[SOLVED] AI FTP arm Fortran scheme case study Excel ant ER CGI algorithm flex js compiler The Numerical Solution of Integral Equations of the Second Kind

The Numerical Solution of Integral Equations of the Second Kind

This book provides an extensive introduction to the numerical solution of a large
class of integral equations. The initial chapters provide a general framework
for the numerical analysis of Fredholm integral equations of the second kind,
covering degenerate kernel, projection, and Nystrom methods. Additional dis-
cussions of multivariable integral equations and iteration methods update the
reader on the present state of the art in this area.

The final chapters focus on the numerical solution of boundary integral
equation (BIE) reformulations of Laplaces equation, in both two and three
dimensions. Two chapters are devoted to planar BIE problems, which include
both existing methods and remaining questions. Practial problems for BIE such
as the set up and solution of the discretized BIE are also discussed.

Each chapter concludes with a discussion of the literature, and a large bib-
liography serves as an extended resource for students and researchers needing
more information on solving particular integral equations.

CAMBRIDGE MONOGRAPHS ON
APPLIED AND COMPUTATIONAL
MATHEMATICS

Series Editors

P. G. CIARLET, A. ISERLES, R. V. KOHN, M. H. WRIGHT

4 The Numerical Solution of Integral
Equations of the Second Kind

The Cambridge Monographs on Applied and Computational Mathematics
reflects the crucial role of mathematical and computational techniques in con-
temporary science. The series publishes expositions on all aspects of applicable
and numerical mathematics, with an emphasis on new developments in this fast-
moving area of research.

State-of-the-art methods and algorithms as well as modern mathematical
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sentation is a prerequisite. It is intended that books in the series will serve to
inform a new generation of researchers.

Also in this series:

A Practical Guide to Pseudospectral Methods, Bengt Fornberg

Level Set Methods, J.A. Sethian

Dynamical Systems and Numerical Analysis, A.M. Stuart andA.R. Humphries

The Numerical Solution of Integral Equations
of the Second Kind

KENDALL E. ATKINSON
University of Iowa

AMBRIDGE
UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, Delhi

Cambridge University Press
The Edinburgh Building, Cambridge C132 8RU, UK

Published in the United States of America by Cambridge University Press, New York

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Information on this title: www.cambridge.org/9780521102834

Atkinson, Kendall 1997

This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written

permission of Cambridge University Press.

First published 1997
This digitally printed version 2009

A catalogue record for this publication is available from the British Library

Library of Congress Cataloguing in Publication data

Atkinson, Kendall E.
The numerical solution of integral equations of the second kind /

Kendall E. Atkinson.
p. cm.

Includes bibliographical references (p. ) and index.
ISBN 0-521-58391-8 (hc)

1. Integral equations Numerical solutions. I. Title.
QA431.A837 1997

551.46 dc20 96-45961
CIP

ISBN 978-0-521-58391-6 hardback
ISBN 978-0-521-10283-4 paperback

To Alice

Contents

Preface page xv

1 A brief discussion of integral equations 1
1.1 Types of integral equations 1

1.1.1 Volterra integral equations of the second kind 1
1.1.2 Volterra integral equations of the first kind 2

1.1.3 Abel integral equations of the first kind 3
1.1.4 Fredholm integral equations of the second kind 3
1.1.5 Fredholm integral equations of the first kind 3
1.1.6 Boundary integral equations 4
1.1.7 Wiener-Hopf integral equations 5
1.1.8 Cauchy singular integral equations 5

1.2 Compact integral operators 6
1.2.1 Compact integral operators on C(D) 7
1.2.2 Properties of compact operators 8
1.2.3 Integral operators on L 2 (a, b) 11

1.3 The Fredholm alternative theorem 13
1.4 Additional results on Fredholm integral equations 17
1.5 Noncompact integral operators 20

1.5.1 An Abel integral equation 20
1.5.2 Cauchy singular integral operators 20
1.5.3 Wiener-Hopf integral operators 21

Discussion of the literature 21

2 Degenerate kernel methods 23
2.1 General theory 23

2.1.1 Solution of degenerate kernel integral equation 26

vii

viii Contents

2.2 Taylor series approximations 29
2.2.1 Conditioning of the linear system 34

2.3 Interpolatory degenerate kernel approximations 36
2.3.1 Interpolation with respect to the variable t 37
2.3.2 Interpolation with respect to the variable s 38
2.3.3 Piecewise linear interpolation 38
2.3.4 Approximate calculation of the linear system 42

2.4 Orthonormal expansions 45
Discussion of the literature 47

3 Projection methods 49
3.1 General theory 49

3.1.1 Collocation methods 50
3.1.2 Galerkins method 52
3.1.3 The general framework 54

3.2 Examples of the collocation method 58
3.2.1 Piecewise linear interpolation 59
3.2.2 Collocation with trigonometric polynomials 62

3.3 Examples of Galerkins method 66
3.3.1 Piecewise linear approximations 66
3.3.2 Galerkins method with trigonometric polynomials 68
3.3.3 Uniform convergence 70

3.4 Iterated projection methods 71
3.4.1 The iterated Galerkin solution 74
3.4.2 Uniform convergence of iterated Galerkin

approximations 75
3.4.3 The iterated collocation solution 77
3.4.4 Piecewise polynomial collocation at Gauss-Legendre

nodes 81
3.4.5 The linear system for the iterated collocation solution 85

3.5 Regularization of the solution 86
3.6 Condition numbers 88

3.6.1 Condition numbers for the collocation method 90
3.6.2 Condition numbers based on the iterated collocation

solution 94
3.6.3 Condition numbers for the Galerkin method 94

Discussion of the literature 98

4 The Nystrom method 100
4.1 The Nystrom method for continuous kernel functions 100

Contents ix

4.1.1 Properties and error analysis of the Nystrom method 103
An asymptotic error estimate 111
Conditioning of the linear system 112

4.1.2 Collectively compact operator approximations 114
4.2 Product integration methods 116

4.2.1 Computation of the quadrature weights 118
4.2.2 Error analysis 120
4.2.3 Generalizations to other kernel functions 122
4.2.4 Improved error results for special kernels 124
4.2.5 Product integration with graded meshes 125

Application to integral equations 132
The relationship of product integration and collocation
methods 134

4.3 Discrete collocation methods 135
4.3.1 Convergence analysis for {rk} {t, } 139

4.4 Discrete Galerkin methods 142
4.4.1 The discrete orthogonal projection operator 144
4.4.2 An abstract formulation 147

Discussion of the literature 154

5 Solving multivariable integral equations 157
5.1 Multivariable interpolation and numerical integration 157

5.1.1 Interpolation over triangles 160
Piecewise polynomial interpolation 163
Interpolation error formulas over triangles 165

5.1.2 Numerical integration over triangles 167
Some quadrature formulas based on interpolation 169
Other quadrature formulas 170
Error formulas for composite numerical integration
formulas 171
How to refine a triangulation 173

5.2 Solving integral equations on polygonal regions 175
5.2.1 Collocation methods 176

The iterated collocation method and superconvergence 178
5.2.2 Galerkin methods 181

Uniform convergence 183
5.2.3 The Nystrom method 184

Discrete Galerkin methods 186
5.3 Interpolation and numerical integration on surfaces 188

5.3.1 Interpolation over a surface 189

x Contents

5.3.2 Numerical integration over a surface 191
5.3.3 Approximating the surface 192
5.3.4 Nonconforming triangulations 204

5.4 Boundary element methods for solving integral equations 205
5.4.1 The Nystrom method 205

Using the approximate surface 207
5.4.2 Collocation methods 213

Using the approximate surface 215
Discrete collocation methods 217

5.4.3 Galerkin methods 218
Discrete Galerkin methods 221

5.5 Global approximation methods on smooth surfaces 222
5.5.1 Spherical polynomials and spherical harmonics 224

Best approximations 228
5.5.2 Numerical integration on the sphere 229

A discrete orthogonal projection operator 232
5.5.3 Solution of integral equations on the unit sphere 235

A Galerkin method 236
A discrete Galerkin method 237

Discussion of the literature 239

6 Iteration methods 241
6.1 Solving degenerate kernel integral equations by iteration 242

6.1.1 Implementation 244
6.2 Two-grid iteration for the Nystrom method 248

6.2.1 Iteration method 1 for Nystroms method 249
Implementation for solving the linear system 254
Operations count 256

6.2.2 Iteration method 2 for Nystroms method 258
Implementation for solving the linear system 261
Operations count 265
An algorithm with automatic error control 266

6.3 Two-grid iteration for collocation methods 267
6.3.1 Prolongation and restriction operators 269
6.3.2 The two-grid iteration method 272

An alternative formulation 280
Operations count 280

6.4 Multigrid iteration for collocation methods 281
6.4.1 Operations count 288

6.5 The conjugate gradient method 291

Contents xi

6.5.1 The conjugate gradient method for the undiscretized
integral equation 291

Bounds on ck 296
6.5.2 The conjugate gradient iteration for Nystroms method 298

The conjugate gradient method and its convergence 299
6.5.3 Nonsymmetric integral equations 301

Discussion of the literature 303

7 Boundary integral equations on a smooth planar boundary 306
7.1 Boundary integral equations 307

7.1.1 Greens identities and representation formula 308
7.1.2 The Kelvin transformation and exterior problems 310
7.1.3 Boundary integral equations of direct type 314

The interior Dirichlet problem 315
The interior Neumann problem 315
The exterior Neumann problem 316
The exterior Dirichlet problem 317

7.1.4 Boundary integral equations of indirect type 317
Double layer potentials 318
Single layer potentials 319

7.2 Boundary integral equations of the second kind 320
7.2.1 Evaluation of the double layer potential 324
7.2.2 The exterior Neumann problem 328
7.2.3 Other boundary value problems 333

7.3 Boundary integral equations of the first kind 338
7.3.1 Sobolev spaces 338

The trapezoidal rule and trigonometric interpolation 341
7.3.2 Some pseudodifferential equations 342

The Cauchy singular integral operator 344
A hypersingular integral operator 346
Pseudodifferential operators 349

7.3.3 Two numerical methods 349
A discrete Galerkin method 351

7.4 Finite element methods 359
7.4.1 Sobolev spaces A further discussion 360

Extensions of boundary integral operators 363
7.4.2 An abstract framework 364

A general existence theorem 367
An abstract finite element theory 372
The finite element solution as a projection 375

xii Contents

7.4.3 Boundary element methods for boundary integral equations 376
Additional remarks 380

Discussion of the literature 381

8 Boundary integral equations on a piecewise smooth planar
boundary 384

8.1 Theoretical behavior 385
8.1.1 Boundary integral equations for the interior Dirichlet

problem 387
8.1.2 An indirect method for the Dirichlet problem 389
8.1.3 A BIE on an open wedge 390
8.1.4 A decomposition of the boundary integral equation 394

8.2 The Galerkin method 397
8.2.1 Superconvergence results 403

8.3 The collocation method 404
8.3.1 Preliminary definitions and assumptions 406

Graded meshes 408

8.3.2 The collocation method 410
A modified collocation method 412

8.4 The Nystrom method 418
8.4.1 Error analysis 421

Discussion of the literature 425

9 Boundary integral equations in three dimensions 427
9.1 Boundary integral representations 428

9.1.1 Greens representation formula 430
The existence of the single and double layer potentials 431
Exterior problems and the Kelvin transform 432
Greens representation formula for exterior regions 434

9.1.2 Direct boundary integral equations 435
9.1.3 Indirect boundary integral equations 437
9.1.4 Properties of the integral operators 439
9.1.5 Properties of K and S when S is only piecewise

smooth 442
9.2 Boundary element collocation methods on smooth surfaces 446

9.2.1 The linear system 455
Numerical integration of singular integrals 457
Numerical integration of nonsingular integrals 460

9.2.2 Solving the linear system 462
9.2.3 Experiments for a first kind equation 467

Contents xiii

9.3.1 The collocation method 472
Applications to various interpolatory projections 474
Numerical integration and surface approximation 474

9.3.2 Iterative solution of the linear system 479
9.3.3 Collocation methods for polyhedral regions 486

9.4 Boundary element Galerkin methods 489
9.4.1 A finite element method for an equation of the first kind 492

Generalizations to other boundary integral equations 496
9.5 Numerical methods using spherical polynomial approximations 496

9.5.1 The linear system for (27r + Pn1C) p = Pn f 501
9.5.2 Solution of an integral equation of the first kind 504

Implementation of the Galerkin method 509
Other boundary integral equations and general
comments 511

Discussion of the literature 512

Appendix: Results from functional analysis 516

Bibliography 519

Index 547

Preface

In this book, numerical methods are presented and analyzed for the solution of
integral equations of the second kind, especially Fredholm integral equations.
Major additions have been made to this subject in the twenty years since the
publication of my survey [39], and I present here an up-to-date account of
the subject. In addition, I am interested in methods that are suitable for the
solution of boundary integral equation reformulations of Laplaces equation,
and three chapters are devoted to the numerical solution of such boundary
integral equations. Boundary integral equations of the first kind that have a
numerical theory closely related to that for boundary integral equations of the
second kind are also discussed.

This book is directed toward several audiences. It is first directed to numer-
ical analysts working on the numerical solution of integral equations. Second,
it is directed toward applied mathematicians, including both those interested
directly in integral equations and those interested in solving elliptic boundary
value problems by use of boundary integral equation reformulations. Finally,
it is directed toward that very large group of engineers needing to solve prob-
lems involving integral equations. In all of these cases, I hope the book is also
readable and useful to well-prepared graduate students, as I had them in mind
when writing the book.

During the period of 1960-1990, there has been much work on developing
and analyzing numerical methods for solving linear Fredholm integral equations
of the second kind, with the integral operator being compact on a suitable space
of functions. I believe this work is nearing a stage in which there will be few
major additions to the theory, especially as regards equations for functions of
a single variable; but I hope to be proven wrong. In Chapters 2 through 6, the
main aspects of the theory of numerical methods for such integral equations
is presented, including recent work on solving integral equations on surfaces

xv

xvi Preface

in W. Chapters 7 through 9 contain a presentation of numerical methods for
solving some boundary integral equation reformulations of Laplaces equation,
for problems in both two and three dimensions. By restricting the presentation
to Laplaces equation, a simpler development can be given than is needed when
dealing with the large variety of boundary integral equations that have been
studied during the past twenty years. For a more complete development of
the numerical solution of all forms of boundary integral equations for planar
problems, see ProBdorf and Silbermann [438].

In Chapter 1, a brief introduction/review is given of the classical theory of
Fredholm integral equations of the second kind in which the integral operator
is compact. In presenting the theory of this and the following chapters, a
functional analysis framework is used, which is generally set in the space of
continuous functions C (D) or the space of square integrable functions L2(D).
Much recent work has been in the framework of Sobolev spaces H (D), which
is used in portions of Chapter 7, but I believe the simpler framework given here is
accessible to a wider audience in the applications community. Therefore, I have
chosen this simpler sframework in preference to regarding boundary integral
equations as pseudodifferential operator equations on Sobolev spaces. The
reader still will need to have some knowledge of functional analysis, although
not a great deal, and a summary of some of the needed results from functional
analysis is presented in the appendix.

I would like to thank Mihai Anitescu, Paul Martin, Matthew Schuette, and
Jaehoon Seol, who found many typographical and other errors in the book. It
is much appreciated. I thank the production staff at TechBooks, Inc. for their
fine work in turning my manuscript into the book you see here. Finally, I also
thank my wife Alice, who as always has been very supportive of me during the
writing of this book.

1

A brief discussion of integral equations

The theory and application of integral equations is an important subject within
applied mathematics. Integral equations are used as mathematical models for
many and varied physical situations, and integral equations also occur as refor-
mulations of other mathematical problems. We begin with a brief classification
of integral equations, and then in later sections, we give some of the classical
theory for one of the most popular types of integral equations, those that are
called Fredholm integral equations of the second kind, which are the principal
subject of this book. There are many well-written texts on the theory and appli-
cation of integral equations, and we note particularly those of Hackbusch [249]
Hochstadt [272], Kress [325], Mikhlin [380], Pogorzelski [426], Schmeidler
[492], Widom [568], and Zabreyko, et al. [586].

1.1. Types of integral equations

This book is concerned primarily with the numerical solution of what are called
Fredholm integral equations, but we begin by discussing the broader category
of integral equations in general. In classifying integral equations, we say, very
roughly, that those integral equations in which the integration domain varies
with the independent variable in the equation are Volterra integral equations;
and those in which the integration domain is fixed are Fredholm integral equa-
tions. We first consider these two types of equations, and the section concludes
with some other important integral equations.

1.1.1. Volterra integral equations of the second kind

The general form that is studied is

x(t) +
J

K(t, s, x(s)) ds = y(t), t > a (1.1.1)t
a

1

2 1. A brief discussion of integral equations

The functions K (t, s, u) and y (t) are given, and x (t) is sought. This is a nonlin-
ear integral equation, and it is in this form that the equation is most commonly
applied and solved. Such equations can be thought of as generalizations of

x'(t) = f (t, x(t)), t > a, x(a) = x0 (1.1.2)

the initial value problem for ordinary differential equations. This equation is
equivalent to the integral equation

t

x(t) = xo + f f (s, x(s)) ds, t > a
a

which is a special case of (1.1.1).
For an introduction to the theory of Volterra integral equations, see R. Miller

[384]. The numerical methods for solving (1.1.1) are closely related to those
for solving the initial value problem (1.1.2). These integral equations are not
studied in this book, and the reader is referred to Brunner and de Riele [96] and
Linz [345]. Volterra integral equations are most commonly studied for functions
x of one variable, as above, but there are examples of Volterra integral equations
for functions of more than one variable.

1.1.2. Volterra integral equations of the first kind

The general nonlinear Volterra integral equation of the first kind has the form

K(t, s, x(s)) ds = y(t), t > a (1.1.3)f
The functions K (t, s, u) and y (t) are given functions, and the unknown is x (s).
The general linear Volterra integral equation of the first kind is of the form

J
K(t, s)x(s) ds = y(t), t > a (1.1.4)

a

For Volterra equations of the first kind, the linear equation is the more commonly
studied case. The difficulty with these equations, linear or nonlinear, is that they
are ill-conditioned to some extent, and that makes their numerical solution
more difficult. (Loosely speaking, an ill-conditioned problem is one in which
small changes in the data y can lead to much larger changes in the solution x.)

A very simple but important example of (1.1.4) is

Jr x(s) ds = y(t), t > a (1.1.5)

This is equivalent to y(a) = 0 and x(t) = y'(t), t > a. Thus the numerical

1.1. Types of integral equations 3

solution of (1.1.5) is equivalent to the numerical differentiation of y(t). For
a discussion of the numerical differentiation problem from this perspective,
see Cullum [149] and Anderssen and Bloomfield [11], and for the numerical
solution of the more general equation (1.1.4), see Linz [345].

1.1.3. Abel integral equations of the first kind

An important case of (1.1.4) is the Abel integral equation

ft
H(t, s)x(s)

ds = y(t), t > 0 (1.1.6)
(tP SP)a

Here 0 < a < 1 and p > 0, and particularly important cases are p = 1 and
p = 2 (both with a = 2). The function H(t, s) is assumed to be smooth
(that is, several times continuously differentiable). Special numerical methods
have been developed for these equations, as they occur in a wide variety of
applications. For a general solvability theory for (1.1.6), see Ref. [35], and for
a discussion of numerical methods for their solution, see Linz [345], Brunner
and de Riele [96], and Anderssen and de Hoog [12].

1.1.4. Fredholm integral equations of the second kind

The general form of such an integral equation is

Ax(t)
J

K(t, s)x(s) ds = y(t), t e D, A 0 0 (1.1.7)
D

with D a closed bounded set in R, some in > 1. The kernel function K (t, s)
is assumed to be absolutely integrable, and it is assumed to satisfy other prop-
erties that are sufficient to imply the Fredholm Alternative Theorem (see The-
orem 1.3.1 in 1.3). For y 0 0, we have A and y given, and we seek x; this is
the nonhomogeneous problem. For y = 0, equation (1.1.7) becomes an eigen-
value problem, and we seek both the eigenvalue A and the eigenfunction x. The
principal focus of the numerical methods presented in the following chapters
is the numerical solution of (1.1.7) with y # 0. In the next two sections we
present some theory for the integral operator in (1.1.7).

1.1.5. Fredholm integral equations of the first kind

These equations take the form

JD
K(t, s)x(s) ds = y(t), t E D (1.1.8)

4 1. A brief discussion of integral equations

with the assumptions on K and D the same as in (1.1.7). Such equations are
usually classified as ill-conditioned, because their solution x is sensitive to small
changes in the data function y. For practical purposes, however, these problems
need to be subdivided into two categories. First, if K (t, s) is a smooth function,
then the solution x (s) of (1.1.8) is extremely sensitive to small changes in y (t),
and special methods of solution are needed. For excellent introductions to this
topic, see Groetsch [241], [242], Kress [325, Chaps. 15-17] and Wing [572].
If however, K (t, s) is a singular function, then the ill-conditioning of (1.1.8) is
quite manageable; and indeed, much of the theory for such equations is quite
similar to that for the second-kind equation (1.1.7). Examples of this type of
first-kind equation occur quite frequently in the subject of potential theory, and
a well-studied example is

J
log It slx(s) ds = y(t), t E F (1.1.9)I

with I a curve in R2. This and other similarly behaved first-kind equations will
be discusssed in Chapters 7 and 8.

1.1.6. Boundary integral equations

These equations are integral equation reformulations of partial differential equa-
tions. They are widely studied and applied in connection with solving boundary
value problems for elliptic partial differential equations, but they are also used
in connection with other types of partial differential equations.

As an example, consider solving the problem

Au(P) = 0, P E D

u(P) = g(P), PET

where D is a bounded region in R3 with nonempty interior, and r is the boundary
of D. From the physical setting for (1.1.10)-(1.1.11), there is reason to believe
that u can be written as a single layer potential:

U(P)
p(Q) dQ, PED (1.1.12)

r IP QI

In this, I P Q I denotes the ordinary Euclidean distance between P and Q. The
function p (Q) is called a single layer density function, and it is the unknown
in the equation. Using the boundary condition (1.1.11), it is straightforward to

1.1. Types of integral equations 5

show that

I p(Q) dQ = g(P), P E F (1.1.13)
Jr IP QI

This equation is solved for p, and then (1.1.12) is used to obtain the solution of

Boundary integral equations can be Fredholm integral equations of the first
or second kind, Cauchy singular integral equations (see 1.1.8 below), or modifi-
cations of them. In the literature, boundary integral equations are often referred
to as BIE, and methods for solving partial differential equations via the bound-
ary integral equation reformulation are called BIE methods. There are many
books and papers written on BIE methods; for example, see Refs. [50], [55],
Jaswon and Symm [286], Kress [325, Chaps. 6, 8, 9], Sloan [509] and the
references contained therein.

1.1.7. Wiener-Hopf integral equations

These have the form

Ax(t) 100 k(t s)x(s) ds = y(t), 0 < t < oo (1.1.14)0Originally, such equations were studied in connection with problems in radiativetransfer, and more recently, they have been related to the solution of boundaryintegral equations for planar problems in which the boundary is only piecewisesmooth. A very extensive theory for such equations is given in Krein [322], andmore recently, a simpler introduction to some of the more important parts ofthe theory has been given by Anselone and Sloan [20] and deHoog and Sloan[163].1.1.8. Cauchy singular integral equationsLet I, be an open or closed contour in the complex plane. The general form ofa Cauchy singular integral equation is given bya(z)O(z) +b(z)0(0 d + I K(z, d _ i(z), z E r(1.1.15)The functions a, b, r, and K are given complex-valued functions, and 0is the unknown function. The function K is to be absolutely integrable; andin addition, it is to be such that the associated integral operator is a Fredholm6 1. A brief discussion of integral equationsintegral operator in the sense of 1.1.4 above. The first integral in (1.1.15) isinterpreted as a Cauchy principal value integral:fF0(0 d = lim fd(1.1.16)E F I I – zI > E}. Cauchy singular integral equations occur
in a variety of physical problems, especially in connection with the solution
of partial differential equations in R2. Among the best known books on the
theory and application of Cauchy singular integral equations are Muskhelishvili
[390] and Gakhov [208]; and an important more recent work is that of Mikhlin
and Pr6Bdorf [381]. For an introduction to the numerical solution of Cauchy
singular integral equations, see Elliott [177], [178] and Pr6Bdorf and Silbermann
[438].

1.2. Compact integral operators

The framework we present in this and the following sections is fairly abstract,
and it might seem far removed from the numerical solution of actual integral
equations. In fact, our framework is needed to understand the behavior of
most numerical methods for solving Fredholm integral equations, including
the answering of questions regarding convergence, numerical stability, and
asymptotic error estimates. The language of functional analysis has become
more standard in the past few decades, and so in contrast to our earlier book [39,
Part I], we do not develop here any results from functional analysis, but rather
state them in the appendix and refer the reader to other sources for proofs.

When X is a finite dimensional vector space and A: X X is linear, the
equation Ax = y has a well-developed solvability theory. To extend these
results to infinite dimensional spaces, we introduce the concept of a compact
operator 1C; and then in the following section, we give a theory for operator
equations Ax = y in which A = I IC.

Definition. Let X and y be normed vector spaces, and let IC: X y be
linear. Then IC is compact if the set

{KxIllxllx<<-1}has compact closure in Y. This is equivalent to saying that for every boundedsequence {x } C X, the sequence {1Cx, } has a subsequence that is convergentto some point in Y. Compact operators are also called completely continuousoperators. (By a set S having compact closure in Y, we mean its closure S is acompact set in y.)1.2. Compact integral operators 7There are other definitions for a compact operator, but the above is the oneused most commonly. In the definition, the spaces X and Y need not be com-plete; but in virtually all applications, they are complete. With completeness,some of the proofs of the properties of compact operators become simpler, andwe will always assume X and Y are complete (that is, Banach spaces) whendealing with compact operators.1.2.1. Compact integral operators on C(D)Let D be a closed bounded set in R”, some in > 1, and define

Kx(t) =
J

K(t, s)x(s) ds, t E D, x E C(D) (1.2.17)
D

Using C (D) with II II,, we want to show that IC : C (D) -* C (D) is both
bounded and compact. We assume K (t, s) is Riemann-integrable as a function
of s, for all t c D, and further we assume the following.

K1. limh,o to (h) = 0, with

w(h) = max max [ IK(t, s) K(r, s) I ds (1.2.18)
t,rED It-rl