Mathematical Foundations for Electromagnetic Theory

IEEE PRESS Series on Electromagnetic Waves ,

I

.

I

The IEEE PRESS Series on Electromagnetic Waves consists of new titl~s as well as reprints and revisions of recognized classics that maintain long-term archival significance in electromagnetic waves and applicatior/s.

i

Series Editor Donald O. Dudley University of Arizona

Mathematical Foundations for Electromagnetic Theory

Advisory Board Robert E. Collin Case Western University

....

Akira Ishimaru University of Washington

Associate Editors Electromagnetic Theory, Scattering, and Diffraction Ehud Heyman Tel-Aviv University Differential Equation Methods Andreas C. Cangellaris UniversIty of Arizona

Donald G. Dudley University of Arizona, Tucson

Integral Equation Methods Donald R. Wilton Universi,ty of Houston Antennas, Propagation, and Microwaves David R. Jackson University of Houston

Series Books Published Collin, R. E., Field Theory of Guided Waves, 2nd ed., 1991 , Dudley, D.O., Mathematical Foundations for Electromagnetic Theory, 1994 Elliott, R. S., Electromagnetic.~: History, Theory, and Applications, 1993 Felsen, L. B., and Marcuvitz, N., Radiation and Scattering of Waves, 1994 Harrington, R. E, Field Computation by Moment Methods, 1993 Tai, C. T., Dyadic Green Functions in Electromagnetic Theory, 2nd ed., 1993 Tai, C. T., Generalized Vector and Dyadic Analysis: Applied Mathematics in Field Theory, 1991

IEEE PRESS Series on Electromagnetic Waves Donald G. Dudley, Series Editor

IEEE PRESS

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lEEE Antenrias ancl Propagation Society, Sponsor

,IEEE PRESS 445 Hoes Lane, PO Box 1331 Piscataway, NJ 08855-1331 1994 Editorial Board William Perkins, Editor ill Chief J. D: Irwin S. V, Kartalopoulos P. Laplante A. LLaub M. Lightner

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J. M. F. Moura I. Pcden E. Sanchez-Sincncio L. Shaw D. J. Wells

Dedicated to Professor Robert S. Elliott

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© 1994 by the Institute of Electrical and EI,ectronics Engineers, Inc. 345 East 47th Street, New York, NY 10017-2394 All rights reserved. No part of this book may be reproduced in any form, nor may it be stored in a retrieval system o~ transmitted in any fonn, without written permission from the publisher. Printed in the United Stales of America 10

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ISBN 0-7803-1022-5 IEEE Order Number: PC03715, Library of Congress Cataloging.in.Publication Data Dudley, Donal:¥jYj

(y -

( 1.68)

We write (1.68) in matrix form as follows:

r

....

= 1,2, ... , M

k

j=l

(Yl,Yl) (Yl, Y2)

y

Fig. 1-2

Sec. 1.6

k = 1, 2, ... , M

0:67)

can have at most N - 1 roots. Therefore, the only solution valid for all r is.B" = 0, n = 1,2, ... , N. We wish to approximate fer) by

E

(0, I)

N

fer) = Lan rn - t

(1.70)

n=l

All possible suc~ linear combinations form a closed linear manifold so that the projection theorem applies. Comparing (1.66), we identify (1.71)

Linear Analysis

24

'Chap. I

Sec. 1.7

Operators in Hilbert Space

25

The solution is given formally by

Define an inner product for £2 (0, :1) by

x=A-Iz

Jol

f(r)g(r)dr

I

(1.72)

The matrix operation is linear. Indeed, given XI, X2, ordinary matrix methods

We then have

1

1 m-l

= o ,r

(Ym, Yn)

r

1 1

(y, Ym)

=

n-Id

m

r

=

1

m+n-

1

i

!

1

r - f(r)dr

I

(1.74)

I

Substitution of (1.73) and (1.74) into (1.69) gives 1

"2

1

1

[1

'I ]

3"

: (1.75)

1 N+I

Inversion of this matrix equation yields the best approximation.

I

j

1

1.7 OPERATORS IN HILBERT SPACE Consider the following transfonnation in R 3 :

!

= CXII~I + CX12~2 + CX13~3 ~2 = CX21 ~ I + CX22~2 + CX23~3 ~3 = CX31~1 + CX32~2 + CX33~3 ~I

I

A(cxx]

, d.73)

•

+ {>x2) = cxAxl + f3Ax2

Ir, l'

~2

~3 f

x =:± [~I

~2

~3 f

where T indicates matrix transpose. We then have

"

Ax

where "

i ;

=z

+ f3Z2

The domain of the operator L is the set of vectors x for which the mapping is defined. The range of the operator L is the set of vectors y resulting from the mapping. The operator L is linear if the mapping is such that for any XI and X2 in the domain of L, the vector CXIXI + CX2X2 is also in the domain and L (CXIX]

+ CX2X2) = CX] LXI + cx2Lx2

0.77)

A linear operator L with domain V L C 1i is bounded if there exists a real number Y such that (1. 78) IILull ~ Yllull for all u

E

VL.

EXAMPLE 1.24 Let Roo be the space of all vectors consisting ~f a countably infinite set of re~1 numbers (components), viz. a

where ak

E

R. If b

=

(ai, a2, a3 ...)

(1.79)

= (tiI, tho th. ...), define an inner product for the space by , 00

z = [~I

= CXZI

and Z2, we have by

The concepts of linearity and inversion for matrices can be generalized to linear operators in a Hilbert space. An operator L is a mapping that assigns to a vector x 'E S another vector Lx E S. We write Lx = y 0.76)

Using the usual matrix notation, we let

I

Zl,

(a,b) =

i

Ladh

(1.80)

k=1

Let the norm for the space be induced by the inner product. We restrict Roo to those vectors with finite norm. Define the right shift operator A R in Roo by

ARa = (0, ai, a2,

...)

The right shift operator A R is linear. The proof is easy and is omitted. In addition, A R is bounded. Indeed,

Sec. 1.7

Linear Analysis

26

i· Therefore, the operator A R is boundetl in Roo. Indeed, the least upper bound on : is unity.

y

EXAMPLE 1.25 On the complex Hilbert space £2 (0, I), we consider the! following integral equation:

10t

:

u(~')k(~, ~/)d( = /(0

A linear operator L with domain V L C 7{ is continuous if given an > 0, there exists a 0 > such that, for every Uo E V L, II Luo ~ Lu II < E, for all u E VI. satisfying lIuo - u II < o. We can interpret this definition to mean that when an operator is continuous, Luo is close to Lu whenever Uo is close to u. There is an important theorem on interchange of operators and limits that follows immediately from the above definition. A linear operator L with domain V L C 7{ is continuous if and only if for every sequence {un }~1 E V L converging to Uo E V L ,

°

E

:.

(1.81)

, This equation can be written in operator notation as follows: Lu

Luo

=/

L =

1

(.

1f

Ik(t

)k(~, ~/)d(

(1.82)

II Lu o - LU n II
00

This first part of the proof shows that, if an operator is continuous, the operator and the limit can be interchanged. In the second part of the proof, we must show that if the operator and limit can be interchanged, the operator is continuous. This part is not essential to our development and is omitted. The interested reader is referred to [12 J. . We now give a theorem linking the boundedness and continuity of operators. A linear operator L with domain V L C 7{ is bounded if and only if it is continuous. The proof is in two parts. In the first part, we show that if the operator is bounded, it is continuous. Indeed, if L is bounded and Uo E V L ,

IILuo - Lull

Ik(~, nI2d~'

= IIL(uo ~ YIIuo -

It follows that:

for all u

E

V L . Then, given any

E

ull

> 0, it is easy to find a 0 :> Osuch that

IILuo - Lull < and finally,

u)1I

E

whenever IIL~II::::

lIuo - ull < 0

Mllull

where M is the bound on the double integral.

•

Indeed, the choice 0 = E/ JI is sufficient. In the second part of the proof, we must show that if an operator is continuous, it is bounded. This part is

I

Sec. 1.7

Linear Analysis

28

not essential to our development and is omitted. The interested reader is referred to [13]. It is straightforward to show that the differential operator L = d/d~ is unbounded. The proof is by contradiction. We suppose that d/d~ is bounded. Then, it is continuous. Therefore, for any Un ~ u, we rhust have Lu = L lim Un = lim LU n n->oo n->oo 1

Un =

11

, Il (IltTlJo I>kLZk. Zj) = (J, Zj), ,

we have lim Un

n->oo

L n->oo lim Un But, lim LU n

n---+oo

l'

=

1,2, ...

"'J [(xIJ (X2 = [(J,ZI)J (J, Z2)

...

.. .

j(~) = -I·W(~) +

I

where

k(~, n = ~(l -

l' k(~, nu(nd~' O::s~::sf

$'),

nl-n

~'

::s ~ ::s I

(1.88)

(1.89)

(1.90)

In operator notation,

(1.85)

j,

~ E

(0, I)

(1.91)

where / is the identity operator. The operator L - Ji-/ is bounded. We leave the proof for .Problem 1.26: We wish to obtain the matrix representation and thereby solve th.e mtegral equatIOn. We define the inner product for the space as in (1.63). For baSIS functions, we choose

Since boundedness implies continuity, 'n

(L - Ji-l)u =

n

= L n->oo lim L(XkZk = lim L L(XkZk k=l n->oo k=l

Zn = sinmr~,

By the linearity of the operator L, we then have

n = 1,2, ...

(1.92)

Then, operating on any member of the basis set, we obtain

:6.86)

(I, - p.l)z" = -1£ sin

mr~ +

1k(~, 1

() silll17r( d(

I

L i

On the real Hilbert space .c 2 (0, I), consider the integral equa-

EXAMPLE 1.26 tion

I~

I;

~f this matrix can be inverted to give the coefficients (Xl, (X2, . : . , substitution ' mto (1.85) completes the determination of u.

n

il

Ij

I

j

" "

lim L(XkZk

ij

Equa~ion (1.87) is a matrix equation that can be written in standard matrix notation as follows:

z·) J '

,

"

: n->oo k=l

Lu

11

II

=0

We expand U in the basis as follows:

=

= 1,2, ...

(1.87)

= (I

zJ')

(Lz 2 ,Zt) (Lz 2, Z2)

= n---+oo lim (-7f sinl17f~)

II

,Il

= 1,2, ...

and this limit is undefined. We therefore have arrived at a contradiction, and we conclude that d / d~ is ~nbounded. i Giveil the concepts of continuity and boundedness of a linear operator, we can show that a bounded linear operator is uniquely determined by a matrix. Indeed, let {Zk}~l be a basis for H. Let L be a bounded linear operator with Lu = 1

U

j

By ~ontinuity of the inner product and the rules for inner products, We obtam

=0

and therefore,

.il )1

k=l

lim "" (XdLzk. Il~OO~ k=l

11,

29

We take the inner product of both sides of (1.86) with a member of the basis set to gi~e

But, if we choose Un as a member of the sequence cosmr~

Operators in Hilbert Space

(1.93)

But, using (1.90), we find that

1 1

o

Sec. 1.7

Linear Analysis

30

' k(;,nsinmr(d;' = (1-;)

1·~

1 1

(sinnrr(d(

+;

; (I-nsinnrr,(d(

~

0

/1l)Zn, Zm) =

=

,

[(n~)2 - /1] (Zn, Zm)

~ [(n~)2 - /1] 8

11m

(1.94)

where the inner product is the usual inner product for £2 (0, I) and 8nm ha~ been defined in (1.33). The matrix representation in (1.88) therefore diagonalize~, and i the inversion yields I Uk

=

2 1 (krr)2 -

/1,

(j,

zd,

k = 1,2, ...

(1.95)

[x, y]

Ix

1= J(Lx, x)

(1.97)

(1.98)

We.empha~i~e that the operator L must be positive for (1.9~b to satisfy the baSIC d~finltlOns of a norm. Indeed, the energy inner product and norm defi.ned In (1.97) and (1.98) must be shown in each case to satisfy the rules for Inner p~od~cts and norms. For positive-definite operators, we can prove the follOWIng Important relationship between norms: 1

IIxil :s -Ixl c (1.96)

• In the above example of representation of an operator by a matrix, the choice of the basis functions resulted in diagonalization of the matrix and, therefore, trivial matrix inversion.I There are many operators, however,' that do not have properties that result ~n this diagonalization. These concepts are better understood after a study of operator properties and resulting Greeh's functions and spectral representations in the next two chapters. , An important collection of operators for which there are e~tabl:ished convergence criteria are nonnegative, positive, and positive-definite ,operators. The reader is cautioned that there is little uniformity of notation concerning these operators in the literature. For the purposes herein, an operator L is nonnegative if (Lx, x) 2: 0, for all x E 'DL. An operator is positive if (Lx, x) > 0, for all x#-O in 'DL. An operatoris positive-definite if (Lx, x) 2: c2 11x1l 2 , for c > 0 and x E 'D L . An operator L is symmetric if (Lx, x) (x, Lx). It is easy to show that nonnegative, positive, and positive-definite operators are symmetric. In fact, any operator haviqg the property that (Lx, x) is real is symmetric. Indeed, '

(1.99)

Indeed,

Therefore,

I

IIxIl2:s~lxF c

1.

i

Taking the square root of both sides yields the desired result. Amon~ the many forms ofconvergence criteria, there are several types that are particularly useful in numerical methods in electromagnetics. For a sequence {un} C H, Un converges to U is written ' (1.100) and means that lim

n---+oo

lIu n - u'l = 0

The statement Un converges in energy to

=

== (Lx, x) = (Lx, x)

= (Lx, y)

With ~his inner product definition, 'DL becomes a Hilbert space HL. The assocIated energy norm in H L is given by

Substitution of (1.95) into (1.85) yields the final result, viz.

(x, Lx)

31

..A speci~1 inner product and norm [17], associated with positive and posltlve-defimte operators, are useful in relating convergence criteria. Define the energy inner product with respect to the operator L by

,

After some elementary integrations, we obtain the general matrix element in ,the square matrix in (1.88), viz.

«L -

Operators in Hilbert Space

U

(1.101)

is written

e

Un~U

(1.102)

and means that lim

n---+oo

IUn -

ul = 0

(1.103)

I

Linear Analysis

32 I

Chap. I

.

The statement Un converges weakly to

U

is written

w

d.104)

:U n - + U

I

lim I(u n -

U,

g)1

11-+00

=0

~

B. Convergence implies vveak convergence. C. Convergence in energy implies LU n ~ f. The weak convergence is for those g, defined by (1.105), in 'HL. If, however, II Lunll

fin 'H. is bounded, then LU n ~ ,

We first prove Property A. We have

= I(L(u n - u), ~n - u)1 ~ IIL(u n - u)lIl1u n - ull = IILu n - Luililu n - ull ~ (IiLu nll + IILulI) lIu n -ull

Since, by hypothesis, II Lunll is'bounded and Un -* u, a limiting operation gives li~ IUn - up = 0 n~oo

To prove Property B, we use the CSB inequality to give

for any g in 'H. Taking the limit yields the desired result, viz. -

U,

c

ul

yields the desired result.

00

1.8 METHOD OF MOMENTS The purpose of this section is to introduce the Method of Moments in a general way and develop various special cases. Emphasis is on convergence and error minimization. If L is a linear operator, an approximate solution to Lu = f is given by the following procedure. For L an operator in 'H, consider Lu -

f =0

(1.106)

where U E V L , f E R L . Define the linearly independent sets {cPdk=l C V Land {wd k=l' where cPk and Wk are called expansion functions and weighting functions, respectively. Define a sequence of approximants to u by , n

Un

=L

akcPk.

1l

= 1, 2, . . .

(1.107)

k=l

A matrix equation is formed in (1.106) by the condition that, upon replacement of U by Un, the left side shall be orthogonal to the sequence {Wk}. We have . (Lu n - f, wm ) = 0, m = 1,2, ... , n (1.108) Substitution of (1.107) into (1.108) and use of (1.25) gives the matrix equation of the Method of Moments [18] ,[ 19], viz.

I(u n - u, g)1 ~ lIu n - ulIlIgll

lim I(u n

1 lIu n -ull ~ -Iun -

,

D. If L is positive-definite, convergence in energy implies convergence.

2

for g E 'H L. This procedure proves the first half of Property C. The proof of the second half is based on the Hilbert space 'HL being dense in 'H and is omitted. (See [17, p. 24-25].) To prove Property D, we write

Taking the limit as n -*

A. If II Lunll is bounded, convergence implies convergence iD energy. ,

l

33

(1.105)

It is straightforward to show the following relationships among the types of convergence:

un

Method of Moments

I

and means that for every g E 1-t

IU -

Sec. 1.8

n

L ak(LcPk. w

g)1 = 0

m)

=

(f, wm ),

m=I,2, ... ,n

(1.109)

k=l

n~oo,

I

To prove Property C, we have I(Lu n - f, g)1

= I(L(u n ~ u), g)1 = I[u n -

u, g]l

~ IUn

-c-

ulhl

where we have used the CSB inequality on Hilbert space 'HL. By hypothesis, we have convergence in energy. Therefore, lim I(Lu" - f, g)1

n-+oo

=0

Note that the exact operator equation (1.106) in a Hi Ibert space'H has been transformed in(o an approximate operator equation on Hilbert space C n , viz. '

Ax =b

(1.110)

where, in usual:matrix form, x

= (a]

a2

(1.111)

.

"

Ii Linear Analysis

34

b

= ( (j, WI)

(i.112)

. (j, W2)

'A

Chap. I

n

Un

I

:

= (j,oo

I

I

Appendix-Proof of Projection Theorem

In proving the Projection Theorem, we begin by noting that the first equality in (A. 1) makes sense. Indeed, IIx - y 11 is bounded below by zero, and therefore has a greatest lower bound. We next show that there exists at least one vector Yo closest to x. We begin by asserting that there exists a vector Yn E M such that by the definition of infimum, (A.2) Taking the limit as n --*

where the energy norm is with respect to the operator L * L. By properties of the adjoint, (1.125) can also be written n

LcxdL¢k, L¢m}

= (f, L¢m),

m=I,2, ...

,11

0.127)

k=1

which is the result in the Method of Least Squares, more usually derived [20] by minimization of

lim IILu n n--->oo .

fII = 0

In this Appendix, we prove the Projection Theorem, considered in S:ection 1.6. We restate the theorem here for convenience. Let x be any ve~tor in the Hilbert space 1t, and let M c 1t be a closed linear manifold. !Then, there is a unique vector Yo E M c 1t closest to x in the sense that I inf

'lix - YII

=

IIx - Yn II = 8

Ilx - yoll

(A.3)

Therefore, we can always define a sequence {Yn} E M such that IIx - Yn II converges to 8. In (1.36), if we replace x by x - Yn and Y by x - Ym, we obtain [21], afte~ some rearrangement,

llYn - Ym 1

2

= 211x - Ynl1 2 + 211x ,

Ym 11 2

-

411x -

~(Yn + Ym)1I 2 2

(AA)

+ Ym)/2 E M, and we may assert that ,:i

A.1 APPENDIX-PROOF OF PROJECTION THEOREM ,

yEM,

lim

n--->oo

q.128)

so that LU n --* f. Unless the operator L * L is positive-definite, n9 thin g can be saidconceming the convergence of Un to u.

8=

we find that

00,

Since M is a linear manifold, (Yn

It is easy to show that (1.126) implies that

37

i (A.l) I

where inf is the greatest lower bound, or infimum. In other words, Yo is closest to x in the sense that IIx - Yoll ~ IIx - yll for all Y in M. Furthermore, the necessary and sufficient condition that Yo is the pnique minimizing vector is that e = x - Yo is in M 1-. The vector Yo is called the projection of x onto M. The vector e is called the projection of x ontb M 1- •

i"

Therefore,

In the limit as m, n --* 00, the right side goes to 28 2 + 28 2 - 48 2 = 0, and we conclude that the sequence {Yn} is Cauchy. Since 1t is a Hilbert space and M is closed, M is a Hilbert space and Cauchy convergence implies convergence. Therefore, Yn --* Yo E M. We next show that Yo is unique [22]. Suppose it is not unique. Then, , we must have at least two solutions Yo and Yo satisfying IIx, - Yo II IIx - Yoll = 8. Then,

lIyo - Yo 11 2 ~ 211x - Yoll 2 + 211x - Yoll 2 - 411x ~ 28 2 + 28 2 Therefore, Yo

= Yo.

-

48 2

=0

~(y~ + yo) 11 2 2 (A.6)

I

Problems

38

Chap. 1

Finally, we show that e =x - Yo E Ml-. We must show that e is orthogonal to every vector in M. Suppose that there exists a vector Z E M that is not orthogonal to e. Then, we would have [23] (e, z)

= (x -

Yo,z)

= A =I 0,

zEM

(A,7)

==

Yo

+

A \I Z11 2 Z

I

e\

= (1,0, ... ,0), e2 = (0, 1, ... ,0), ... , en = (0,0, ...',1)

is linearly independent. Is the same conclusion valid in C n ? Is the set of' vectors a basis for C n ? n

n

k=1

k=\

(l:= akXb y} = L adxb y} 1.10. Show that C(a, (3) with inner product defined by (1.27) is a real inner product.

".

IIx - zoll2

= (x -

space.

A' A Yo - - z2· x - Yo - - z2)

IIzll " 2 A = Ilx - Yo II - IIzll 2 (x

1.11. Consider the linear space of real continuous twice differentiable functions

IIz11

A

- Yo, z) -

IIz11 2(z, x

- Yo)

1~12

+ 11211 2

IAI 2 = IIx - Yo II - IIzll 2 2

(A.9)

Therefore,

1

Ilx - zoll .

1.8. Show that in R n the set of vectors

(A;8)

Then,

I

39

Problems

1.9. Given the basic definition of an inner product space, show that

We define a vector Zo EM such that Zo

Chap. 1


oo LU n is undefined. The problem is that L is unbounded. This result is an example of the fact that a Fourier series cannot always be differentiated term by term.

Test the operator A for boundedness. I

1

-k coskTf~

It is well-known [16] that

00

(a, b)

n=O

~

2

Using well-known series summation results [161, show that, although L is unbounded, in this case the operator and limit can be interchanged.

i Problems

42

,

.

Chap. 1

1.26. Show that the operator in (1.91) with kernel defined in (1.90) is bounded.

(u,~) =

References

43

1.31. Consider the real Hilbert space £2(0,1) with inner product '

I

1.27. Consider Hilbert space £2 (-1, 1) with inner product ,

Chap. 1

11 u(Ov(~)d~

where fU;), g(O

where all functions are real-valued. Consider the following function

f

1f(~)g(~)d~ 1

(j, g) =

I

(0:

E

£2(0, 1). Suppose that

f(~) = 1 - i

2

(a) By the method of best approximation, approximate f(~) by i(~), where

i(~) is a linear combination constructed from the orthonormal set

(JEkcoskn~}t=o in £2(0, 1). In the orthonormili set, and 2 for k # O. (b) Calculate the norm of the error IIf(O - i(~)II.

Ek

is 1 for k

=

0

1.32. Consider Euclidean space R 4 . Define vectors a and b in R 4 by a = (aI,

, a4)

b = (fJI,

, f34)

Define an inner product for the space by 2rr

(u, v) = ,

where members of the space

1 0

~re

4

u(Ov(~)d~

(a, b) =

L.:>kf3k k=l

Consider those vectors a in R 4 restricted by

complex functions.

(a) Show that the sequence '

is an orthogonal sequence. (b) Produce an orthonormal (O.N.) sequence from the orthogonal sequence. (c) Using the members of the O.N. sequence contained on -N ~ ~ ~ N, where N is a positive integer, find the best approximation to f(~)

.

2a) ,

=;sm (~-2-

aER

in the sense given in Section 1.6, Best Approximation. 1.30. Consider the real Hilbert space £2(-1, 1). For define an inner product

(j, g) =

1 -1 /

f(~)g(~)

f(o.

g(O E £2('-1, 1),

d~

JI=T2

Determine whether this dcfini'tion results in a legitimate inner product.

(a) Show that all vectors with this restriction form a linear manifold M. (b) Find all vectors b that are members of M 1- •

REFERENCES [I] Stewart, G.w. (1973), Introduction to Matrix Computations. New York: Academic Press, 54-56. [2] Hardy, G.H. (1967), A Course in Pure Mathematics. London: Cam~ bridge University Press, 1-2. [3] De Lillo, N.J. (1982), Advanced Calculus with Applications. New York: Macmillan, 32. ' [4] Stakgold, I.: (1967), Boundary Value Problems of Mathematical Physics. Vol, I. New York: Macmillan, tal-102. [5] Mac Duffee; c.c. (1940), Introduction to Ahstract Algehra. New York: Wiley. chapter VI.

References

44

[6] Greenberg, M;D. (1978), Fhundations ofApplied Mathematics.! Englewood Cliffs, NJ: Prentice-Hall, 319-320. [7] Helmberg, G. (1975), Int~oduction to Spectral Theory in Hilbert Space. Amsterdam: N0 rth-Holland,20--2l. [8] Ibid., Appendix B. , [9] Shilov, G.E. (1961), An Introduction to the Theory of Linear Spaces. Englewood qiffs, NJ: Prentice-Hall, sect. 83-85. I [10] Stakgold, 1. (1979), Green's Functions and Boundary Vatue 'Problems. New York: Wiley-~nterscience, 36-41. : [11] Lue)1berger, D.G. (1969),: Optimization by Vector Space Methods. New iork: Wiley, 5~57. ', [12] op.cit. Helmberg, 73. [13] Akhiezer, N.L, and LM. Glazman (1961), Theory ofLinear Operators ' in Hilbert Space. New York: Frederick Ungar, 33,39. [14] Dudley, D.G. (1985), Error minimization and convergence in n~mer­ ical methods, Electromagnetics 5: 89-97. [15] Butler, C.M., and D.R. Wilton (1980), General analysis of narrow strips and slots, IEEE Trans. Antennas Propagat. AP-28: 42-48., [16] Abramowitz, M., and LA. 'Stegun (Eds.) (1964), Handbook of Mathematical Functions, National Bureau of Standards, Applied Mathematics Series, 55, Superintendent of Documents, U.S. Government Printing Office, Washingtqn, DC 20402, 1005. , [17] Mikhlin, S.G. (1965), The Problem of the Minimum of a Quadratic Functional. San Francisco: Holden-Day. [18] Harrington, R.E (1968), field Computation by Moment Methods. New York: Macmillan. [19] Harrington, R.E (1967), "Matrix methods for field problems," hoc. IEEE 55: 13~149. [20] op.cit. Stakgold (1967), v61. 2, sect. 8.10. [21] Krall,A.M. (1973), Linear Methods ofApplied Analysis. ReadIng, MA: Addison-Wesley, 181-182. [22] Ibid., !182. [23] op.cit. Akhiezer and Glazman, 10. ,

l

r

\

, Chap. 1

2 The Green's Function Method , ,

,

2.1 INTRODUCTION In this chapter, we begin our study of linear ordinary differential equations of second order. Our goal is to develop a procedure whereby we can solve the differential equations using fundamental solutions called Green's functions. We begin >yith a brief discussion of the delta function. We follow with a description of the Sturm-Liouville operator L and Its properties. We de~ne three types of Sturm-Liouville problems and investigate their propertIes. In all three types, we examine the role of the operator L and its ' adjoint operator L *. These operators are used to define the Green's func- : ~ion and the adjoint Green's function, respectively. Our study culminates: 10 a procedure fo~ applying the Green's function and/or the adjoint Green's, function in solving the differential equation Lu = f. 2.2 DELTA FUNCTION The concept of the delta function arises when we wish to fix attention on ' the value of a function f(x) at a given point xo. Mathematically, we seek an operator T that transforms a function f (x), continuous at xo, into f (xo), the value of the function at xo. In equation form, we require T such that T [f(x)] = f(xo)

(2.1)

:

!

The Green's Function Method

46

Chap. 2

E

< X < Xo

+E

lim : (2.2)

otherwise Note that, regardless of the value of E, the area under the pulse is 'unity. Indeed, if (a, b) is any interval containing (xo - E, Xo + E),

l

b

.

P€ (x - xo)dx

=

a

l

XO

+€ I

Xo-€

-

2E

dx

=I

Delta Function

Taking the limit as

We begin by considering the pulse function P€ (x - xo), defined by ;

Xo -

Sec. 2.2

€-+o

l

E ---+

0, we have

b

= f(xo),

f(x)P€(x - xo)dx

a

Xo

EXAMPLE 2.1 Let f(x) = x 2, Xo continuous at x = 0 and we have

. (2.3) lim

,->0

1" a

=

0, and Xo

E (a,

= ,->0 lim[~ 2E

f(x)p,(x - xo)dx

An important property of the pulse function is that it is even about Xo, viz.

I'

I

l

I'

\

b

f(x)p€(x - x.o)dx

a

:

= -I

2E

l

+€

b

f(x)P€(x - xo)dx

Since f(O)

(2.6)

~Pe(X-Xo)

,->0

1"

f(x)p,(x - xo)dx = lim - 1 ,->02E

a

Since

f

(rr/3)

=

so that

-1-----+----:---+----+-----" x

2-1

+8

Pulse function p,(x - xo) and function f(x).

(i +

• E (a,

b). In this case,

dx

E) - sin

(i -

E)]

I~ =

•

Expression (2.7) fonns the cornerstone of our definition of the delta function, as follows:

1

I·

2 X dX]·

1/2, we have again verified (2.7).

h

f(x)

Fi~.

1+' cosx 1-'

1

f(x)8(x - xo)dx

a

Xo

1

= !~ 12E [sin

,--------=,

1/28

_,

= 0, w,e have verified (2.7).

'

=f

b). In this case, f(x) is

l'

EXAMPLE 2.2 Let f(x) = cosx, Xo = rr/3, and Xo f (x) is continuous at x = rr /3 and we have lim

f is the mean value of f(x)

(2.7)

2

(2.5)

f(x)dx

Xo-€

By the mean value theorem for ihtegrals [I], if on the interyal x E (xo - E, Xo + E),

l

xo

(a, b)

= ,-.0 lim(E ) = 0 3

(2.4)

This property can be proved by interchanging x and Xo in (2.2). The details are left for the problems. Multiplying the pulse function by f (x) and integrating over any interval containing the pulse gives (Fig. 2-1)

E

The integration followed by the limiting operation in (2.7) transfonns f(x) to f (xo), the value of the function at Xo.

,

~

47

=

lim €-+o

l

l

b

f(x)P€(x - xo)dx

(2.8)

= f(xo)

(2.9)

a

b

f(x)8(x - xo)dx

for any Xo in the interval (a, b). Note in (2.2) that as E becomes smaller, the pulse function becomes narrower and higher while maintaining unit area. If the limit in (2.7) could be taken under the integral, we would have

8(x -xo) 4: lim [p, (x -xo)] €-+O

(2.10)

I

i 48

The Green's Function Method

Sec. 2.2

Since this limit does not exist, the interchange of limit and integration in (2.8) is not valid. We have therefore placed an "s" over the equality in (2.10) to indicate symbolic equality only. The delta function 8 (x - xo) has two remarkable properties. Symbolically, it is a function that is zero everywhere except at x = xu, where it is undefined. Second, when integrated against a function f that is continuous at xo, it yields the value of the function at xo. We note that (2.9) defines the operator T we were seeking in (2.1). Indeed, comparing (2.1) and (2.9) yields b (2.11 ) T = (.)8(x - xo)dx

1

....

a.

where (.) indicates the position of the function upon which T operates. From the basic definition of the delta function in (2.9), we obtain some additional relations. If we set Xo equal to zero, we find

l

b

f(x)8(x)dx

= f(O)

Delta Function

49

Suppose we wish to solve the equation Lu

=f

(2.15)

where L is a diffe~ential operator. Formally, the solution is given by multiplying both sides of (2.15) by the inverse operator, viz.

or (2.16) Since L is a differential operator, we shall assume that its inverse is an integral operator with kernel g(x, ~), so that u(x)

=

f

g(x,

~)f(~)d~

(2.17)

i

(2.12) I

Substitution into (2.15) gives

! I

Also, if in (2.9) we set f(x)

\

l

a

= 1, we obtain b

,

8(x - xo)dx

=1

f(x)

=

f

I

(2.13)

=L

Finally, from (2.4) and (2.8), we conclude symbolically that

= i

8(x - xo) :b 8(xo - x)

L[u(x)]

(~.14) I I

In concluding our development of the delta function and its properties, we remark that there are certain difficulties with the definitions. Indeed, any function that is zero everywhere except at one point must producci zero when Riemann integrated over any interval containing the point. The result in (2.13), for example, is therefote unacceptable in the Riemann sense. To interpret the integral, it seems that we must return to the basic defirtition in (2.8). The mathematical acceptability of integrals involving the delta function have, however, been fdrmalized in the Theory of Distributions, introduced by Schwartz [2]. In the theory, the delta function is called a generalized function, and the integral in (2.9) is said to exist in the distributional sense. Although the theory is beyond the scope of this book, the interested reader can find introductory treatments in [3],[4]. ' The central role played by'the delta function in the solution to certain differential equations becomes apparent in the following argument.

I I

f

g(x,

~)f(~)d~

Lg(x,

~)f(~)d~

(2.18)

where we have assumed, without proof, that we can move the operator L inside the integral. But, from the properties of the delta function, we have

f(x)

for x

E (a, b).

=

l

b

8(x -

~)f(S)d~

(2.19)

Comparing (2.18) and (2.19), we identify (2.20)

Presumably, if we can solve (2.20), then the solution to (2.15) is given explicitly by (2.17). The kernel g(x, ~) is called the Green's function for the problem. It is the purpose of this chapter to formalize and structure the introductory ideas above. The result will be the solution to a class of linear ordinary differential equations of second order by the Green's function method.

I

The Green's Function Method

50

Chap. 2

d 2u(x) du(x)' ao(x)--2-+at(x)--+a2(x)u(X)-AU(X) dx dx

Sturm-Liouville Operator Theory

EXAMPLE 2.3

2.3 STURM-LIOUVILLE OPERATOR THEORY Consider the following linear, ordinary, differential equation of order:

Sec. 2.3

s~cond

Consider Bessel's equation of order v, given by 1 I -u " - -u X

(v 2

+ x- 2 - k 2) u = f

(:2.21)

where A is, in general, a complex parameter independent of x. The fun6tions ao, at, and a2 are real and assumed to have the following properties [5],[6]:

ao =-1

= -O/x) = (v/x)2

al

a. a2, datJdx, and d2ao/dx2 are continuous in a ::'S x ::'S b ".. b. ao f:. 0 in a < x < b

a2

To transform to Sturm-Liouville form, we use (2.23)-(2.25) and obtain

In (2.21), we also require that u(x) be twice differentiable and that i f (x) be piecewise continuous. We may always recast this differential eqJation in Sturm-Liouville form, as follows:

q(x) =

I,

(2.22)

w(x)

so that 1 I I --(xu) X

(2.23)

= h p [fX at (t) dt]

(2.24)

w(~) = _ p(x)

(2.25)

.

i

ao(t)

ao(x) I

We may verify these transformations by substituting (2.23)-(2.25) ihto (2.22) to produce (2.21). The details are left for the problems. We r~wtite (2.22) in operator notation as follows: I

=f

2

x

EXAMPLE 2.4 ferent form, viz.

=x

2 2) k u =f

+ (V-x 2 -

•

-x 2U "

-

xu I -

[k ( x) 2 -

A = _v 2

ao

= _x 2

al

= -x

a2 = _(kx)2

L

1 'd[ p(x)d] + q(x) =-w(x) dx

Using (2.23)-(2.25), we obtain

(2.27)

dx

For the rem~inder of this chapter, without loss of generality, L will always mean the Sturm-Liouville operator in (2.27). I

v 2) u =

f•

(2.30)

We note that (2.30) is obtained simply by multiplying both sides of (2.28) by x 2 . In this case, we identify

(2.26)

where we identify the Sturm-Liouville operator L, viz.

(2.29)

Consider Bessel's equation, given by (2.28), in a slightly dif-

I

(L - A)U

V

2"

p(x) = x

The necessary coefficient transformations are given by [7]

p(x)

(2.28)

where "prime" indicates differentiation with respect to x. Comparing to (2.21), we identify

= f(x),

d [ p(x)-dU(X)] + . q(x)u(x) - AU(X) = f(x) - -1- w(x) dx dx

51

q(x) = _(kx)2

p(x) = x

1

w(x) = x

The Green's Function Method

52

,

i Chap. 2

so that in Sturm-Liouville form,

.

-x(xu')' - (kx)2 u

+ v 2u = f

.!

,

We note, in particular, that the weighting function w(x) differs from that in ;Example 2.3. ' , ,

•

It might appear that the distinction between (2.29) and (2.31) is tri;ial since the latter can be obtained from the former by dividing by x 2 . However, the difference in the weighting functions between (2.29) and (2.31) changes the Hilbert.space, and makes a major difference in spectral representations associated ~ith the radial portion of the Helmholtz equation in cylindrical ' coordinates [8], as we shall find in' Chapter 4. ,

.

EXAMPLE 2.5 follows:

Consider Legendre's equation on the interval x -(1 - x 2 )u"

+ 2xu' -

n(n

Stunn-Liouville Problem of the First Kind

E (-I, I),

For the first form of the Sturm-Liouville problem, we consider,(L - A)ll = over a finite interval x E (a, b) and for real A and real f. For -00 < a < b < 00, consider the Hilbert space L2(a, b) with real inner product

f

(u, v)

L).)l

l

b

u(x)v(x)w(x)dx

(2.34)

=

f,

a<x 0, w(x) >

c. A is real and independent of x

:Using (2.23)-(2.25), we obtain q(x)

In addition, we require u(x) E V L C Ida, b), where V L is the domain of the operator L. Because we are dealing with second-order differential operators, the domain is restricted to those functions that are twice differentiable. Finally, we require that u(x) satisfy two boundary conditions as follows:

=0

p(x)=l-x

2

w(x) = 1

'so that in Sturm-Liouville form,

- [(I -

x 2 )u']' - n(n

°for a ::::: x ::::: b

+ I)u

=

f

(2.33)

• The Sturm-Liouville form of the second order differential equati,on, given by (2.22), plays a central role in the solution of electromagnetic boundary value problems. We distinguish three forms of the SturmLiouville problem, which we consider in the next three sections.

B I (u)

= a = allu(a) + a12u'(a) + aI3u(b) + aI4u' (b)

(2.38)

B2(U)

= f3 = a2l 11 (a) + a22 u' (a) + a23u(b) + a24u'(b)

. (2.39)

where, for SLP I, a, f3, and aij are real. Typically, in (2.38), if a is nonzero, the boundary condition is said to be inhomogeneous. If a = 0, the boundary condition is homoger'eous. . There are several important special cases contained in the boundary conditions in (2.38): and (2.39). A boundary condition is unmixed if it involves conditions bn u(x) at one boundary only. If SLPI involves an

chap. 2

The Green's Function Method

54

unmixed condition at one end of the boundary and an unmixed condition at the other end, we refer to. this caSe as SLPI with unmixed conditions.'. The most general case of unmixed bqundary conditions is al3 = al4 = a21 = a22 = 0, so that , BI (u) = a = allu(a)

B2(U)

= f3

:::i: a23u(b)

, + a\2u (a)

(2.40)

+ a24u'(b)

(2.41) I

The relations in (2.38) and (2.39) are said to be initial conditions, if dll' = a22 = 1 and:all other aij coeffidents are zero, so that .

I

I

Bl(U)

= a = u(a)

(2.42)

B2(U}

= f3 = u'(a)

(2.43)

The two conditions in (2.38) and (2.39) are periodic if the value of the function u(x) at one boundary is identical to the value at the other boundary, and if the value of the derivative u' (x) at one boundary is identical to the I value at the other boundary. To produce the periodic conditions, we require all = -a13= a22 = -a24 = I land all other coefficients zero, so that

\;

(2.45)

:

k

E

E

(0, 1): I

R

with two homogeneous unmixed boundary conditions

= u(1) = 0

•

The operator Lin SLPI has aformal adjoint, which we constn:lctby the following procedure. For u,.V E £2(a, b), we form i (Lu, v) ::::::

l

a

b

{ - -1- d

(Lu, v)

= Ja{b u(x) {-

I d [dV(X)] W(x) dx p(x)~

['p(x)-dU(X)] w(x) dx' dx .

: + q(x)u(x) } v(x)w(x)dx

+ q(x)v(x) } w(x)dx

_ {P(X) [V(X) d~~X) _ u(x) d~~)]}

b

l

a

(2.47)

We write this result in inner product notation as (Lu, v) = (u, L *v)

+ J(u, v)

I:

(2.48)

where J (u, v) is called the conjunct and is given by J(u, v)

= -p(u'v -

uv')

(2.49)

The operator L *, produced in the integration by parts operation, is called the formal adjoint to L. We note that

=L

(2.50)

When (2.50) is true, we say that L is formally self-adjoint. We conclude, in general, that the Sturm-Liouville operator for SLPI is formally selfadjoint. In our search for a solution, or solutions, to (2.35), we shall first assume that the boundary conditions in (2.38) and (2.39) are homogeneous. We then follow with the extension to the inhomogeneous case. Accordingly, if u(f) is to be a solution to (2.35), we require the following I restrictions: a. u

We identify p(x) = w(x) = 1, q(x) = 0, A. = k 2 , a = 0, b = I, a = f3 = O. In the boundary conditions in (2.3~) and (2.39), all coefficients aij = 0, except all = a23 = 1. We find that all requirements for SLPI are satisfied.

55

Integrating by parts twice, we obtain

(2.44)

Consider the following differential equation on x

u(O)

Sturm-Liouville Problem of the First Kind

L*

= u(b) u'(a) = u'(b) u(a)

EXAMPLE 2.6

Sec. 2.4

E

£2(a,b)

b. u E V L

c. u satisfies two boundary conditions, B j (u)

= 0, B2(U) = 0

These restrictions define a linear manifold M L C £2(a, b). The proof is left for the problems. We next consider the function vex) in (2.48). We place the following restrictions on v (x):

a. v

E £2(a,b)

b. v

E

VL*

I

(2.46) I

II i

c. v satisfies two ad;oint hnundary conditions, Rt(v) = 0, RI(I') =

°

The Green's Function Method

56

!

Chap. 2

Since vex) is unspecified in the ~riginal problem statement in (2.35), we are free to choose the adjoint boundary conditions in any manner we wish, consistent with the integration by parts operation in (2.48). We define the adjoint boundary conditions to by those conditions Bt(v) = 0, B~(v? = 0 that when coupled with the boundary conditions on u(x), result 1\1 the vanishing of the conjunct, viz. ,

Sec. 2.4

Substitution into the conjunct gives J(u, v)

I:

In this case, (2.51) is satisfied if we choose adjoint boundary conditions

1

:

In =

(2.51 )

0

These res,trictions on vex) defin~ a linear manifold ML* C L2(a, b). At present, it is not clear that it is ,Possible to define the a?j~int boun?~ry conditions such that (2.51) is satisfied. We next show explicitly the adJomt boundary condition result for the unmixed, initial, and periodic cases. We have defined the unmixed boundary case in (2.40) and (2.41). For the homogeneous case, they become

B2(U) = IX23,U(b)

la =

+ IX24U'(b) =

a

.

IX

23 v(b)'+ v' (b)] - p(a)u(a) IX24 '

[~v(a) + V1(d)] IX \2

(2.61)

+ IX24V'(b)

= 0

(~.56)

We note that in the unmixed boundary case, the boundary conditiohs on vex) in (2.55) and (2.56) are identical to those on u(x) in (2.52) and (2.53!. Therefore, for the case of unmixed boundary conditions, the linear ?1 am fold ML is the same as the line~r manifold ML*. A formally self-adjoint operator with ML = ML* is said to be self-adjoint. We shall. find s~bse­ quently that self-adjoint problems possess remarkable properties. I For homogeneous initial conditions, we have u(a) = 0

I ~.57)

/l'(a) =

(2.58)

0

(2.62)

- u(a) [p(a)v'(a) - p(b)v'(b)]

In this case, (2.51) is satisfied if we choose adjoint boundary conditions Bt(v)

'

(t·

B;(v) = IX2jv(b)

-pCb) [u'(a)v(b) - u(a)v'(b)]

= p(a)v(a) -

p(b)v(b)

= p(a)v'(a) -

p(b)v'(b)

=0

(2.63)

=0

(2.64)

(2.54)

In this case, (2.51) is satisfied if we choose the following adjoint boundary . ! conditions: . I 55 ) Bt(v) = IXII v(a) + IX\2v'(a) = 0

..

0

= u'(a) [p(a)v(a) - p(b)v(b)]

(2.53)

0

I

=:p(b)u(b) [

B;(v) = v'(b) =

+ pea) [u'(a)v(a) - u(a)v'(a)]

We use these expressions in the ~onjunct to eliminate u' (a) and u' (b), ~iz. b

(2.60)

h

i

J(J, v)

(2.52)

I

0

We note that for initial conditions, the boundary conditions on vex) in (2.60) and (2.61) are not the same as those on u (x) in (2.57) and (2.58). Therefore, M L '# ML*, and the initial condition case is never self-adjoint. For periodic:conditions, we substitute (2.44) and (2.45) into the conjunct to give

l

I

Bt(v) = v(b) =

b

J(i,t, v)

J(u, v)

(2.59)

= -pCb) [u'(b)v(b) - u(b)v'(b)]

I

\

57

Stunn-Liouville Problem of the First Kind

Bi(v)

We note that for the general form of L in (2.37) and for periodic' conditions, the boundary conditions on vex) in (2.63) and (2.64) are notthe same as those on u(x) in (2.44) and (2.45). However, if the operator L is such that pea) = pCb), the conditions are identical and the problem becomes self-adjoint. To produce the solution to SLPI by the Green's function method, we define two auxiliary problems: the Green's function problem and the adjoint Green's function problem. The Green's function problem is defined as follows: •

,

L).g(x,O =

I

8(x -~) w(x)

a < ~ < b

,

(2.65)

B)(g) =

0

(2.66)

B2(g) =

0

(2.67)

\

The Green's Function Method

58

Chap. 2

I

where w(x) is the weight functiori defined in (2.25) and (2.27) and appearing in the inner product definition in (2.34). We note that, by definition, the boundary conditions on g are identical to the boundary conditions on u. The adjoint Green's function problem is defined as follows: LAh(x,~)

=

8(x - ~) ~(x) ,

(2.68)

Bt(h)

=0

B;(h)

=0

,

Sec. 2.4

Sturm-Liouville Problem of the First Kind

°

Bt(h) = 0, B2(h) = reduces the second term in (2.74) to zero. The extension to the inhomogeneous case, however, is now available. We simply apply the given b