

, Ôïî

,

where ß , à , and å are arbitrary constants. 4.

Ö ð Ö ñ

= ð

ò

Ú

Exact solution: Ý

ñ

1

, ð

2 Ö Ö

(Þ , Ô ) =

ð Û 2 Ü

è é Õ ê

+ ó ( )ô ñ

(Ô ) ë$ß + ì ê

–

+ ÷ ( )ô

õ ö

ê

ñ

ø

(Ô ) (Ô )

(Ô ) = exp ëì

í Ôïî á

õ ö

.

+è

−

é Õ ê

(Ô , å 2 ) í

(Ô ) ë$à + ì Ôïî

ê

ù

(Ô ) (Ô )

í Ôïî

,

,

where ß , à , and å are arbitrary constants.

© 2002 by Chapman & Hall/CRC Page 736

737

B.5. SECOND-ORDER NONLINEAR EQUATIONS

5.

=

Ö ð

ò

Ö ñ

ð

ñ

2

1

,

1Ú

+ô

ð

Ö

ð

Û 2 Ü Ö

There are solutions of the form 6.

=

Ö ð

ò

Ö ñ

,

Ú

ð

ñ

Ý

Ö Ö

(Þ , Ô ) = cos( å

2Ú ò

ñ

ñ

(Ô ) (Ô ) ø

á

(Ô ) = exp ëì ê

+ sin( å

í Ôïî

õ ö

3Ú ò

1

, ñ

ð

2

.

ð Û 2 Ü

Ö Ö

(Ô ).

é Õ û

) + ÷ ( ) sin( ü Û

ê

−

(Ô ) + è

è é Õ ú

ñ

Û 2 Ü Ö

–

+ô

ð

Ö

ð

(Þ , Ô ) =

) ê (Ô ) ë$ß + ì Þ

2

1

,

+ ó ( ) cos( ü

ð Û 2 Ü

ð

Exact solution:

Ý

2

1

õ ö

). Û

) ê (Ô ) ëà + ì Þ

(Ô , − å 2 ) í

ê

(Ô ) (Ô ) ù

,

í Ôïî

,

Ôïî

where ß , à , and å are arbitrary constants. 7.

=

Ö ð

ò

Ö ñ

ð

1Ú ñ

2

1

,

Ö

+ cos( ü

ð

ð

Û 2 Ü Ö

There are solutions of the form Ý 8.

=

Ö ð

ò

Ö ñ

ð

ñ

Ö

ð

ñ

ð

2 Ö

Û 2 Ü Ö

) ú (Ô ) + sin( å Þ

+ sin( ü

ð

Þ

) Û

3Ú ò

ñ

1

, ð

Exact solution in multiplicative form: Ý

=

Ö ð Ö ñ

ð

ò

Ú ñ

1

, ð

2 Ö Ö

Û 2

Ö ð

(Ô , å ) í

=

å ú

,

Ôýî

ø

Ö

2

–

ð

Û 2

ÿ Ö ð Ö

1
. Exact solution in multiplicative form: Ý

.

á

(Þ , Ô ) = ú (Þ ) exp ëì

2

,

ð

Ö

ð Û 2 Ü

)û (Ô ).

where the function ú = ú (Þ ) satisfies the linear ordinary differential equation (Þ ) ú 9.

2 Ö

.

ð Û 2 Ü Ö

ò

1

,

2Ú

(Þ , Ô ) = cos( å

2

(Û ) ó

,

Ú

) Û

(Þ , Ô ) =

exp ëÀå 

Þ

Õ Õ

.

. Ü

Û

þ#þ

+ì

á

(Ô , å 2 , 0) í

Ôýî

,

where  is an arbitrary constant. 2
. Exact solution in multiplicative Ý form: (Þ , Ô ) = ( ß è é Õ + à è − é Õ ) ú (Ô ), where ß and à á areâ arbitrary constants, and the function ú = ú (Ô ) satisfies the ordinary differential 2 2 2 equation ú þ = ú . Ô , å , 4ß à å ú ã 3
. Exact solution in multiplicative form: Ý (Þ , Ô ) = [ ß sin( å Þ ) + à cos( å Þ )] ú (Ô ), where ß and à á areâ arbitrary constants, and the function ú = ú (Ô ) satisfies the ordinary differential 2 2 2 equation ú þ = ú Ô , −å , −å (ß + à 2 )ú 2 .



10.

Reference: Ph. W. Doyle (1996), the case Ö ð Ö ñ

= ð

ò

Ú ñ

,

2 Ö

ð

Ö

Û 2

,

Ö ð Ö

Û

–Û

ã

≡ 0 was considered.

 

2 Ö Ö

ð

Û 2

,2 ð

– 2Û

Ö ð Ö

Û

+ Û

2

2 Ö

Exact solution in multiplicative form: Ý (Þ , Ô ) = (  2 Þ 2 +  1 Þ +  0 ) ú (Ô ), where  0 ,  1 , and  2 are á arbitrary constants, and the function â differential equation ú þ = ú Ô , 2 2 ú ,  1 ú , 2 0 ú .



Reference: Ph. W. Doyle (1996), the case

 

.

ð Û 2 Ü Ö

ú

= ú

(Ô ) satisfies the ordinary

ã

≡ 0 was considered.

© 2002 by Chapman & Hall/CRC Page 737

738

B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES 2

B.5.5-2. Equations of the form 2

11. Ö

ð

2

= ò

Ö ñ

,

Ú Û

Ö ð Ö

2

, Û



Ö

+

ð Û 2 Ü Ö

á

=

 2





Ý

, ñ

,Ô , , Þ



Exact solution in additive form:

Ý â

Ö ð



+ Ü

, 



Õ





2

Õ

ã

.

. ð

Ö ñ

2

,

(Þ , Ô ) = ú (Þ ) + û (Ô ),

where the functions ú (Þ ) and û (Ô ) are determined by solving the second-order nonlinear ordinary differential equations (  is an arbitrary constant) â á

,ú Þ

û

2

12. Ö

ð

2

=

ò

Ö ñ

ð

1

,



ñ

ð

2

1

,

Ö ð Ö ñ

Ö

ð

ð Ö

Õ â

Õ Õ ã Ô , û þ ã

Ö ð

ð Ö



1
. Exact solution in multiplicative form: Ý

+

þ#þ

ú

−

2

,

2

,ú

þ

−

þ#þ 

Ö

+à

è é Õ

,

= 

. .

ÿ Ö ð



(Þ , Ô ) = ( ß



2

–

2

û

=



− è

Ü

) ú (Ô ),

é Õ

á â arbitrary constants, and the function ú = ú ( Ô ) satisfies the ordinary differential where ß and à are þ  ú , å 2 , 4ß à å 2 ú 2 . equation ú þ#þ  = ú Ô , ú  ã

2
. Exact solution in multiplicative form: Ý

(Þ , Ô ) = [ ß sin( å

) + à cos( å Þ

)] ú (Ô ), Þ

where ß and à are á â arbitrary constants, and the function ú = ú ( Ô ) satisfies the ordinary differential þ  ú , −å 2, −å 2(ß 2 + à 2 )ú 2 . equation ú þ#þ  = ú Ô , ú  ã

2

13. Ö

ð

2

=

ò

Ö ñ

ð



2

ñ

, Ö

ð Ö

2

,

Ö



2

–

Ö ð





Exact solution in multiplicative form: Ý

where  0 ,  1 , and  differential equation ú 2

14. Ö

ð

2

=

ò

Ö ñ

ð

1

2

–2 ð

2

2Þ

+

2

1 Ö

ð

ð

Ö

+

2 Ü

ô õ ö ò

2 

ñ

2 ð Ö ñ

2

= ð

ò

1 

ñ

2 

0)ú

2 Ö

ð Ö



+

1

, ð

2 ð Ö Ö

2 Ü

(Þ , Ô ) = + cos( ü 

ò

2



There are solutions of the form Ý

(Þ , Ô ) = cos( å Þ

ð Ö

.

2 Ü



(Ô ), = ú

(Ô ) satisfies the ordinary ú



ñ

+

2 Ü

ô

–

õ ö ò

3 

ñ

1

, ð

−

ð

2 ð Ö

2 Ü

+ sin( ü



) ú (Ô ) + sin( å

Ö

ð Ö

2 Ü

.



(Ô ).

é Õ û

Ö

2

1

,



(Ô ) + è

è é Õ ú

)

2 Ö

ð



Ý

Ö

Ö

1Þ

1

,

There are solutions of the form

15.





(Þ , Ô ) = ( 

+

Ö ð

ã

, ñ

Ö

,2

are arbitrary constants, and the function á â =ú Ô , 2 2 ú ,  1 ú , 2 0 ú .

2  þ#þ 



ð Ö

Þ



) ò

3 

ñ

1

, ð

2 ð Ö Ö

2 Ü

.



)û (Ô ).

© 2002 by Chapman & Hall/CRC Page 738

739

B.6. THIRD-ORDER NONLINEAR EQUATIONS

B.6. Third-Order Nonlinear Equations B.6.1. Stationary Hydrodynamic Boundary Layer Equations 1.

2 Ö ð Ö



Ö Ö



2

– ð

Ö

Ö ð Ö



Ö

ð Ö



3

=

2

Ö 

ð Ö



.

3



The system of equations of stationary laminar boundary layer on a flat plate (Schlichting 1981, Loitsyanskiy 1996), 

1

 

1

+

   

 

2

1

   

+

 

1

=

2

2 



1

,

  2

= 0,

 

can be reduced to this equation by introducing the stream function  in accordance with the relations  1 =   and  2 = −    !  " (  and  are the longitudinal and transverse coordinates,  1 and  2 are the longitudinal and transverse components of the fluid velocity, and  is the kinematic fluid viscosity). Ý

1
. Suppose = Then the function

Ý

(Þ , # ) is a solution of the stationary hydrodynamic boundary layer equation. Ý

=

1

Ý â

1 

where ú (Þ ) is an arbitrary function and  of the equation.



+

2Þ 

1,

2, 



1 2#

+ ú (Þ ) + 

3,

and 

4

3, 

ã

4,

are arbitrary constants, is also a solution

Reference: Yu. N. Pavlovskii (1961), L. V. Ovsyannikov (1978).

2
. Exact solutions involving arbitrary functions: Ý Ý

Ý

(Þ , # ) = 

1#

(Þ , # ) = 

1#

+ ú (Þ ), 2

+ ú (Þ )# +

4

2 ú

1

(Þ ) + 

2,

Þ +  1  2 + +  3, + ú (Þ ) [# + ú (Þ )]2 (Þ , # ) = ú (Þ ) exp(−  1 # ) + $  1 Þ +  2 , Ý

(Þ , # ) =

6$

1

Ý

(Þ , # ) =

#



(

exp % − 

1

(' , # ) = 6 $ (

(' , # ) = −6 $

1) 3

1'  

( Þ )& + 

2ú

tanh * + 

1) 3

1'

−

2#

2,

tan * + 

= *

2,

1# 

= *

+

3#



3ú '

1# '

(' ) + $ 

2'

−2 ) 3

+ ú (' ),

−2 ) 3

+ ú (' ),

+

4,

where  1 ,  2 ,  3 , and  4 are arbitrary constants and ú (' ) is an arbitrary function. The second solution is specified in Zwillinger (1998) and the fourth and fifth were obtained by Ignatovich (1993). 3
. Exact solution: (' , # ) = (

where the functions equations ø

= (# ) and ø

ù

'

(# ) + (# ), ù

ø

(1)

= (# ) are determined by the system of ordinary differential ù

( + þ )2 − ø ø +þ +þ − ø

ù

ø ø ù

þ#þ + +, þ#þ + +,

= = $

ø $

ù

þ#þ#+,þ + +, þ#þ#+, þ + +,

, .

(2) (3)

The order of equation (2) can be reduced by two. Assume that a solution = (# ) of equation (2) is ø ø known. Then equation (3), which is linear in , has two linearly independent particular solutions ù

ù

1

= 1, ù

2

= (# ) ø

© 2002 by Chapman & Hall/CRC Page 739

740

B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES

The second particular solution is apparent from comparing equations (2) and (3). The general solution of equation (2) can be represented in the form (see Zaitsev and Polyanin 1995): (# ) = ù

1 

+

2

= (# ), ø

+ ø



û ì

ø

1

= û

ø

3

í

exp  −

( + þ )2

−ì #

ø

1 $

û ø

ì

í #

, -

(4) í

ø

#

. -

It is not difficult to check that equation (2) has the particular solutions (# ) = 6 $ (# +  )−1 , ø

+

(# ) = ø



−0

. /

(5)

, $

where  and 0 are arbitrary constants. With reference to (1) and (4), one can see that the first solution in (5) leads to the third solution in Item 2
with ú (' ) = const. Substituting the second expression in (5) into (1) and (4) yields another solution.



2.

Reference: A. D. Polyanin (2001b, 2001c). 2 Ö ð Ö



Ö Ö



2

– ð

Ö

Ö ð Ö



Ö

ð Ö



3

=

2

Ö 

ð Ö



+ ó ( ).

3





The equation of laminar boundary layer with pressure gradient. 1
. Suppose ( 1961)

= (

(' , # ) is a solution of the equation in question. Then the functions (Pavlovskii (

=1

1

(

2

,1 '

+ ú (' ) 3 +  , #

where ú (' ) is an arbitrary function and  is an arbitrary constant, are also solutions of the equation. 2
. Nonviscous solutions (independent of the viscosity $ ): (' , # ) = 1 (

(

(' , # ) =

#

4

1# 

2ì 2

(' ) í ø

3
. Exact solution for (' ) =

5 '

ø

= 6

4

4

1

and 

2

2 ú

+ ú (' ),

(' ) − 2 ì ø

(' ) í '

+ î

2,

are arbitrary constants.

+ : (' , # ) = ' (

where the functions 6 equations

1

1î

1

+ ú (' )# +

where ú (' ) is an arbitrary function and 

1) 2

+ '

(# ) and

= 

(6 6



(# ) are determined by the system of ordinary differential

)2 − 6 þ+ − 6

+þ

+þ 

(# ) +  (# ), 6

6 

= =

þ#þ + +, þ#þ + +,

$

6

þ#þ#+,þ + +,

$



þ#þ#+, þ + +,

+5 , + .

(1) (2)

The order of the autonomous equation (1) can be reduced by one. If a particular solution 6 (# ) of equation (1) is known, the corresponding equation (2) can be reduced to a second-order linear equation by the change of variable 7 (# ) =  þ+ . If 6 (# ) = 1 8 5 # +  , equation (2) can be integrated in quadrature, because its two particular solutions are known if  = 0, namely,  1 = 1 and  2 = 1 2 1 + # . 28 5 # 4
. Exact solution for (' ) = 5

ø

(

(' , # ) = ú (' ) . /

+

. 9 :

−

: 5

2; 0

2ú

(' )

. 9 :

−

+ /

−$ 0

'

+

2$ 0 ;

2 #

+

2$ ;

0

ln | ú (' )|,

where ú (' ) is an arbitrary function and 0 is an arbitrary constant.



Reference: A. D. Polyanin (2001b, 2001c).

© 2002 by Chapman & Hall/CRC Page 740

741

B.6. THIRD-ORDER NONLINEAR EQUATIONS


3.

2







– 

2






=

2

2




( ) 









+  ( ) + ( ).

2 

This equation can be used to model turbulent boundary layer. Exact solution:

=  ( ) +  ( ), where the functions  ( ) and  ( ) are determined by the system of ordinary differential equations (

   

)  + 

   

− (

(

   

)  + 

   

−

)2 +  = 0, 

+  = 0.

   

B.6.2. Nonstationary Hydrodynamic Boundary Layer Equations 1.

2





+

 

2







2


– 





3


=

2

 

.

3

This is the equation of nonstationary laminar boundary layer on a flat plate; and are the longitudinal and transverse coordinates and is the stream function (Schlichting 1981, Loitsyanskiy 1996).



= ( , ,  ) is a solution of the equation is question. Then the function (see 1  . Suppose Vereshchagina 1973)

1



=



, +  ( ,  ),  +

 



( ,  ) 



+  ( ),

where  ( ,  ) and  ( ) are arbitrary functions, is also a solution of the equation. 2  . Exact solutions:







=

! 1

=

! 1

=

6" + ! 1 ! 2 + + +  ( ,  ) [ +  ( ,  )]2



=

2

! 1

= −6 "

1

+  ( ,  ) +

exp # − !

! 1

= 6"

+  ( ,  ),

1% 3

2

! 1

−!

4!

( ,  ) +

 



tan & + 









( ,  )

&

, &

! 1

=

! 1

where  ( ,  ) is an arbitrary function of two arguments and ! 3  . Exact solution: where the functions ' variables



= '

( , ,  ) = =

( ,  ) and ( 2  

  

'



 2 



+

(

+

 '



2 '





+

! 2

 

+  ( ,  )

2% 3

+  ( ,  )

2% 3

 

( ,  )

,

, ,

1 , ! 2 , ! 3 , and ! 4

( ,  ) + ( ( ,  ),



are arbitrary constants. (3)

( ,  ) are determined by the simpler equations with two (

)

'

,

=

,

( ,  ) + "

3

,

( ,  ) 

 



+!

3

( ,  ) 







( ,  ) 

 



( ,  )$ + !

2

tanh & +

1% 3

2 

1





*



 

( 

2

−'



−' 



'

2

2 (

2

3

= "



3 

= "

'

3

 

(

3

,

(4)

.

(5)

© 2002 by Chapman & Hall/CRC Page 741

742

B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES TABLE B4 Exact solutions of equation (4) Function ' = ' ( ,  ) (or general form of solution)

No 1

= '

4

=

5

'

'

! 1

= '

=

! 1



= '

3

6

=  ( ) '

2

+! 

exp # − ,

, 





+,



(  )$ −  +  (  ) + " 

+ ! 1 exp[− , + ,  ( )] −  +  ( ) + " 1 + ! 2 exp(− ,  ) -

! 1, ! 2, ,

−

-

7 8

=

'

'

=

−1 % 2

(& ), '

= +, &

1 2& $

(& ) −

#10

,

( ) is an arbitrary function, , , are any numbers



! 1, ! 2, -

3 4

−1 % 2 

− 20

-

.  )2

+ ('

.  . , ' / 

= &

( ) is an arbitrary function, are any numbers

 ,

,

( ) is an arbitrary function ( ) is an arbitrary function, ! 1 , , are any numbers

 ,

+ ,  ( )] + 1 −  +  ( ) + " +! 2

( ) is an arbitrary function ( ) is an arbitrary function, ! 1 is any number



+  ( )

1

6" +  +  ( ) +  ( )

exp[− ,

Remarks (or determining equation)

.  )2

+ (0

.

−' −0

.  . ' / 0

= ./ .

.  ./ " ' /  .

=

" 0

././ .

Equation (4) is independent of (5). If a particular solution ' = ' ( ,  ) of equation (4) is known, then the corresponding equation (5) can be reduced by the change of variable 2 = 3 4  to the 3 second-order linear equation 2  

−' 

2 

=



2 "



2

2 

−

' 

. 2



(6)

Exact solutions of equation (4) are listed in Table B4. The ordinary differential equations in the last two rows are autonomous and, therefore, admit reduction of order. Table B5 presents solutions of equation (6) that correspond to the solutions of equation (4) specified in Table B4. One can see that in the first three cases the solutions of equation (6) are expressed in terms of solutions to the classical heat equation with constant coefficients. There are other three cases where equation (6) is reduced to a separable equation. 4  . Exact solution:

( , ,  ) = 5

( ) =

18

#15

( ) 6

! 1

exp ( "

7 

+ 9 ( ) 6 2 ,

7

28

$ 6 :

−;

< 1 )

+,

−;

< 2 )

+,



+  ( ) + ;

9

( ) =

! 2

exp ( "

where  ( ) is an arbitrary function and !

,

1, ! 2, ;

2



( ) 



( ) 



 



,

, 



,

, < 1 , < 2 , and , are arbitrary parameters.

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B.6. THIRD-ORDER NONLINEAR EQUATIONS

TABLE B5 Transformations of equation (6) for the corresponding exact solutions of equation (4) [the number in the first column corresponds to the number of the exact solution ' = ' ( ,  ) in Table B4] No

Transformations of equation (6)

1

= = ( > ,  ), 2

= + +1B 1 = ( C , D ), = ( + ! 1 ) + E

2

2

C

3

= 2

4

= 2

−3 >

= 

−1 % 2

(& , D ), =

5  . Exact solution:

( , ,  ) =

= −,

( ) = 5

= &

= &

+,

−1 % 2 

3 @ 3 F

exp ( "

2 ,

3 @+ 3

( ) 

( ) =

! 2

exp ( "

,

= ln  D

−;

) + ,

, >0

-

)=5 is any,

'

− 12 S 6 6

9

1 2

6

= '

9

+ S

1 2

6

=  ( ) 6 '

−2

= '



#10

=

6 5

−

−1 % 2

' 5





+ S

( )

(& ) −

1 2& $

(& ),

9

=Q T

( )] +  +  ( ) 9

=Q T

( )] +  +  ( ) 

−  :

 +  (

+ ,

) −" ( )

, 

= &

= +, &

+;

; +

( )] +  +  ( ), ( )] +  +  ( )



+,

4, 2 

'



+,

cosh[ ,

 :

+, +,



sinh[ ,

+ S

+

, >0

sin[ , cos[ , +

− 12 S

+

S

6 5

=9 =9

'

, >0

-

 1 (

,

+

+

S

6 5

 1 ( 5

>0

-

)= > 0,

5

−S 6

 1 (

= ; ( ) +  ( ) '

+

Determining equation (or determining coefficients)

2U

− 5 −2 0

−5 + , 

.

.  . ' /

S M

,

S M

,

=Q ,

3 4

=

2| U | M S T

,

−1 % 2 

=Q 9

2U

2

 1 (

)

,

= T

S 2

,

= T

S 2

,, = S 2 T

+( 0

.  )2 − 0

+ ('

.  )2

−'

T

M

M

S 2 M

M 0

./ .

=

" 0

.  . ' /

=

.  ./ " ' /  .

././ .

Exact solutions of the Riccati equation for ; = ; ( ) with various  1 ( ) can be found in Polyanin and Zaitsev (1995). The substitution ; =  +/ V  brings this equation to a second-order linear equation for  (  ):  +W + −  1 (  )  = 0. In particular, if  1 (  ) = const, we have ! 1 cos( <  ) − ! 2 sin( <  ) ; ( ) = < for  1 = − < 2 < 0, ! 1 sin( <  ) + ! 2 cos( <  ) ! 1 cosh( <  ) + ! 2 sinh( <  ) ; ( ) = < for  1 = < 2 > 0. ! 1 sinh( <  ) + ! 2 cosh( <  ) On substituting solution (4) with arbitrary  1 ( ) into equation (3), we obtain 2 

 

The transformation = 2

= C

X

2 "

 

1 ( ) X



=

2

2

+

#;

(  ) +  (  )$

( ) +

(C , D ) +







 2 (

X

( ) ( ) 

X

) ( ) +!



,  X

2,

− ; ( ) 2 +  2 ( ).



 =

2 

D

=

X 2 

( ) 

(5)

+!

1,



( ) = exp 

;

( ) 



,

takes (5) to the classical constant coefficient heat equation =  Y

Z\[

] ^`_

a b

Y

Z\[

] ^`_

c b



D

=

2 "

 

=

C 2

.

The ordinary differential equations in the last two rows in Table B6 (see the last column) are autonomous and, hence, can be reduced in order.

Suppose ( , ,  ) is a solution of the nonstationary hydrodynamic boundary layer equation with  ( ,  ) =  1 ( ) +  2 ( ). Then the function 1

= ( +  ( ), ,  ) −  + ( ) ,

where

 +W +

+  1 ( )  = 0,

is also a solution of this equation.

© 2002 by Chapman & Hall/CRC Page 745

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B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES

( , ,  ) =  ( ,  ) 6 

 :

,

> 0: -

−  :

+  ( ,  ) 6

1

+

( )



2 , 2  ( ,  )  where f ( , g ) is an arbitrary function of two arguments.

e

m

1p

(u , n , g ) = o

exp q

ln |  ( ,  )| 

=Q

,

5 h . Exact solutions for  ( , g ) = i ( ) j k l , > 0: p e 1p m ( , n , g ) = o exp q 12 g/r s t (u ) sinh[





,



,

+ S

6

( ,  ) = −

+ S

4  . Exact solution for  ( ,  ) =  ( ) 6

d

v

p

1 2 e g/r s

+ f (u , g )] + n

(u ) = 2

(u ) x

i

+y

u

=

1,

6 h . Exact solution for z (u , g ) = i (u ) j −k l , < 0: e p 1p m (u , n , g ) = o exp q − 21 g/r s t (u ) sin[ e

m

1p

(u , n , g ) = o

exp q − 21

v

+ f (u , g )] + n

g/r s

+ f (u , g )] + n

(u ) = 2

(u ) x

i

+y

u

=

1,

7 h . Exact solution for z (u , g ) = m

(u , n , g ) = f (u , g ) j +

1p v v

g

−}

{ j k | ~



2

{

2f e

u

−  −} l ~

j k |

(u , g )

ln | f (u , g )| x w

p

v

(u , g ) x

g

(u , g ) x f

w

−"

+ u

2 p e

+

1p

+ n

w

‚

u

,

ln | f (u , g )| ,

€

is an arbitrary constant.

8 h . Exact solution for z (u , g ) = z (g ): (u , n , g ) =

, u

p

2"

where f (u , g ) is an arbitrary function of two arguments and m

, u

: l

p

−

,

.

e

d

2" where f (u , g ) is an arbitrary function of two arguments. w

u

(u , g ) x

f w

v p

t

(u , g ) x f

w

g

v

(u ) cos[ t

g

v

p e

,

,

e

d

2" where f (u , g ) is an arbitrary function of two arguments. w

f w

v p

t

g

v

(u ) cosh[ t

,

,

e

2"

+ f (u , g )] + n

−"

p

(ƒ , g ) x

+ f (g )n + t (g )u , ƒ

ƒ

p

=

+

„ u

, n

where f (g ) and t (g ) are arbitrary functions, „ and are arbitrary parameters, and ( ƒ , g ) is a function ‚ satisfying the second-order linear parabolic equation p

v ‚

The transformation

v

‚

= 2

g

+

#„ f

(g ) −

p

( g )$ t

v ‚ v

(† , g ) −

1p f

(g ) +

= ƒ

z w

9 h . Exact solution for z (u , g ) = z (g ): y 1j

−  + ( , ) ~ ~ ‡ | l

−

‚ ƒ 2

1p

f … l

(g ) +

1p

(g ). z p

1p

(g ) x g ,

= g

v

2 "

= †

p 2

v

(u , n , g ) =

2 v v

brings it to the customary heat equation

m

2 "

2 v v

2

− ƒ

w

(g ) −

#P„ f

( g )$ t

x g

.

† 2

p

− { (g ) f (u , g ) − v

v

g w

f

(u , g ) x u

+ { (g )n + "

where f (u , g ) is an arbitrary function of two arguments and y

1, y 2,

and

u p

, {

(g ) = w

z

(g ) x g

+y

2,

are arbitrary constants.

© 2002 by Chapman & Hall/CRC Page 746

747

B.6. THIRD-ORDER NONLINEAR EQUATIONS

10 h . Exact solution for z (u , g ) = z (g ): m

(u , n , g ) = f (u , g ) j

p

 ~

−  ~

+ t (u , g ) j

+ ˆ (u , g ) + { (g )n ,

where is any number, f (u , g ) is an arbitrary function of two arguments, and the other functions are defined by y

(u , g ) = t

2

~

l

(u , g ) f

1p

(u , g ) = ˆ

2‰

" j

− u

Š

(g ) x {

w

1p

(g ) ln | f (u , g )| + {

v

m

m

j

= j

‰

=

2

~

−‰

(y l 2

~

+y

1j ~ 

−

2j

=

−

y 1j

v

p l #y 1

sin(

~ 

g

v

)+y ƒ

p m

)+

~ 

p

)$ + ƒ

v

p

sin( ƒ

2

− 2"

+y g

+

2)

v g

v

+ƒ u

(u , g ) x

+ƒ u

12 h . Exact solutions for z (u , g ) = m

=− m

=

„ u

Ž

3 ƒ

6"

+y

+y 1

2

2ƒ

+y „

exp − €

3.

2

; l

. u

= ƒ

(g ) x g , z

n

+ f (u , g ),

(g ) x g , z

w

= ƒ

= ƒ

n

n

+ f (u , g ), + f (u , g ),

p

1, y 2,

+

1ƒ

−

v Ž

g

v

f w

2 ƒ

2„

(u , g ) x

+y

+

2ƒ

, u

and

are arbitrary constants.

v g

v

= n + f (u , g ), ƒ

f w

,

= n + f (u , g ),

(u , g ) x u

1, y 2,

and „ are arbitrary constants.

where f (u , g ) is an arbitrary function of two arguments and y

P

~

: Ž

ƒ  "

−" u

w

where f (u , g ) is an arbitrary function of two arguments and y

2‰ j

p

+ƒ u

+y g

(g ) x g , z

w

(u , g ) x f

f w

(g ) x z

w

ln | f (u , g )| x

g w

v

(g ) = {

w

(u , g ) x f

w

cos(

2

g

v

11 h . Exact solutions for z (u , g ) = z (g ):

,

gŒ‹

ƒ

Reference: A. D. Polyanin (2001b). 2’

+

‘ ‘ “‘ ”

2’ ’

‘

‘ ‘ ”

‘ •

–

2’ ’

‘

‘ ”

‘ ‘ •

2

2’

= ‘

‘ ”

Š–

‘

‘ ”

€



2

+ — ( , ). ‹

•

‘ ”

“

This equation describes the flow of a non-Newtonian fluid in a two-dimensional nonstationary boundary layer with a pressure gradient. Here, m is the stream function and the function z = z ( ) ‚ depends on the rheological properties of the fluid. For power-law fluids, z = „ | | ˜ −1 . ‚

‚

1 h . The assertion of Item 1 h , equation 1 in Subsection B.6.2, remains valid for this equation. 2 h . The equation admits the nonviscous solutions presented in Item 2 h , equation 2 in Subsection B.6.2, where z (u , g ) must be replaced by i (u , g ). 3 h . Exact solution for i (u , g ) = i (g ): m

where 2

= 2

(u , n , g ) = { (g )u +

(n , g ) x 2

w

, n

(n , g ) is a function satisfying the second-order equation 2 v v

2

g

− { (g ) v v

n

= v

v n

Š

z

2 v

€

v

 n

+ i (g ). ‹

(1)

The transformation 2

= (ƒ , g ) + ‚

w

i

(g ) x g , ƒ

= n

+ w

{

(g ) x g

© 2002 by Chapman & Hall/CRC Page 747

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B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES

brings equation (1) to the simpler equation v ‚

= g

v

v ƒ

v

z Š

v ‚ €

 ƒ

v

. ‹

(2)

Equation (2) admits exact solutions with the forms [for any z = z (™ )] p

(ƒ , g ) = ‚

(ƒ , Ÿ ) = ‚

(ƒ , Ÿ ) = ‚

( › ), š

+ š ( › ),

{ ƒ  

= ›

( › ),

Ÿ š

= ›

= ›

„ ƒ

+p

„ ƒ

+

ƒ ¡  

p

g Ÿ

Ÿ

p š

=

equation š

equation

1 2š

œ 

equation

œ  œ 

„ z

(„

=

„ z

(„

−

1 2› š

where { , „ , y , and ¢ are arbitrary constants. The first two equations for š parametric form.

)+y ;

š

ž… š

ž…

ž…

= [ z (š

+{ )+y ; ž…

)] ž… ,

= š ( › ) can be solved in

4 £ . Exact solution for ¤ (¥ , Ÿ ) = ¤ (Ÿ ): ¦

(¥ , § , Ÿ ) =

(¨ , Ÿ ) x ™

w

+ © (Ÿ )§ + ª (Ÿ )¥ , ¨

¨

=§ +„ ¥

,

where © (Ÿ ) and ª (Ÿ ) are arbitrary functions, „ is an arbitrary parameter, and ™ (¨ , Ÿ ) is a function satisfying the second-order nonlinear parabolic equation «

« ™

«

+ Ÿ

( Ÿ ) − ª ( Ÿ )®

¬­„ ©

«

« ™

= ¨

«

« ¨

=

( › , Ÿ ) − © (Ÿ ) + ± °

leads to the simpler equation

(Ÿ ) ² Ÿ ,

«

¤

= Ÿ

«

« ›

´¶µ

‹

=¨ −± ›

« °

«

 ¨

€

The transformation ™

™ «

z Š

·

° « ¸ ¹

− © ¯… (Ÿ ) + ¤ (Ÿ ).

¬­³ ©

( Ÿ ) − ª ( Ÿ )®

² Ÿ

. º

For exact solutions of this equation, see Item 3 » . 5 » . Exact solution for ¼ (½ , Ÿ ) = ¾ (Ÿ )½ + ¿ (Ÿ ): À

(½ , Á , Ÿ ) =

(Ÿ )Á + Ä (Ÿ )ÅL½ + ±

ÂPÃ

(Á , Ÿ ) ² Æ

, Á

where Ä (Ÿ ) is an arbitrary function, à = à (Ÿ ) is determined by the Riccati equation

=

and Æ

Æ

2

+Ã

à ÈÇ

= ¾ (Ÿ ),

(Á , Ÿ ) satisfies the second-order equation É É

É Æ

= Ÿ

É

É Á

´Êµ

·

É

É Æ ¹ Á

+ º

( Ÿ ) Á + Ä ( Ÿ )Å

Â1Ã

Æ É

Á

− à (Ÿ ) Æ + ¿ (Ÿ ).

The transformation Æ

=

where

Ë

Ë

1 (Ÿ )

´Ì

(Í , Î ) + ±

(Ÿ ) = exp ´

±

Ã

¿

(Ÿ ) ²

Ë

(Ÿ ) (Ÿ ) ² Ÿ º

,

Ÿ º

= Î

Ë 2 ±

(Ÿ ) ²

+Ï , Ÿ

Í

=Á

Ë

(Ÿ ) + ± Ä

Ë

(Ÿ ) (Ÿ ) ² Ÿ

+Ð ,

, leads to the simpler equation É

É É Ì Î

= É

É Í

´Êµ

·

É Ì Í

¹

º

.

For exact solutions of this equation, see Item 3 » .

© 2002 by Chapman & Hall/CRC Page 748

749

B.7. FOURTH-ORDER NONLINEAR EQUATIONS

B.7. Fourth-Order Nonlinear Equations B.7.1. Stationary Hydrodynamic Equations (Navier–Stokes Equations) 1.

Ñ Ò

Ñ

Ñ Ó

Ñ Ô

(Õ

)– Ò

Ñ Ò

Ñ

Ñ Ô

Ñ Ó

(Õ

)= Ò

Ö

Õ

Õ

, Ò

Õ

2

= Ò

Ñ

Ò

+

2

Ñ Ô

2 Ñ

Ò

2

.

Ñ Ó

The two-dimensional equations of steady-state motion of a viscous incompressible fluid (stationary Navier–Stokes equations) × ×

1 Ø

Ø ×Ù ×

1 Ø

×

1

+×

2 Ø

2

+×

2 Ø

Ø Ù ×

1

Ø

Ø ×Ú Ø ×Ú

+ Ø

Ø Ù

1

=−

2

=−

2

1Û

ØÜ Ø Ù

1Û

ØÜ

+Ý Þ ×

1,

+Ý Þ ×

2,

Ø Ú

=0

Ø Ú

are reduced to the equation under consideration. To this end, one introduces the stream function ß by the formulas × and ×

2

= − àá

àã

and eliminates the pressure , using cross differentiation, from the first two equations.

1

= àá

àâ

Ü

1 » . Exact solutions in additive form: À

(½ , Á ) = À

(½ , Á ) = À

(½ , Á ) = À

(½ , Á ) = (½ , Á ) = À

where ä

1 , ççç

,ä

5,

3

ä 1Á

2

ä 1½ ä 1

+ä

2

2Á

+ä

exp( å 1 exp( å

ä 1

½

ä

½

Á

3Á

+ä

2½

exp(− å

+ä

3Á

)+ä

)−æ )+æ

+ä

4,

+ä

4Á

2 2

2Á

+ä

+ä +ä

å ½ å ½

+ä

5,

+ä

3Á

+æ

4

,

å ½

exp( å Á ) + æ å 2 exp(− å Á ) + æ

+ ä 3, Á + ä 3, Á

2

å

and å are arbitrary constants.

2 » . Exact solutions: À À

(½ , Á ) = ( Ï À

(½ , Á ) =

À À

½

Ï

è

− é (  + êLë )

(½ , Á ) =

ÂPÏ

sinh(î

(½ , Á ) =

ÂPÏ

sin(î

(½ , Á ) = Ï

−é 

+ Ð )è

è

é  +ï ë

½

½

+æ

+ä ,

å ½

+ Ð (Á + ì

)2 + ä ( Á + ì ½

) + Ð cosh(î ) + Ð cos(î

+Ð

ë

è ð

+æ

)Åè ½

)Åè ½

+

ñ Á

−é 

−é  æ

+ ñ

å

½ æ

+ æ

å

(å

å

)+æ 2

(î 2

å

(ì

2

+ 1)½ + í ,

+ å 2 )½ + ä ,

− î 2 )½ + ä ,

(î − ñ )½ + ä , ñ

=ò ó

å 2

+ î 2,

where Ï , Ð , ä , í , ì , î , and å are arbitrary constants. 3 » . Exact solution: À

where the functions ô = ô (Á ) and õ differential equations

(½ , Á ) = = õ

(Á ) are determined by the system of fourth-order ordinary

ô  Ç ô ÇÇ  Ç ô ÇÇ  õ

(Á )½ + õ (Á ), ô

−ô −ô

ô ÇÇÇ   ÇÇ Ç  õ

= =

, .

(1) (2)

+Ï , +Ð ,

(3) (4)

æ ô ÇÇÇÇ  æ õ

 ÇÇ ÇÇ 

Integrating yields the system of third-order equations (ô õ

Ç Ç ô

)2 − ô Ç − ô

ô ÇÇ  õ

 ÇÇ 

= =

æ ô ÇÇÇ  æ õ

 ÇÇ Ç 

© 2002 by Chapman & Hall/CRC Page 749

750

B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES

where Ï and Ð are arbitrary constants. The order of the autonomous equation (3) can be reduced by one. It is not difficult to verify that equation (1) has the particular solutions (Á ) = Ã Á + ö , (Á ) = 6 æ (Á + Ã )−1 ,  (Á ) = Ã è − é + å æ , ô ô ô

(5) (6) (7)

where à , ö , and å are arbitrary constants. In general, equation (4) can be reduced by the change of variable nonhomogeneous linear equation +ô

æ ÷ ÇÇ 

÷ Ç

−ô

+Ð

Ç ÷

= 0,

where ÷

÷

=

=õ Ç

Ç õ

to the second-order

.

(8)

The corresponding homogeneous equation (with Ð = 0) has two linearly independent particular solutions Ë ú if ô ÇÇ  ≠ 0, ô ÇÇ  Ë ú Á 1 ÷ 1 = ø ÷ 2 = ÷ 1 ù , where = exp û − ù ô Á ¹ . (9) ô if ô ÇÇ  = 0, æ ÷ 12 (The first solution is apparent from comparing equations (1) and (8) with solutions of equations (8) and (2) are given by ÷

as

=

ä 1÷ 1

+ä

2÷ 2

+ä

÷ 1 ú Ë Á

3û ÷ 2ù

−÷

÷ 2 ú Ë Á ¹

1ù

,

= õ

ú ÷

ù

Á

+ä

= 0.) The general

Ð

4,

ä 3

=−

Ð æ

. (10)

The general solution of equation (2) corresponding to the particular solution (6) is represented

(Á ) = ä ü 1 (Á + Ã )3 + ä ü 2 + ä ü 3 (Á + Ã )−1 + ä ü 4 (Á + Ã )−2 , where ä ü 1 , ä ü 2 , ä ü 3 , and ä ü 4 are arbitrary constants (these are expressed in terms of ä 1 , ä 2 , ä 3 , and ä 4 ). The general solutions of (2) corresponding to the particular solutions (5) and (7) are determined from (9) and (10). õ

4 » . Exact solution of a more general form: À

where the functions ô = differential equations

=

( ý ) and õ ô

ô þ Ç ô þ\ÇÇ þ

−ô

Çþ ô þ\ÇÇ þ

−ô

õ

(½ , Á ) =

õ

( ý )½ + õ ( ý ),

õ

ô

=Á +ì ý

½

,

( ý ) are determined by the system of fourth-order ordinary

ô þ\ÇÇþ\Ç þ

= æ (ì

2

+ 1) ô

þ\ÇÇþ\Çþ\ Ç þ

þ\ ÇÇþ\ Ç þ

= æ (ì

2

+ 1) õ

þ\ ÇÇþ\ ÇÇ þ\þ

,

(11) + 4ì

+

æ ô þ\ÇÇþ\Ç þ

(ì

2ì ô + 1)

.

ô þ\ÇÇ þ

2

(12)

Integrating yields the system of third-order equations (ô õ

þÇ

)2 − ô

ô þ\ÇÇ þ

2

= æ (ì

+ 1) ô

+Ï ,

þ\ÇÇþ\Ç þ

(13)

ú 2ì ô ô þ\ÇÇ þ ý +Ð , (14) ù +1 are arbitrary constants. The order of the autonomous equation (13) can be reduced Çþ ô þ Ç

−ô

õ

þ\ ÇÇ þ

2

= æ (ì

+ 1) õ

+ 4ì

þ\ ÇÇþ\ Ç þ

+

æ ô þ\ÇÇ þ

ì 2

where Ï and Ð by one. It is not difficult to verify that equation (11) has the particular solutions (ý ) = ô

à ý

+ö ,

(ý ) = 6 æ (ì ô

−é þ

2

ý

= Á

+ì ½

,

−1

+ 1)( ý + Ã ) ,

(ý ) = à è + å æ ( ì 2 + 1), where à , ö , and å are arbitrary constants. In general, equation (14) can be reduced by the change of variable nonhomogeneous linear equation. ô

ÿ


÷

= õ

Çþ

to a second-order

Reference: A. D. Polyanin (2001d).

© 2002 by Chapman & Hall/CRC Page 750

751

B.7. FOURTH-ORDER NONLINEAR EQUATIONS

2.

Ñ Ò Ñ

Ñ Ó

Ñ Ô

(Õ Ò

)–

(Õ

Ñ Ò Ñ

Ñ Ô

Ñ Ó

)= Ò

Ö

Õ



Õ

+ ( ), Ò

Õ

Ó

2

= Ò

Ñ

Ò

2

+

2

Ñ Ô

Ñ

Ò

.

2

Ñ Ó

This equation describes plane flow of a viscous incompressible fluid under the action of a transverse force (À is the stream function). The case ô (Á ) = Ã sin( å Á ) corresponds to A. N. Kolmogorov’s model which is used to describe subcritical and transcritical (laminar-turbulent) modes of flow. 1 » . Exact solution in additive form for arbitrary  (Á ): À



1 ( , Á ) = − 2æ

where ä with

ù

0

Ë

( Á − ý )2 ( ý )

1, ä 2, ä 3, ä 4,

ú

+ä ý

−é 

1è

+ä

2

2Á

+ä

+ä

3Á

+æ

4

å

Ë

, 

−é þ

(ý ) = è

ù

è

é þ 

(ý )

ú ý

,

and å are arbitrary constants.

Example. In the case  ( ) =  cos(  ), which corresponds to ( ) = Ú

1 = 2 = 4 = 0 and  Ú = − Ý that Ú 

ß ( , )=−  2( 2 + Ý 2  2) Ù Ú

sin(  

)+Ý Ú

sin(  

cos( 

Ú

), it follows from the previous formula Ú

) +

− Ú



, Ù

where  and are arbitrary constants. This solution was indicated by Belotserkovskii and Oparin (2000); it describes the flow with a periodic structure.

2 » . Exact solution in additive form for  ( ) = 

where ä

1

and ä

( ,  ) = 2

ä 1è

−é ë

+ä

è

é 

+Ð

Ï

−

2

Ï

å 3(ä 2

+æ

) å

−é  è

é  è

+

: Ð

−æ

å 3(ä 2

) å

−é  è

−æ å



,

are arbitrary constants.

3 » . Generalized separable solution for arbitrary  ( ): ( ,  ) =  ( ) +  ( ), 

where the functions  =  ( ) and  =  ( ) are determined by the system of fourth-order ordinary differential equations  

 

− −

    

  

 

       

= =

 

 

       

, +  ( ).

(1) (2)

Integrating yields the system of third-order equations ( 

)2 −  

 



−

   

   

= =









       

+ , +ù 

(3) ( )

ú 

+ ,

(4)

where  and  are arbitrary constants. The order of the autonomous equation (3) can be reduced by one. It is not difficult to verify that equation (1) has the particular solutions   

( ) =   +  , ( ) = 6  ( +  )−1 ,  ( ) =   −  +   ,

(5) (6) (7)

where  ,  , and  are arbitrary constants. In general, equation (4) can be reduced by the change of variable nonhomogeneous linear equation 



 

+ 



−

 

+

= 0,

where 

=



, 

=

= 

ù





to the second-order



( )

ú 

+ .

(8)

© 2002 by Chapman & Hall/CRC Page 751

752

B. METHODS OF GENERALIZED AND FUNCTIONAL SEPARATION OF VARIABLES

The corresponding homogeneous equation (with solutions: 

1 

for for

   

= 

≠ =  

+ , + ,

   

=

2 

= 0) has two linearly independent particular 

ú

1ù 



,

2 1 

(The first solution is apparent from comparing equations (1) and (8) with solutions of equations (8) and (2) are given by = 

3.

1 &

'

(

1 1 "

(*

' &

+"

2 2

)–

1

+

1 





2 

#

1

−



2 

1 

#



,

$ %

1 &

'

(&

(* '

)=

(

+

$ % *

, *

(

(



=



+" 

' &

,



. !

4.

$ % &

&

ú 

ù

= 0.) The general 

#

1

= *

1

= exp û −

where

'

+ !

(&

2

1 2 &

(

'

(

2

.

' ) ' ) ' ' ' ) in polar The equations of steady-state flow' of a ' viscous incompressible fluid (stationary Navier–Stokes equations) written coordinates (- = . cos / , 0 = . sin / ) reduce to this equation. The radial and tangential components of the fluid velocity are expressed in terms of the stream function 1 by the formulas 2 3 = 13 à54 and 2 = − à54 3 .

(8 , 9 ) = 7

where "

1,

;5;5;

,"

6

à6

1 » . Exact solution in additive form: 

+"

19 "

28

1 +2

:

+"

2

38

+"

à

ln 8 + "

4

5,

are arbitrary constants.

5

2 » . Exact solution:

(8 , 9 ) = < (8 )9 + = (8 ). 7

The functions < = < (8 ) and = = = (8 ) are determined by the system of ordinary differential equations − < >  L( < ) + < [L( < )] > =  8 L2 ( < ), − = >  L( < ) + < [L(= )] > =  8 L2 (= ),

(1) (2)

where L( < ) = 8 −1 (8 < >  ) > . Exact solution of system (1)–(2):
 o + Ž 2  − (  t i 21 +  t ‘ 22 )  = 0, SIAM J. Math. Anal., Vol. 7, pp. 230–263, 1976. Bˆocher, M., Die Reihenentwickelungen der Potentialtheory, Leipzig, 1894. Brenner, H., Forced convection-heat and mass transfer at small Peclet numbers from particle of arbitrary shape, Chem. Eng. Sci., Vol. 18, No. 2, pp. 109–122, 1963. Brychkov, Yu. A. and Prudnikov, A. P., Integral Transforms of Generalized Functions, Gordon & Breach Sci. Publ., New York, 1989. Budak, B. M., Samarskii, A. A., and Tikhonov, A. N., Collection of Problems on Mathematical Physics [in Russian], Nauka, Moscow, 1980. Burde, G. I., The construction of special explicit solutions of the boundary-layer equations. Steady flows, Q. J. Mech. Appl. Math., Vol. 47, No. 2, pp. 247–260, 1994. Butkov, E., Mathematical Physics, Addison-Wesley, Reading, Mass., 1968. Butkovskiy, A. G., Characteristics of Systems with Distributed Parameters [in Russian], Nauka, Moscow, 1979. Butkovskiy, A. G., Green’s Functions and Transfer Functions Handbook, Halstead Press–John Wiley & Sons, New York, 1982. Carslaw, H. S. and Jaeger, J. C., Conduction of Heat in Solids, Clarendon Press, Oxford, 1984. Clarkson, P. A and Kruskal, M. D., New similarity reductions of the Boussinesq equation, J. Math. Phys., Vol. 30, No. 10, pp. 2201–2213, 1989. Colton, D., Partial Differential Equations. An Introduction, Random House, New York, 1988. Courant, R. and Hilbert, D., Methods of Mathematical Physics, Vol. 2, Wiley–Interscience Publ., New York, 1989. Crank, J., The Mathematics of Diffusion, Clarendon Press, Oxford, 1975. Davis, B., Integral Transforms and Their Applications, Springer-Verlag, New York, 1978. Davis, E. J., Exact solutions for a class of heat and mass transfer problems, Can. J. Chem. Eng., Vol. 51, No. 5, pp. 562–572, 1973. Deavours, C. A., An exact solution for the temperature distribution in parallel plate Poiseuille flow, Trans. ASME, J. Heat Transfer, Vol. 96, No. 4, 1974. Dezin, A. A., Partial Differential Equations. An Introduction to a General Theory of Linear Boundary Value Problems, Springer-Verlag, Berlin-New York, 1987. Ditkin, V. A. and Prudnikov, A. P., Integral Transforms and Operational Calculus, Pergamon Press, New York, 1965. Doyle, Ph. W., Separation of variables for scalar evolution equations in one space dimension, J. Phys. A: Math. Gen., Vol. 29, pp. 7581–7595, 1996. Doyle, Ph. W. and Vassiliou, P. J., Separation of variables for the 1-dimensional non-linear diffusion equation, Int. J. Non-Linear Mech., Vol. 33, No. 2, pp. 315–326, 1998. Elrick, D. E., Source functions for diffusion in uniform shear flows, Australian J. Phys., Vol. 15, No. 3, p. 283–288, 1962. Faddeev, L. D. (Editor), Mathematical Physics. Encyclopedia [in Russian], Bol’shaya Rossiiskaya Entsyklopediya, Moscow, 1998. Faminskii, A. V., On mixed problems for the Corteveg–de Vries equation with irregular boundary data, Doklady Mathematics, Vol. 59, No. 3, pp. 366–367, 1999.

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