Introductory Applications of Partial Differential Equations: With Emphasis on Wave Propagation and Diffusion

INTRODUCTORY APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS

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INTRODUCTORY APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS With Emphasis on Wave Propagation and Diffusion G. L. Lamb, Jr. The University

of

Arizona

A WILEY-INTERSCIENCE PUBLICATION

JOHN WILEY & SONS, INC. New York · Chichester · Brisbane · Toronlo · Singapore

This text is printed on acid-free paper. Copyright © 1995 by John Wiley & Sons, Inc. All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to the Permissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158-0012. Library of Congress Cataloging in Publication Data: Lamb, G. L. (George L.), 1931Introductory applications of partial differential equations with emphasis on wave propagation and diffusion / G. L. Lamb, Jr. p. cm. Includes bibliographical references and index. ISBN 0-471-31123-5 1. Differential equations, Partial. I. Title. QA347.L32 1995 53Γ.1133Ό1515353—dc20 94-33111

10 9 8 7 6 5 4 3

To Joan

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CONTENTS Preface 1

One-Dimensional Problems—Separation of Variables

1

1.1 1.2

Introduction One-Dimensional Heat Conduction

1 4

1.2.1

6

1.3 1.4

Steady State Solutions Time Dependent Heat Flow—Separation of Variables

6 10

1.4.1 1.4.2 1.4.3

11 13 17

1.5 1.6 1.7 1.8

1.9 2

3

xi

Boundary and Initial Conditions

Elementary Solutions Synthesis of Elementary Solutions—Fourier Series Changing the Boundary Conditions—Insulated Ends

Steady State Heating by a Localized Source—Delta Function Inhomogeneous Boundary Conditions Inhomogeneous Heat Equation—Source Terms Wave Equation—Vibrating String

21 28 33 38

1.8.1 1.8.2 1.8.3 1.8.4

39 41 45 46

General Wave Equation Solution by Separation of Variables Energy Flow on the String Source Terms—Inhomogeneous W a v e Equation

d'Alembert's Solution of the Wave Equation

49

Laplace Transform Method

60

2.1 2.2 2.3

Vibrating String Diffusion Equation Miscellaneous Examples

60 65 74

2.4

2.3.1 Impulse Acting on a String 2.3.2 Long-Time Behavior 2.3.3 Total Heat Flow through an End Point Source Problem—Preview of Green's Function

74 75 77 79

T w o and Three Dimensions

85

3.1 3.2

85

Introduction Steady State Temperature Distribution in Rectangular Coordinates—Laplace's Equation

86 vii

viii

CONTENTS 3.3 3.4

Time Dependent Diffusion in Rectangular Coordinates Waves on a Membrane—Rectangular Coordinates 3.4.1 3.4.2

3.5

3.6

3.7

4

Normal Modes Guided Waves

4.2 4.3

4.4 4.5 4.6

99 101

Orthogonal Curvilinear Coordinates

105

3.5.1 3.5.2 3.5.3

108 108 111

Cylindrical Coordinates Spherical Coordinates Oblate Spheroidal Coordinates

Spherical Symmetry

114

3.6.1

Spherical Standing Waves

115

3.6.2

Spherical Traveling Waves

116

Circular and Cylindrical Symmetry

118

3.7.1 3.7.2 3.7.3 3.7.4

118 121 126 131

Steady State Temperature in a Pie-Shaped Region Laplace's Equation in an Annular Circle Vibrating Membrane Steady State Temperature in a Cylinder

Green's Functions 4.1

93 98

142

Introductory Example

142

4.1.1

145

Reciprocity

General Procedure for Constructing Green's Functions in One Dimension One-Dimensional Steady Waves

155 158

4.3.1 4.3.2

160 162

Scattering of Waves on a String Significance of Boundary Terms

Method of Images A Non-Self-adjoint Green's Function Green's Function for a Damped Oscillator

165 167 174

4.6.1

177

A Generalization

4.7

One-Dimensional Diffusion and Wave Motion

179

4.8 4.9

4.7.1 Diffusion Equation 4.7.2 Wave Equation Two and Three Space Dimensions—Green's Theorem Green's Function in Free Space

179 183 188 191

4.9.1

192

4.10 4.11 4.12 4.13

Boundary Condition at Infinity

Two-Dimensional Problems Inversion and the Method of Images for a Circle Eigenfunction Expansion Methods Modified Green's Functions—One Dimension

199 205 210 221

CONTENTS

5

Spherical Geometry

226

5.1 5.2

Solution of Laplace's Equation Source Terms and the Multiple Expansion

226 235

5.2.1

236

5.3 5.4

6

240 245 247 248 250 251

Fourier Transform Methods

255

6.1 6.2

255 258

Fourier Sine and Cosine Transforms Examples

6.2.2 6.2.3 6.3 6.4 6.5 6.6 6.7

8

Axial Multipoles

Inversion—Green's Function for a Sphere Spherical Waves 5.4.1 Spherical Bessel Functions 5.4.2 Radiation from a Point Source 5.4.3 Reduction to a Plane Wave 5.4.4 Scattering of a Plane Wave by a Sphere

6.2.1

7

ix

Green's Function for Steady Waves on a Semiinfinite String Temperature Distribution in a Quarter Plane d'Alembert Solution for a Semi-infinite String

258 260 262

Convolution Theorems Complex Fourier Transforms

266 272

6.4.1

276

Approach to the Boundary

Fourier Transforms in Two and Three Dimensions Circular Symmetry, Fourier-Bessel Transform Green's Functions for Time Dependent Wave Equation in One, T w o , and Three Dimensions 6.7.1 One Dimension 6.7.2 Two Dimensions 6.7.3 Three Dimensions

281 286 291 291 293 294

Perturbation Methods

297

7.1 7.2 7.3

First Order Corrections Equal Frequencies (Degeneracy) Variational Methods

297 303 307

7.3.1 7.3.2

307 312

Differential Equation Approach (Rayleigh's Method) Integral Equation Approach

Generalizations and First Order Equations

317

8.1 8.2

317 325 326

Classification of Second Order Equations Uniqueness and General Properties of Solutions 8.2.1 Laplace's Equations

X

CONTENTS

8.2.2 8.2.3 8.2.4 8.3 8.4 9

Uniqueness Diffusion Equation Wave Equation

First Order Equations Burgers' Equation

327 327 328 331 343

Selected Topics

352

9.1 9.2 9.3 9.4

352 356 361

9.5 9.6 9.7 9.8 9.9

Oscillating Heat Source on a Beam Temperature Distribution in a Pie-Shaped Region Babinet's Principle Comparison of Wave Motion in One, T w o , and Three Dimensions—Fractional Derivatives Modified Green's Function for a Sphere Oscillation of an Inhomogeneous Chain Point Source Near the Interface between T w o Half Spaces Waves in an Inhomogeneous Medium A Hybrid Fourier Transform

9.10

Invariants of the Linear Parabolic Equation

387

364 370 373 376 379 382

Appendix A

Fourier Series

391

Appendix Β

Laplace Transform

410

Appendix C

Sturm-Liouville Equations

426

Appendix D

Bessel Functions

436

Appendix Ε

Legendre Polynomials

450

Appendix F

Tables of Sums and Integral Transforms

462

References

466

Index

467

PREFACE This book has evolved from a one-semester undergraduate course in applied martial differential equations that the author has given over the past decade. Students in this course are rarely encountering partial differential equations for he first time but rather are looking for the background to and synthesis of the scattered exposure to the subject already received in previous physics and/or ;ngineering courses. Frequently they are beginning graduate students who find hat such a course is actually a prerequisite for some of their graduate engileering courses. Usually they have a working knowledge of Fourier series and Laplace transform methods. These topics, along with a brief summary of SturmJouville theory and solutions of the Bessel and Legendre equations, are sumnarized here in appendices. The course provides a number of opportunities for he student to see the usefulness of various topics that are often introduced in in unmotivated way in a previous course in ordinary differential equations. For nstance, the Cauchy-Euler equation is now found to provide a very simple neans for examining diffusion and vibration in inhomogeneous systems. Lord iayleigh's use of this solution to examine a model for the reflection of a wave torn an inhomogeneous medium {Theory of Sound, Dover, 1945, Vol. 1, pp. >35-239) can certainly be read with profit by students at this stage in their :ducation. Although simple derivations of the diffusion equation and the wave equation ire included, some prior knowledge of heat conduction and wave motion is mplicitly assumed. I have attempted to develop a presentation that interweaves he physics and mathematics so as to indicate their interdependence. The book nay thus be classified as neither physics nor mathematics or on the other hand, is a combination of both, depending upon one's own convictions and prejuiices. A course at the level of this text is taken by students whose mathematical ibilities and physical understanding are changing quite rapidly. This fact, courted with the varied interests and backgrounds of the students, makes it difficult ο choose an appropriate level and mode of presentation. The physical settings lave thus been drawn from the fields of mechanical vibration and heat coniuction since an understanding of these topics requires only the simplest physcal intuition and is almost immediately available to students in all engineering iisciplines. I have attempted to begin at the beginning and gradually include opics that require more extensive calculation. In keeping with the approach hat engineering students usually find most natural, the development proceeds From the particular to the general. Thus, maximum principles, mean value xi

xii

PREFACE

theorems, and uniqueness of solutions as well as the general classification of partial differential equations are not considered until Chapter 8. In a number of texts the solutions of partial differential equations are left in a ponderous form that adds little to one's intuitive understanding of the physical processes being analyzed. Wherever possible, I have tried to use the solutions to obtain some specific information about the physical system being considered. It is my feeling that an important aspect of a student's mathematical training is the ability to translate physical situations into mathematical terms. Thus, the wording of many of the problems included here, while perhaps lengthy in some instances, is designed to provide opportunities for the student to carry out this process of translating a physical situation into a corresponding mathematical formulation. In addition, it has been found that students with less than an optimal background can learn considerably from being led through the subsections into which many problems at this level may be decomposed. I am indebted to the many students who, through their questions and comments, have aided significantly in my attempt to develop a pedagogic exposition of this subject. Finally, I wish to thank Gary Geernaert for providing the opportunity for me to assimilate the wonders of modern-day word processing and to my wife Joan for implementing these possibilities so efficiently. G. L . L A M B , J R . Tucson, Arizona January 1995

INTRODUCTORY APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS

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One-Dimensional ProblemsSeparation of Variables Quantities such as temperature, the displacement of a vibrating string or membrane, or the air pressure in a sound wave are examples of scalar quantities that may be evaluated at a specific location in space (of one, two, or three dimensions) as well as at some given time. They thus require more than one independent variable for their specification. The differential equations that must be solved to determine these quantities, for example, the temperature, are relations among the partial derivatives of the temperature with respect to these independent variables and are thus referred to as partial differential equations. The most commonly occurring partial differential equations describe relations among the same relatively small number of physical quantities such as density, displacement, and the change of these quantities that occur in various physical contexts. Hence the same equations recur in various fields, and to obtain a satisfactory introductory perspective of the subject, relatively few partial differential equations need to be considered. In fact, we shall find that there are only three basic types of equations. Success in solving partial differential equations by analytical means has been due in large measure to the fact that the solution can be expressed in terms of solutions of one or more ordinary differential equations. This is accomplished by the method of separation of variables, to be discussed in this chapter, by the method of characteristics, or by the use of various integral transform techniques. These latter two topics will be considered in subsequent chapters.

1.1

INTRODUCTION

Before taking up any detailed consideration of the methods of solution, we first introduce the three basic types of equations. 1. W e will show later (Section 1.8) that the simplest examples of wave motion on a string can be described by solutions of the equation 1 du 2 dx c ι Λdt " Ο Bu 2

2

2

2

λ

0.1.1) 1

2

ONE-DIMENSIONAL PROBLEMS-SEPARATION OF VARIABLES

where u(x, t) represents the displacement of the string from its equilibrium position along an JC axis, c is a constant (shown later to be the velocity at which waves propagate along the string), and t is the time. Henceforth, partial derivatives will frequently be indicated by a subscript notation, that is, u„ = d u/dx , u = d u/dxdy, etc. The equation for the string given above is then written u„ — ( l / c ) « „ = 0. 2

2

2

xy

2

2. The simplest examples of diffusion of heat along a beam will be shown in Section 1.2 to be described by solutions of the equation «„-7" «» = 0

(1.1.2)

2

where u (x, t) now refers to the temperature on the beam and y ~ is a constant that is related to the specific heat and thermal conductivity of the medium. One's intuitive familiarity with wave motion and diffusion leads to the expectation that the two equations listed above describe two fundamentally different types of processes. For instance, a motion picture of the type of wave motion described by (1.1.1) could be run backward and the result would represent an equally likely form of wave motion (the wave merely proceeding in the opposite direction). A typical diffusion process would look fundamentally different, however, when the film is run in reverse. The "undiffusing" would not conform to any phenomenon observed in nature. This fundamental difference is evident in the governing equations as well, since the second time derivative in the wave equation (1.1.1) is unchanged when t is replaced by - / while in the diffusion equation (1.1.2) the sign of the first time derivative is reversed. This seemingly minor change in the equation accounts for the completely different type of behavior exhibited by solutions of the time-reversed diffusion equation. 2

3. The third type of partial differential equation usually occurs in problems that contain no time dependence. As we shall see later in Section 3.4, the equation for the steady displacement of a membrane sagging under its own weight is (1.1.3)

u„ + u„=-pg/T

Here u (x, y) is the static displacement below an xy plane of a membrane that is stretched with a tension Τ while pg is the force of gravity per unit area on a membrane having density p. The combination of derivatives appearing in (1.1.3) is usually abbreviated V w and is referred to as the Laplacian of u. For g = 0 we obtain the equation V u = 0, known as Laplace's equation. Without any force of gravity acting on the membrane (g = 0) we would expect no sagging of the membrane to occur. The satisfaction of Laplace's equation is a statement of this fact. A nonzero value of the Laplacian V w is thus a measure of the sagging or bulginess of the surface of the membrane, u(x, y). In three space dimensions the Laplacian V K = 4- u + u frequently has the 2

2

2

2

yy

a

1.1

INTRODUCTION

3

interpretation of the " l u m p i n e s s " of some density function u(x, y, z). T h e universality of the physical concepts associated with the Laplacian leads one to expect this quantity to appear quite frequently when processes are described by partial differential equations. A more analytical discussion of these three types of equations as well as techniques for determining how other equations may be reduced to one of these three categories will be considered in Chapter 8. Another classification of equations, essentially according to the difficulty of effecting their solution, is the classification into equations for which the coefficients of the various partial derivatives are (1) constants, (2) functions of the independent variables, or (3) functions of the dependent variable. The first two categories are linear equations while the third category yields a nonlinear equation. This classification is also used for ordinary differential equations, and since these equations are presumably already familiar to the reader, we first review these three categories in this context. The ordinary differential equation that describes the oscillation of a frictionless mass-spring system is (1.1.4) where y is the displacement of the system from equilibrium, m is the mass of the oscillator, and —ky is the restoring force acting on the system. If m and k are constant, we have a differential equation with constant coefficients and a familiar example of simple harmonic oscillation. If, on the other hand, k = at, that is, the spring " c o n s t a n t " k varies with time (e.g., the small amplitude oscillation of a pendulum for which the length varies inversely with time), then the equation for the displacement is my" + aty = 0 where the dots indicate time derivatives. T h e variable coefficient makes this equation more difficult to solve than the equation with constant coefficients. If the term k has a dependence upon the displacement itself, such as k = /8y so that my = j3y , then the equation for y is nonlinear. Nonlinear ordinary differential equations are in general the most challenging to solve. Nonlinear partial differential equations present an even greater challenge and, except in a few isolated instances, will not be treated in this book. 2

The categories outlined above for ordinary differential equations are readily taken over to describe partial differential equations. As we shall see in Section 1.8, the vibration of a string of density ρ that is stretched to a uniform tension Τ and is encased in a rubberlike material that provides an elastic restoring force —ky is governed by the equation py„ - Ty

xx

=

-ky

(1.1.5)

where y is the displacement of the string from equilibrium. T h e three categories of equations mentioned above arise once again if w e set the restoring force

4

ONE-DIMENSIONAL PROBLEMS-SEPARATION OF VARIABLES

term k equal to any of the three possibilities: a constant, -a(x, t), or -β(y). One could, of course, have the completely general situation -β(χ, t, y) y. The solution of partial differential equations for these latter three categories can provide very formidable tasks and, except for a few special cases, will not be taken up in this text. From prior experience with ordinary differential equations the reader should already have noted that the solution of an equation may be relatively simple, while satisfaction of boundary or initial conditions may be somewhat cumbersome. Indeed, an advantage of the Laplace transform technique is that it expedites this process for initial-value problems. The role played by boundaries is even more dominant in partial differential equations. The shape of the boundary and the conditions imposed along the boundary impose severe restrictions upon our ability to solve partial differential equations analytically.

1.2

ONE-DIMENSIONAL H E A T CONDUCTION

As a first example of a partial differential equation associated with a simple physical system, we consider heat conduction along a beam of constant crosssectional area A. W e assume that the temperature is uniform over each cross section but may vary along the beam. W e also assume that heat does not radiate out from the sides of the beam, that is, we assume that the sides are perfectly insulated. In the most elementary situation, the heat Η (in some unit such as the calorie) contained in a material of mass Μ that is at a uniform temperature u is expressed as Η = cMu, where c is a constant (the specific heat). If the linear density of the beam, p(x), varies along the beam, then the mass between χ = a and χ = b may be written Μ = A\ p(x) dx. Similarly, if the temperature of the beam varies both with position and time, then the heat on the beam between χ = a and χ = b is h

a

(1.2.1) If we ignore any dependence of ρ (or c) upon either Jtorw and assume that they are constant, the change in H (t) with time is given by ah

(1.2.2) The change in H can take place because of a flow of heat along the beam into or out of the region a < χ < b as well as because of source terms (such as a radioactive source embedded in the beam) that introduce additional heat into the region between a and b. Heat flow along the beam, J(x, t), measured, say, in calories per square ab

1.2 ONE-DIMENSIONAL HEAT CONDUCTION

5

centimeter per second, is most easily analyzed by considering flow in the positive χ direction. Then heat enters the region at χ = a and leaves at Λ: = b. For a beam of area A we have dH

ab

- A(J

out

t) - J φ, t)] + A t s(x, t) dx

= A[J(a, = -A

+ sources

- J )

in

\

I

Jo

OX

dx + A

s(x,t)dx

(1.2.3)

Ja

where s (χ, ή is a heat source density in calories per second per cubic centimeter. The replacement of the difference J (a, t) — J φ, t) by an integral over J (x, t) is the one-dimensional version of the divergence theorem that would be used if we were deriving the equation for heat diffusion in three dimensions. Equating the definition of dH ldt from (1.2.2) with the expression for it obtained in Eq. (1.2.3) we have x

ah

f

{pcu, + J - s) dx = 0 x

(1.2.4)

Since the interval a < χ < b is arbitrary, the vanishing of the integral implies the vanishing of the integrand itself, and we obtain 1

pcu, + J

x

= s(x, t)

(1.2.5)

This is the equation for the conservation of heat on the beam. T o obtain a partial differential equation for the temperature u(x, f), we must introduce an assumption concerning the dependence of the heat flow J(x, t) upon the temperature. Since heat flows from hotter to cooler regions, we can expect J to depend upon the gradient of the temperature, u (x, t), in some way. T h e simplest relation (due to Fourier and substantiated by considerable experimental evidence) is the linear one x

J(x, t) = -Ku (x, x

t)

(1.2.6)

The proportionality parameter K, known as the conductivity, may depend upon both χ and u. Combining this assumption for J(x, t) with the conservation 'To conclude otherwise would lead to a contradiction. Since u(x, t) and the entire integrand in (1.2.4) is presumably a continuous function of x, a nonzero, say positive, value of the integrand at some point x would imply an interval about x in which the integrand is also positive. By placing the limits of the integrand a and b in this interval, we would have a nonzero value for the integrand in (1.2.4), thus providing a contradiction. 0

0

6

ONE-DIMENSIONAL PROBLEMS-SEPARATION OF VARIABLES

equation (1.2.5) we have pcu, — {Ku )

x x

(1.2.7)

= s(x, t)

This result is the partial differential equation for one-dimensional heat conduction. If there are no sources, so that s(x, t), = 0 and Κ is a constant, we obtain γ

2

=

(1.2.8)

Klpc

which is the equation quoted in (1.1.2). 1.2.1

Boundary and Initial Conditions

As with ordinary differential equations, the association of a partial differential equation with a specific experimental situation requires the imposition of certain additional conditions. For the heat conduction situation just described we will be required to give information concerning what happens at each end of the beam at all times, the boundary conditions, as well as to specify an initial condition, u(x, 0), the temperature distribution along the entire beam at some time labeled t = 0. The most common boundary conditions are that either u or J, where / = —Ku , be specified at each end. If an end is perfectly insulated, then J = 0 at that end and thus u = 0 there. Specification of u or u is referred to as a Dirichlet or a Neumann boundary condition, respectively. If the heat flow at an end, say at χ = 0, is proportional to the temperature at the end, then a linear combination of u and u is specified there, that is, κ ( 0 , t) = aJ(0, t) = -aKu (0, t) where α is a proportionality constant. The two cases mentioned previously, that is, « or « , = 0, than arise in the limits a -* 0 and a -* oo, respectively. x

x

x

x

s

Note that a background or reference temperature may always be ignored in problems governed by the diffusion equation (1.2.8). If the ends of the beam are maintained at temperature u , for instance, then we may set u = M + ϋ(χ, t) and determine the function I9(JC, r), which is zero at each end. Since only derivatives of u appear in the diffusion equation (1.2.8), the temperature increment d satisfies the same diffusion equation as that satisfied by u. After obtaining ϋ we need only add the constant u to obtain the temperature u(x, t). 0

0

0

1.3

STEADY STATE SOLUTIONS

Whenever a new equation is encountered, it is usually instructive to examine that equation and its solutions in some limiting cases that are readily understood. A simple limit of this sort for the heat equation is the so-called steady state situation in which there is no time dependence, that is, w, = 0. When we consider the case of constant conductivity K, the heat equation (1.2.7) reduces, in the steady state situation, to the form (1.3.1)

1.3 STEADY STATE SOLUTIONS

7

which is merely an ordinary differential equation. Since a source might be expected to heat the beam and thus render any steady state situation impossible, some restriction must obviously be satisfied in order to have a steady state temperature distribution on the beam. Requirements that apply to an entire system may frequently b e obtained by integrating the governing equations over the extent of the system. If we consider the steady state heat equation (1.3.1) as it applies to a beam of length L and integrate the equation over the length of the beam, we obtain

~K \ ιΐχχ dx = \ s(x) dx = Η Jo

-K[u (L) x

Jo

- u (0)] x

= Η

(1.3.2)

where Η (in calories per second) is the rate at which heat is being deposited on the entire beam. According to the definition of heat flow 7 given in (1.2.6), the result obtained in (1.3.2) may be written - 7(0) = Η

J(L)

(1.3.3)

We thus obtain the physically reasonable result that for a steady state situation to prevail, the rate of the heat flow out the ends of the beam must equal the rate at which heat is supplied to the entire system by the source s(x). Note that since heat is assumed to flow off of the beam, 7(0) represents a heat flow in the negative χ direction and thus - 7 ( 0 ) is a positive quantity for heat flow out of the beam at χ = 0. If there is no source term s(x), then Η = 0 in (1.3.3) and the steady state situation requires that the boundary condition satisfy 7(0) = J(L). W e thus find that for steady state problems the boundary values of the derivative of u may not be imposed arbitrarily. If we consider a steady state problem in which the temperature itself is specified at each end rather than the heat flow, then the condition on 7 given in (1.3.3) will be automatically satisfied by the steady state solution that is obtained.

Example 1.1. The end points of a beam of constant conductivity Κ are located at χ = 0 and L and are maintained at temperatures 0 and u , respectively. An amount of heat Q calories per second is continually supplied over the length of the beam with a density s(x) = (7rQ/2L)sin irx/L. [Note that J&sC*) dx = Q.] Determine the steady state temperature of the beam. From (1.3.1) we require the solution 0

= - a sin ττχ/L,

a = icQI2KL

(1.3.4)

subject to the boundary conditions u(Q) = 0, u(L) = w„. This ordinary differential equation is sufficiently simple that it may be solved by direct integration to yield u = a(L/ir) sin 2

irx/L + ax + b

(1.3.5)

8

ONE-DIMENSIONAL PROBLEMS-SEPARATION OF VARIABLES

where a and b are constants of integration. (We may also think of the first term as a particular solution and the last two terms as the solution of the homogeneous equation u = 0.) Applying the boundary conditions we find «(0) = b = 0 and then u(L) = aL = u . Recalling the definition of a from (1.3.4) we obtain the solution a

0

u(x) = (LQI2*K)un

(1.3.6)

τχ/L + u x/L 0

Determination of the derivative of this result and evaluation at χ = 0, L show that the steady state condition (1.3.3) is satisfied. As a final result, note also that if there is no external source (Q = 0), the steady state temperature due to the boundary conditions for this problem is the linear relation u(x) = u x/L. 0

Example 1.2. The boundary conditions on both ends of the beam considered in Example 1.1 are changed. At χ = L the insulated end condition u (L) = 0 is imposed. If the source term remains the same, that is, s(x) = (vQ/2L)un τχ/L, determine the value that must be imposed upon u (0) so that a steady state condition is maintained. Then show that with these values of the u imposed at the ends, the temperature of the beam is only determined to within a constant. For the steady state condition (1.3.3) to be satisfied with u (L) = 0 and Η = β s(x) dx = Q, we require u (0) = QIK. To determine u(x) we integrate (1.3.4) once and obtain u = ( β 12Κ)cos τχ/L + c . The integration constant c, is found to equal Q/2K when we impose either of the boundary conditions «^(0) = QIK or u {L) = 0. A second integration yields x

x

x

x

x

x

x

x

«(*) = (LQI2icK)un τχ/L + Qx/2K + c

2

(1.3.7)

The constant of integration c is thus left undetermined. 2

Problems 1.3.1

(a) Determine the steady state temperature on a beam if κ(0) = = 0, and the source is

six)

=

(s , 0

0 < χ < L/2

CO,

L/2 < χ < L

u(L)

(b) Determine the ratio of the heat flow off the beam at χ - 0 to that at χ = L, that is, \J(0)\/J(L). (c) Where is the temperature a maximum? (d) Sketch the temperature profile on the beam. 1.3.2

Reconsider the previous problem when the boundary at χ = L is insulated, that is, u (L) = 0. x

(a) Show that all heat flows off the end at χ = 0, that is, J(0) - # κ , ( 0 ) = s L/2. (b) Sketch u(x). 0

=

1.3

1.3.3

STEADY STATE SOLUTIONS

9

A beam of length L is located between χ — a and χ = b = a + L. The conductivity of the beam varies linearly along the beam and is given by Κ (χ) = K x/L. The sides of the beam are insulated and the ends are maintained at zero temperature. The beam is subjected to a steady, spatially uniform heat source Q calories per second over its entire length. The equation for the temperature on the beam is thus 0

0

d_ dx

du

K(x)-

Qo L

dx

(a) Use the relation J — -K(x) duldx to determine the amount of heat flowing out each end of the beam and show that the total heat flow out both ends is Q . Q

(b) Show that if b = 2a so that K(b)IK{a) 2 In 2 - 1

Jib) \J(a)\ 1.3.4

1 - In 2

= 2,

s

1.26

The ends of a beam of length 2L are maintained at zero temperature at χ = ±L. A source deposits Q calories per second over the length of the beam with a heat density 0

s(x)

=

Show that the steady state temperature on the beam is

u(x)

=

QQL

7Γ

2 ~ Z

-I s

,

n

I

II

-

(This problem is a limiting case of a time dependent problem considered in Section 9.1.) 1.3.5

A beam of length L is maintained at zero temperature at χ = 0. At χ = L there is a constant heat flux J onto the beam. At each point along the beam there is a heat loss vu(x) where ν is a constant and u(x) is the temperature on the beam at that point. This loss term may be interpreted as a negative heat source s(x) = — vu(x). Hence the steady state temperature on the beam is governed by 0

=

(vlK)u

Show that the heat flux off of the beam at χ = 0 is equal to J

0

sech

10

1.3.6

ONE-DIMENSIONAL PROBLEMS-SEPARATION OF VARIABLES

A beam of length 2L is kept at zero temperature at the ends located at χ = ±L. The beam is heated by the steady source Q(x) = (H /2a) sech (xla). 0

2

(a) Show that the steady state temperature on the beam is H a , cosh Ua u(x) = —— In — - — — 2Κ cosh xla 0

(b) Show that in the limit alL « 1, in which the source may be described as a "point s o u r c e , " the temperature becomes

«—>·

0 < χ < L

u(x) = -L

< χ < 0

(c) Sketch the temperature u(x) obtained in (b) as well as the function Q(x) in the limit alL « 1.

1.4 TIME DEPENDENT HEAT F L O W - S E P A R A T I O N O F VARIABLES The one-dimensional steady state examples of the previous section involved a single independent variable and thus required only the solution of an ordinary differential equation. When the temperature on the beam depends upon both space and time, a partial differential equation of the type given in (1.2.7) or (1.2.8) must solved. Such equations may frequently be solved by relating them to ordinary differential equations in each of the independent variables. One technique for carrying out this simplification is known as the method of separation of variables. In adopting this approach to solving a partial differential equation we put aside, temporarily, the goal of obtaining a complete solution that satisfies both the initial and boundary conditions of a specific problem. We merely attempt to obtain some sort of solution in terms of functions X(x) and 7 ( 0 , each of which depends upon only one of the two independent variables χ and t. How u(x, t) is to be decomposed into two such functions is a combination of guesswork and experience. For the linear equations that we shall encounter in this text, the decomposition is usually made in terms of the product expression u(x, ή = X(x)T(t). For nonlinear equations the combination may be quite different (cf. Problem 1.4.10). An advantage of the separation-of-variables method is that partial derivatives of u(x, t) are replaced by ordinary derivatives of X(x) and T(t), for example, the partial derivatives that appear in the diffusion equation (1.2.8) become u,(x, t) = XdT/dt and u^ix, t) = (d X/dx )T. We have no a priori expectation 2

2

1.4

TIME DEPENDENT HEAT FLOW-SEPARATION OF VARIABLES

11

that such solutions can be combined so as to satisfy the boundary and initial conditions of any given problem. H o w Fourier series techniques are to be used to carry out the superposition of these elementary solutions so that initial and boundary conditions can be satisfied for linear partial differential equations is the main topic of this chapter.

1.4.1

Elementary Solutions

Using the forms of u„ and u, that result from the product representation for u(x, t), the diffusion equation "«-7~ ", = 0

(1.4.1)

X"T

(1.4.2)

2

can be written as = y- XT' 2

where, as noted above the primes indicate a derivative of a function with respect to the associated independent variable. On dividing this equation by XT we obtain

*1 -

LIL „

2 2

X ~ y

d-4.3)

Τ

W e have now derived an equation that separates the spatial and temporal aspects of the problem. Since χ and / are independent variables, we may assign values to each of them at will. For equality of both sides of the equation to persist for any choice of χ and t, each side must be a constant, the so-called separation constant. Although there are some instances in which useful solutions are obtained by allowing the separation constant a to be imaginary (cf. problem 1.4.9), we shall assume in the following that it is real and write it in the form ±a . The proper choice of sign will be determined subsequently. Equating each side of (1.4.3) to this constant, we obtain the pair of relatively simple ordinary differ2

2

2

An alternate way to arrive at this result is to differentiate (1.4.3) with respect to one of the variables, say x, and obtain 2

dx X ~

7

2

dx Τ ~

since X is a function of χ alone, d/dx(X"/X) = dldx (X"/X) and integration with respect to χ yields X"/X = const. Similar usage of a time derivative leads to T'/T = const, and by (1.4.3) these two integration constants must be equal.

12

ONE-DIMENSIONAL PROBLEMS-SEPARATION OF VARIABLES

ential equations X" =F a X

= 0

2

Γ Τ (αγ)

2

(1.4.4)

Τ = 0

For the two possible choices of sign as well as a = 0, the solutions of these equations are 2

X = Ae

+a :

X = A cos ax + Β sin

= 0:

X = Ax + B,

2

ax

2

a

2

+

Τ =

-a :

Be~ , ax

ax,

T

=

C

Τ =

Ce ' (ayY

e

-

{

a

y

)

2

'

(1.4.5)

c

where A, B, and C are integration constants. For each of these three cases, the function u = XT is now a very specialized solution of the diffusion equation (1.4.1). We note in passing that the possibility of dividing by zero in the separated form (1.4.3) is taken care of by the fact that X" vanishes whenever X does. Thus, we merely obtain an indeterminate form for the ratio X"/X. The choice of sign associated with a as well as specific values of α are determined when we impose the further restriction that the elementary solution u(x, t) = X(x)T(t) satisfy the boundary conditions for a given problem. In the product form for u(x, t), this boundary information is incorporated into the function X(x). If, for instance, we are considering diffusion of heat on a beam of length L that is maintained at zero temperature at the end points χ = 0 and L, then we impose the requirements X(0) = X(L) = 0. Let us turn immediately to the second of the three solutions listed in (1.4.5). This choice will be seen to be the proper one. The reason why the other two must be discarded will be given below. For the second solution we have X(0) = A = 0 and X(L) — A cos <xL + Β sin aL = 0. From the condition at χ = 0 we have A = 0. The condition at χ = L then reduces to X = Β sin aL = 0 and we thus see that either Β = 0 or sin aL = 0. With the first choice we would have a zero value for both A and Β and thus X = 0. The whole solution u = XT would then be zero, an uninteresting result. The second choice, namely sin aL = 0, does allow for nonvanishing solutions since it is satisfied by aL = nw, where η = 0, ± 1 , ± 2 , . . . . For each value of η we then obtain a different possible expression for X(x) as well as a corresponding expression for T\t), namely, 7\t) = C exp[—(nicy/L) t]. The solution for u(x, t) is then 3

2

2

u„(x, t) = X„T„ = b e- ' ' {nin LY

n

sin (ητχ/L)

(1.4.6)

'Since - o = (iaf, the first two forms of the solution may be transformed into each other by using β*" = cos ax + i sin ax and then redefining the constants of integration. Also, if we set Β = Β Ία in the second solution and then let a -* 0, we obtain X = A + Β 'x, Τ = C, which has the form of the third solution. 2

1.4

TIME DEPENDENT HEAT FLOW-SEPARATION OF VARIABLES

13

where we have labeled the solutions by the value of η and have combined the integration constants BC into a single constant b„ that can be different from each value of n. The value η = 0 may be discarded since it merely yields the solution u = 0 . Negative values of η may also be discarded. They provide no solution that could not be obtained by merely changing the sign of b„. The specific values of a that enable us to obtain the nonzero solutions X„ ( i . e . , a = a„ — ηπ/L in the present instance) are frequently referred to as eigenvalues. The solutions X„ themselves are referred to as eigenfiinctions. We now briefly consider the other two possible solutions that were listed in (1.4.5). When the first solution is required to vanish at χ = 0, we obtain A + Β = 0. To vanish at χ = L the solution must also satisfy Ae~ + Be~ = 0. This pair of homogeneous algebraic equations for A and Β has no solution for real a other than A = Β = 0. As noted in footnote 3 , solutions may be obtained for purely imaginary values of α but they are merely the same as those obtained from the second solution in (1.4.5). Similarly, for the third solution in (1.4.5) we obtain X(0) = Β = 0 and X(L) = AL + Β = 0. Here again the two constants A and Β must vanish. Hence, only the second solution in (1.4.5) is useful in solving this problem. It is worth examining the solution in (1.4.5) a bit further and noticing that it conforms to our intuition. Since heat conduction is a diffusion process, we expect that any temperature variations will tend to smooth out as time progresses. The more fine grained the variations in the temperature, the more rapidly the variation should smooth out. In the solutions to the diffusion equation listed in (1.4.6), the larger the value of n, the more rapidly the spatial term oscillates. As expected, the associated time factor exp[-(nny/L) t] decays more rapidly in time the larger the value of n. aL

aL

2

1.4.2

Synthesis of Elementary Solutions—Fourier Series

The solutions given in (1.4.6) describe very specialized situations. They show that an initial temperature distribution of the form u (x, 0) = b &in nirx/L decays in time as u„(x, t) = b„ e x p [ - ( n w y / L ) 1 ] s i n nirx/L. T o approach the more interesting problem of describing the temporal variation of essentially arbitrary types of initial temperature distributions, w e exploit the fact that the partial differential equation (1.4.1) is linear. As with linear ordinary differential equations, a sum of solutions to a given equation is still a solution to that equation. In the case of a linear partial differential equation we can have an infinite number of solutions such as those given in (1.4.6) for η = 1 , 2 , 3, . . . . W e can thus write a solution to the original partial differential equation as a sum of all of these solutions. This procedure yields n

n

2

u(x, t) =

Σ n= 1

u„(x, t) =

Σ η= I

b„e~' '" sin ^

(1.4.7)

{1

L

14

ONE-DIMENSIONAL PROBLEMS-SEPARATION OF VARIABLES

where (1.4.8)

In so doing, we have written the solution in the form of a Fourier series. Each term in the series has the time-varying coefficient b„ exp(-n vt). At / = 0 the coefficients are just b„ and, as will now be shown, can be determined by using the techniques available from the theory of Fourier series and summarized in Appendix A. When the solution (1.4.7) is considered at t = 0, we have 2

00 u(x, t) = Σ b un— n=i L

(1.4.9)

n

and from the results given in Eq. (A.23) we have Κ = 7 ( u(x, 0)sin ^ dx L Jo L

(1.4.10)

We now consider two examples of the use of Fourier series methods for solving an initial-value problem in heat conduction. Example 1.3. Determine the temperature distribution at any time on a beam of length L if the initial temperature distribution is u(x, 0) = w sin irx/L and the ends are kept at zero temperature. Also show that after a long time (i.e., as / -+ <x) half of the initial heat on the beam will have flowed off at each end of the beam. The coefficients b„ are obtained by evaluating the integral in (1.4.10) for the given form of u(x, 0). The calculation is particularly simple in the present case when use is made of the trigonometric identity 4 sin ϋ = 3 sin ϋ - sin 3d. The two integrations required in evaluating (1.4.10) are in the form of orthogonality integrals given in Eq. (A.2) with d = -L and ρ = L. Since the integrand is even in χ we have 3

0

3

(1.4.11) and similarly (1.4.12)

1.4 TIME DEPENDENT HEAT FLOW-SEPARATION OF VARIABLES

15

while all other h„ vanish due to orthogonality. In this example, then, the solution (1.4.7) is not an infinite series at all but merely «(*, ') = "o

e - sin ψ

- - e-"" sin

(1.4.13)

l

This solution is readily shown to predict results that are in conformity with our intuition. As an example, note that the symmetry of the initial temperature distribution about the midpoint of the beam suggests that half of the initial heat on the beam will flow off each end of the beam. From the definition given in (1.2.1), the total amount of heat initially on the beam is 4

7TJC

4

sin — dx = — pcu LA \f sin' L 3π

W(0) = pcu A

3

0

(1.4.14)

0

Jo

The total amount of heat flowing off one end, say at χ = L, is obtained by integrating the expression for J(L, t) given by (1.2.6) overall time. Using the solution just obtained in (1.4.13) to calculate u (L, t), we have x

J

OO

/»oo

dt

ο

J(L, t) dt = -KA

u (L, x

3*KAuiJo C

4L 2

— 3tt

(

e

t) - „ _

e

-

n

d

t

(

1

4

1

5

)

JO

pcu LA 0

which is just one-half of the total heat initially on the beam. An initial temperature profile that leads to a Fourier series containing an infinite number of terms is provided by the problem w(0, /) = u(L, t) = 0, u(x, 0) = UQ which can be used to describe a beam initially at a uniform temperature M h i d s suddenly set to zero temperature. From (1.4.10) the coefficients b„ are given by t n a t

a s

{ s e i ,

0

2M f . ηπχ 2u b„ = — I sin dx = — (1 - cos ηπ) L Jo L ηπ L

0

f4«o 7Γ

0

η = 1, 3 , 5, . . .

j

(1.4.16)

0,

η = 2 , 4 , 6, . . .

The vanishing of the terms with even values of η is evident on the basis of simple symmetry considerations. Graphs of u(x, 0) and the functions sin ητχ/L 4

Sct f = irx/L and use the result {J sin ξ άξ = \. 3

16

ONE-DIMENSIONAL PROBLEMS-SEPARATION OF VARIABLES U(X,0) I

0

Figure 1.1. Initial temperature profile and first few mode shapes.

L/2

0

L/2

L

L

for the first few values of η are shown in Figure 1.1. The even values of η are seen to correspond to functions that are odd with respect to the midpoint of the beam at χ = L/2. Since the initial temperature distribution is even with respect to the midpoint, the product u (x, 0) sin ητχ/L is odd with respect to the midpoint χ = L/2 for η = 2, 4 , 6 The integrals in (1.4.16) over the range 0 < χ < L thus vanish as a result of this antisymmetry. The first few terms in the series for u(x, t) are thus 4«ο / _„, . irx 1 _,„. . 3irx 1 _,„. . 5τχ \ u(x, t) = — [e "' sin — + - e "' sin — + - e " sin — + . . . π \ L 3 L 5 L / 3

5

(1.4.17) where again ν = ( π γ / L ) . The exponential time dependence of the coefficients in this series provide rapid convergence for vt > 1. For vt « 1, however, the series converges only as n , which is quite slow. In Section 2.2 we will develop a completely different representation of the solution that does converge rapidly at these early times. According to the form of the solution given in (1.4.17), the temperature at 2

- 1

5

Atx = L/2 and as / approaches zero, the solution in (1.4.17) approaches u(L/2, 0) = 4«ο/τ(1 - j + 5 — · · ·) = 4 M O / I T , TT/4 = H . The series representation for τ that is encountered here is known to be veiy slowly convergent. In identifying the value of the series evaluated at t = 0 with the initial condition of the problem, it should be noted that we are assuming the validity of the interchange lim Zu„(x, t) = Σ lim u„(x, t). In the present instance this interchange may be justified by using Abel's test (cf. Section A.3). 5

0

1.4

TIME DEPENDENT HEAT FLOW-SEPARATION OF VARIABLES

17

any given point along the beam is represented by a sum of terms each of which has its own characteristic rate of decay. Note in particular that at the locations χ — LIZ and χ = 2L/3 along the beam the second term in the series vanishes at all times. These two points on the beam would thus be the most appropriate locations at which to obtain experimental information. Since the second term in the series is absent at either of these locations, the decrease in temperature would be given in terms of the single exponential term 4(u /n) e~"' sin πχ/L after the brief initial time beyond which the term proportional to e~ ' could be neglected. 0

25v

1.4.3

Changing the Boundary Conditions—Insulated Ends

If an end of the beam is covered with some insulating material (asbestos was customarily allowed at one time), then no heat flows out of the beam at that end and J = —Ku = 0 there. W e thus have the boundary condition u = 0 at a perfectly insulated end. T o incorporate this information into the solution of a heat diffusion problem, we must return to the solution of the ordinary differential equation satisfied by X(x) in the second of (1.4.5), namely, X(x) = A cos ax + Β sin ax, and apply the new boundary condition. In particular, if both ends are insulated, we set X'(0) = X'(L) = 0. For the given form of X(x) we obtain x

x

X'(0)

= aB = 0

X'(L)

= a(-A

sin aL + 5 cos aL)

(1.4.18)

which yields Β = 0 and a = ητ/L where η = 0, 1, 2, . . . . The solution is thus X(x) = A cos πχ/L. Note that the solution η = 0 must be retained in the present instance since the cosine does not vanish for this value of n. T h e solution obtained for η = 0 is also provided by the third of the three solutions listed in (1.4.5). The elementary solutions for a beam with two insulated ends are thus u„(x, t) = a^-" "' 2

cos ~ , la

η = 0, 1, 2, . . .

(1.4.19)

with ν — (iry/L) . The term η = 0, since it is time independent, has an important physical significance. The insulated ends on the beam prevent heat from escaping from the system. As t -» oo, any initial temperature distribution will thus have smoothed out to a constant value over the entire beam. The value of that constant temperature is determined by the conservation of heat on the beam. This result is evident physically and also from an examination of the coefficients in the Fourier series. From Eq. (A.24), these coefficients 2

18

ONE-DIMENSIONAL PROBLEMS—SEPARATION OF VARIABLES

are given by «o = 7 \

i

u(x, 0) dx = (u(x, u{x

ί

0)>

'

«• = 7 «(*, 0)COS — dx, L So Jo L ' H

a

η = 1, 2, 3 . . .

X

(1.4.20)

The coefficient OQ is seen to have the interpretation of being the average temperature on the beam.

Problems 1.4.1

Determine the eigenvalues and eigenfunctions for the boundary value problem y" + k y = 0, y'(0) = y(L) = 0. Sketch the first three eigenfunctions. 2

1.4.2

A beam of length L has the initial temperature distribution 0 < χ < L/2 L/2 < χ 0 having the boundary condition « ( 0 , t) = M COS Ωί. Show also that similar considerations apply to the imaginary part. 0

(d) Show that if the heat flow at χ = 0 is given by 7 ( 0 , t) = r) = 7 c o s Of, then 0

u(x, t) =

Joy

cos

Ωί

-

y/W2x 7

-Ku (0, x

20

1.4.10

ONE-DIMENSIONAL PROBLEMS-SEPARATION OF VARIABLES

A more elaborate example of the separation-of-variables procedure is provided by the nonlinear equation xx

yy

~

u

u

Sin U

=

(a) Introduce a solution in the form u = 4 tan" and use the identity sin 4a = 4 tan a (I - tan a)/(l + tan to show that X(x) and Y(y) are related according to 2

γ ll

ν 2

XX"

2

a)

2

Y^Y "

+ —— =

+ YY" + ——

2(X'

+ Y' )

2

+ Υ

2

2

-

X

2

(0 (b) Differentiate this result with respect to both χ and y to obtain the separated equations (Υ"ΐγγ

(X"/xy

(c) Integrate these equations to obtain (X')

= -\mX

(Υ'Ϋ

= \mY?

2

+ aX

A

+ β

2

+aY

2

(iii)

+ b

where α, β, a, and b are constants of integration. (d) Substitute (ii) and (iii) into (i) to obtain b -

-β

maa

(e) Show that the special case m = 0, α = 1, β = -A

2

= a yields

_i cosh (x + C|)

u = 4 tan"

y + c

2

= 2ir - 4 t a n

- 1

[(y + c ) sech (x +