Computational Methods for Linear Integral Equations

Transcription

1 Computational Methods for Linear Integral Equations

2 Prem K. Kythe Pratap Puri Computational Methods for Linear Integral Equations With 12 Figures Springer Science+Business Media, LLC

3 Prem K. Kythe Professor Emeritus of Mathematics University of New Orleans New Orleans, LA USA Pratap Puri Department of Mathematics University of New Orleans New Orleans, LA USA Library of Congress Cataloging-in-Publication Data Kythe, Prem K. Computational methods for linear integral equations / Prem K. Kythe, Pratap Puri. p. em. Includes bibliographieal referenees and index. ISBN ISBN (ebook) DOI / Integral equations-numerieal solutions. 1. Puri, Pratap, II. Title. QA431.K '.45--de AMS Subject Classifieations: 45XX, 45-04, 45A05, 45B05, 45D05, 45EXX, 34A12, 45C05, 45E05 Printed on acid-free paper Springer Science+Business Media New York Originally published by Birkhăuser Boston in 2002 Softcover reprint of the hardcover 1 st edition 2002 Ali rights reserved. This work may not be translated or eopied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC exeept for brief excerpts in connection with reviews or scholarly analysis. Use in conneetion with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. ISBN SPIN Produetion managed by Louise Farkas; manufacturing supervised by Joe Quatela. Camera-ready copy provided by the authors

4 Contents Preface xi Notation xv 1 Introduction Notation and Definitions Classification Kinds and Types Convolution of Kernels Function Spaces Convergence Inverse Operator Nystrom System Other Types of Kernels N ondegenerate Kernels Degenerate Kernels Neumann Series Resolvent Operator Fredholm Alternative Eigenvalue Problems Linear Symmetric Equations Ritz Method Method of Moments Kellogg's Method Trace Method Residual Methods Collocation Method v

5 vi CONTENTS LeastcSquares Method Galerkin Method Degenerate Kernels Replacement by a Degenerate Kernel Bateman's Method Generalized Eigenvalue Problem Applications Equations of the Second Kind Fredholm Equations Systems ofintegral Equations Taylor's Series Method Volterra Equations Quadrature Methods Block-by-Block Method Classical Methods for FK Expansion Method Product-Integration Method Quadrature Method Deferred Correction Methods A Modified Quadrature Method Collocation Methods Elliott's Modification Variational Methods Ga1erkin Method Ritz-Galerkin Methods Special Cases Fredholm-Nystrom System Iteration Methods Simple Iterations Quadrature Formulas Error Analysis Iterative Scheme Krylov-Bogoliubov Method Singular Equations Singularities in Linear Equations Fredholm Theorems Modified Quadrature Rule Convolution-Type Kernels Volterra-Type Singular Equations

6 CONTENTS vii 7.6 Convolution Methods Convolution of an SVK Convolution of an SVK Abel's Equation Sonine's Equation Logarithmic Kernel Asymptotic Methods for Log-Singular Equations Iteration Methods Atkinson's Scheme Brakhage's Scheme Atkinson's Direct Scheme Kelley's Algorithm Singular Equations with the Hilbert Kernel First-Kind Equations Second-Kind Equations Finite-Part Singular Equations Weakly Singular Equations Weakly Singular Kernel Taylor's Series Method Lp-Approximation Method Product-Integration Method Atkinson's Method Generalization Atkinson's Modification Asymptotic Expansions Splines Method From Singular SK2 to IDE B-spline Method From Log-Singular to Cauchy-Singular A Class of Singular Integrals Weakly Singular Volterra Equations Cauchy Singular Equations Cauchy Singular Equations of the First Kind Approximation by Trigonometric Polynomials Cauchy Singular Equations of the Second Kind From CSK2 to FK Gauss-Jacobi Quadrature Solution by Jacobi Polynomials Collocation Method for CSK A Special Case Generalized Cauchy Kernel Collocation Method for CSK Canonical Equation

7 viii CONTENTS 10 Sinc-Galerkin Methods Sine Function Approximations Conformal Maps and Interpolation Approximation Theory Convergence Sinc-Galerkin Scheme Computation Guidelines Sine-Collocation Method Single-Layer Potential Double-Layer Problem Equations of the First Kind Inherent Ill-Posedness Separable Kernels Some Theorems Numerical Methods Quadrature Method Method of Moments Regularization Methods Other Regularizations Comparison From FK1 to FK Equations over a Contour Volterra Equations of the First Kind Quadrature Method Method of Linz Product-Integration Method Abel's Equation Iterative Schemes Hanna and Brown's Scheme Van den Berg's Scheme Inversion of Laplace Transforms Laplace Transforms General Interpolating Scheme Inverse in Terms of Legendre Polynomials Inverse in Terms of Shifted Legendre Polynomials Method of Bellman, Kalaba, and Lockett Solution of the System Inverse in Terms of Laguerre Polynomials Inverse in Terms of Chebyshev Polynomials Inverse in Terms of Jacobi Polynomials Inversion by Fourier Series Dubner and Abate's Method

8 CONTENTS ix Error Analysis Examples Durbin's Improvement Error Analysis Examples Inversion by the Riemann Sum Approximate Formulas A Quadrature Rules A.1 Newton-Cotes Quadratures A.2 Gaussian Quadratures A.3 Integration of Products A.4 Singular Integrals A.5 Infinite-Range Integrals A.6 Linear Transformation of Quadratures A.7 Trigonometric Polynomials A.8 Condition Number A.7 Quadrature Tables B Orthogonal Polynomials B.1 Zeros of Some Orthogonal Polynomials C Whittaker's Cardinal Function C.1 Basic Results C.2 Approximation of an Integral D Singular Integrals D.l Cauchy's Principal-Value Integrals D.2 P.Y. of a Singular Integral on a Contour D.3 Hadamard's Finite-Part Integrals D.4 Two-Sided Finite-Part Integrals D.5 One-Sided Finite-Part Integrals D.6 Examples of Cauchy P.Y. Integrals D.7 Examples of Hadamard's Finite-Part Integrals Bibliography Subject Index

9 Preface This book presents numerical methods and computational aspects for linear integral equations. Such equations occur in various areas of applied mathematics, physics, and engineering. The material covered in this book, though not exhaustive, offers useful techniques for solving a variety of problems. Historical information covering the nineteenth and twentieth centuries is available in fragments in Kantorovich and Krylov (1958), Anselone (1964), Mikhlin (1967), Lonseth (1977), Atkinson (1976), Baker (1978), Kondo (1991), and Brunner (1997). Integral equations are encountered in a variety of applications in many fields including continuum mechanics, potential theory, geophysics, electricity and magnetism, kinetic theory of gases, hereditary phenomena in physics and biology, renewal theory, quantum mechanics, radiation, optimization, optimal control systems, communication theory, mathematical economics, population genetics, queueing theory, and medicine. Most of the boundary value problems involving differential equations can be converted into problems in integral equations, but there are certain problems which can be formulated only in terms of integral equations. A computational approach to the solution of integral equations is, therefore, an essential branch of scientific inquiry. Overview The basic terminology and notation are adopted from Porter and Stirling (1993), Baker (1978), and Mikhlin and Smolitskiy (1967). Generally, while discussing eigenvalue problems the terms 'eigenvalues' and 'characteristic values' are used interchangeably. But in the literature on integral equations we find that these terms xi

10 xii PREFACE represent different quantities. Some authors use the notation A for eigenvalues only and others the notation J-l for eigenvalues or characteristic values such that A J-l = 1. To avoid this ambiguity we have maintained a precise distinction between these two terms by denoting eigenvalues by A and characteristic values by J-l. With this notation the eigenvalues A of the kernel of an integral equation coincide with those of the corresponding differential equation. Similarly, we have used a distinct notation k for the kernel function and K for the kernel operator. A general description of the topics covered in the book is as follows. In Chapter 1, besides an outline of the subject of linear integral equations, definitions, and notations used, the concept of the Nystrom method is introduced at the outset with some examples. Eigenvalue problems are discussed in Chapter 2 and different methods are presented with examples. Chapters 3 through 6 cover Fredholm and Volterra equations of the second kind, where classical methods, like the expansion method, product-integration method, quadrature method, collocation method, Galerkin methods, and iteration methods are discussed in detail. Singular equations are introduced in Chapter 7, and different methods and iteration schemes to solve them numerically are presented. Weakly singular equations are discussed in Chapter 8 and a variety of methods for their numerical solution is presented. Singular equations of the Cauchy type are studied in Chapter 9 with different methods for their numerical solution. Chapter 10 deals with the application of Whittaker's cardinal function, and the sinc-galerkin method is presented for solving singular equations. Chapter 11 covers equations of the first kind. Their inherent ill-posedness makes them so special and difficult. Chapter 12 deals with the numerical inversion of Laplace transform. This topic is important in itself as it finds its use in numerous applied and engineering problems. There are four appendixes: Appendix A discusses the subject of numerical integration and presents different useful quadrature rules. Appendix B lists properties and results for orthogonal polynomials. The definitions of and results on Whittaker's cardinal functions are given in Appendix C. Singular integrals, including Cauchy's principal-value (p.v.) and Hadamard's finite-part integrals and their examples are presented in Appendix D. A large number of examples are solved throughout the book with computational results presented in tabular form. Salient Features The book is designed as a new source of classical as well as modern topics on the subject of the numerical computation of linear integral equations. We have not only discussed the underlying theory of integral equations and numerical analysis

11 PREFACE xiii of numerical integration and convergence but also provided Mathematica files which become a readily available source of computation and verification of numerical results. The main features of the book are as follows: 1. The notation is kept straightforward; it makes a distinction between the kernel function and the kernel operator as well as between eigenvalues and characteristic values. 2. The Nystrom method which forms the basis of the numerical approach is discussed in the very beginning with enough examples where a distinction between the Nystrom points and the quadrature points is carefully explained. 3. All quadrature rules used in the book are discussed in detail toward the end (see Appendix A). 4. Eigenvalue problems are treated in a separate chapter in the beginning of the book. S. Different kinds of linear integral equations are covered in the following order: Fredholm and Volterra equations of the second kind, singular equations, weakly singular equations, singular equations of the Cauchy type, and integral equations of the first kind. 6. Because numerical approximations fail significantly in the neighborhood of singularities of the kernel function, the solution of singular integral equations by the sinc-galerkin methods, using Whittaker's cardinal function, is presented in a separate chapter. 7. Integral equations of the first kind occupy a special place since they present an inherent ill-posedness that makes their numerical approximations difficult. Some well-known methods to deal with this situation are discussed in detail. 8. The numerical inversion of Laplace transforms is a special case of integral equations of the first kind. Because of the importance of this topic in research and industry, different numerical and computational methods that yield desirable and verifiable results are presented in a separate chapter devoted to this topic, which is usually not considered in books on integral equations. 9. The notation used in the book is presented in the very beginning in tabular form for ready reference. 10. Enough computational details are provided in our examples so that the numerical results can be verified and duplicated. 11. A bibliography toward the end of the book, though not exhaustive, covers more references than cited in the book. The purpose is to provide a sufficiently large repertoire of references for the readers to use if they so desire. 12. A subject index at the end of the book should be very useful.

12 xiv PREFACE Intended Readers The book, written primarily for graduate students, can be used as a textbook or a reference book depending on need and interest. No familiarity with any programming language, like Fortran or C, is needed. A good knowledge of numerical integration and some working knowledge of Mathematica are required. Since almost all important aspects of the numerical computation of linear integral equations along with numerical methods and auxiliary Mathematica files to generate numerical approximations are contained in this book, it definitely takes the form of a handbook that the interested reader can readily use. This is basically a graduate-level book. The readers are naturally graduate students and researchers in applied mathematics, physics, engineering, and industry. It is hoped that all such readers will find the handbook very useful. Computational Aspects It has been our experience that Mathematica is sufficient to carry out any and all computational aspects of the methods presented in the book. Some computer codes in Fortran are available in the public domain. Most of them deal with the numerical solution of Fredholm integral equations of the second kind. About 130 examples are solved in this book. Mathematica 4.0 has been used to verify most of the solutions. Mathematica files are available at the authors' websites: Acknowledgments The authors thank the Copy Editor and the Senior Production Editor at Birkhauser Boston for the fine work of editing, and the people at TechType Works, Inc., Gretna, Louisiana, for typesetting the manuscript. New Orleans, Louisiana December 2001 Prem K. Kythe Pratap Puri

13 Notation A list of the notation and abbreviations used in this book is given below. a.e. arg AxA A\B II Am AT CSKI CSK2 Ck(D) Coo (D) C(,h,x) C Coo d),(x, s) D D(>.) {ed:l e.g. E Eq(s) f(x) f(x) FKl FK2 FK3 almost everywhere argument of a complex number product of sets A and B complement of a set B with respect to a set A closure of a set A mth trace of the kernel k(x, s); (= f: km(x, s) ds) transpose of a matrix A Cauchy singular equations of the first kind Cauchy singular equations of the second kind class of continuous functions with k continuous derivatives on D infinitely differentiable functions on a domain D Whittaker's cardinal function (cardinal interpolant of ) complex plane extended complex plane Fredholm minor domain Fredholm determinant basis for example error equation(s) (when followed by an equation number) free term complex conjugate of a function f(x) Fredholm equation of the first kind Fredholm equation of the second kind Fredholm equation of the third kind xv

14 xvi NOTATION F*G FoG f :F :Fe :Fs nfm(" ;x) g(x,s) h H HSKI HSK2 H(x) H 2 [ ] Hn(x) He< HI Hn(x) i.e. iff IDE 1 lln 12N I <;S Jo(x) k(x, s) k*(x, s) k[nl(x, s) k>..(x, s) kn(x, s) kt K K* lim (x) Lip[a, b] L-1 L L2 Lp Ln(X) L~e<)(x) type 1 convolution type 2 convolution vector Fourier transform Fourier cosine transform Fourier sine transform generalized hypergeometric function regular (continuous) part of a kernel step size resolvent operator defined by L -1 = 1 +,\ H first-kind singular equation with Hilbert kernel second-kind singular equation with Hilbert kernel Heaviside unit step function Hadamard transform of Hermite polynomials of degree n HOlder condition for 0 < a ::::; 1 Lipschitz condition Hermite polynomial of degree n that is if and only if integro-differential equation identity operator one-dimensional sinc-interpolant two-dimensional sinc-interpolant identity matrix imaginary part Bessel's function of first kind and order zero kernel of an integral equation adjoint (conjugate) kernel ( = k(x, s)) degenerate (separable) kernel resolvent of the kernel k(x, s) nth iterated kernel transposed kernel integral operator operator associated with k* (also l j (x) ) Lagrange interpolating polynomial Lipschitz condition on [a, b] inverse integral operator (= (I -,\ K) -1) operator (= 1 -,\ K) function space function space, p?:: 1 Laguerre polynomials generalized Laguerre polynomial

15 NOTATION xvii Pi(X) {pi(x) } Pn Pn(X) P;'(X) p~l/)(x) p.:;,f3 PN Qj(F), Qj r[ nj r" ~ lr n lr+ S Sj Si S S(m, h)(z) S;"(z) SKI SK2 SFKI SFK2 SVKl SVK2 T Tn (x) (u,v) (u,v) -< u, v >- U U(t - a) Un (x) Vn(x) VKI VK2 VK3 Laplace transform L{f(t)} = F(s) inverse Laplace transform orthogonal polynomials set of linearly independent basis functions bounded linear projections; interpolation projections Legendre polynomials of degree n shifted Legendre polynomial Gegenbauer polynomial Jacobi polynomials of degree n projection operator quadrature rule j = 1, 2,...,N residual for the nth approximate solution n spectral radius real part Euclidean n-space set of positive real numbers variable of the Laplace transform quadrature points sine integral square {(x, s) : x, s E [a, b] x [a, b]} sinc function space of splines of order P with knot sequence z singular equation of the first kind singular equation of the second kind singular Fredholm equations of the first kind singular Fredholm equations of the second kind singular Volterra equations of the first kind singular Volterra equations of the second kind triangle {(x, s) : a:::; s :::; x :::; b} Chebyshev polynomials of the first kind of degree n inner product of u and v inner product with a weight function of u and v discrete inner product of u and v unit disk Izl < 1 unit step function Chebyshev polynomials of the second kind of degree n interpolating polynomial of degree n Volterra equation of the first kind Volterra equation of the second kind Volterra equation of the third kind weights in quadrature rules Nystrom points transpose of a matrix x ( 4.1)

16 xviii x Z(x) Oij A Ai P, V /'l, ~j p(a) p(a) u(a) <I> 1fJi(X) \7 2 NOTATION Banach space fundamental solution Kronecker delta (= 0 if i =1= j; = 1 if i = j) numerical parameter; also, eigenvalue ith eigenvalue characteristic value (= 1/ A) index regular (continuous) part of k(x, s); also, index Chebyshev points; also, Gauss points resolvent of a set A condition number of a matrix A resolvent of a set A; also, spectrum of a matrix A unknown function in an integral equation approximate value of vector of approximate solutions 's; also, blocks complete set of functions; also, ith eigenfunction Laplacian ox2 + oy2 Cauchy principal-value (p.v.) integral Hadamard (finite-part) integral summation with the first term halved summation with the first and last terms halved n n operation defined by L ai bi = L ai (x) bi (s) i=l i=l