Kanwal Linear Integral Equations

Theory and Technique
1. Auflage 2014
ISBN: 978-1-4832-6250-5
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 310 Seiten, Web PDF

Linear Integral Equations: Theory and Technique is an 11-chapter text that covers the theoretical and methodological aspects of linear integral equations. After a brief overview of the fundamentals of the equations, this book goes on dealing with specific integral equations with separable kernels and a method of successive approximations. The next chapters explore the properties of classical Fredholm theory and the applications of linear integral equations to ordinary and partial differential equations. These topics are followed by discussions of the symmetric kernels, singular integral equations, and the integral transform methods. The final chapters consider the applications of linear integral equations to mixed boundary value problems. These chapters also look into the integral equation perturbation methods. This book will be of value to undergraduate and graduate students in applied mathematics, theoretical mechanics, and mathematical physics.

Autoren/Hrsg.

Kanwal, Ram P.

Inhaltsverzeichnis

1;Front Cover;1
2;Linear Integral Equations: Theory and Technique;4
3;Copyright Page;5
4;Table of Contents;8
5;PREFACE;12
6;CHAPTER 1. INTRODUCTION;16
6.1;1.1 Definition;16
6.2;1.2 Regularity Conditions;18
6.3;1.3 Special Kinds of Kernels;19
6.4;1.4 Eigenvalues and Eigenfunctions;20
6.5;1.5 Convolution Integral;20
6.6;1.6 The Inner or Scalar Product of Two Functions;21
6.7;1.7 Notation;22
7;CHAPTER 2. INTEGRAL EQUATIONS WITH SEPARABLE KERNELS;23
7.1;2.1 Reduction to a System of Algebraic Equations;23
7.2;2.2 Examples;25
7.3;2.3 Fredholm Alternative;29
7.4;2.4 Examples;36
7.5;2.5 An Approximate Method;38
7.6;Exercises;39
8;CHAPTER 3. METHOD OF SUCCESSIVE APPROXIMATIONS;41
8.1;3.1 Iterative Scheme;41
8.2;3.2 Examples;46
8.3;3.3 Volterra Integral Equation;50
8.4;3.4 Examples;50
8.5;3.5 Some Results about the Resolvent Kernel;52
8.6;Exercises;54
9;CHAPTER 4. CLASSICAL FREDHOLM THEORY;56
9.1;4.1 The Method of Solution of Fredholm;56
9.2;4.2 Fredholm's First Theorem;58
9.3;4.3 Examples;64
9.4;4.4 Fredholm's Second Theorem;66
9.5;4.5 Fredholm's Third Theorem;72
9.6;Exercises;75
10;CHAPTER 5. APPLICATIONS TO ORDINARY DIFFERENTIAL EQUATIONS;76
10.1;5.1 Initial Value Problems;76
10.2;5.2 Boundary Value Problems;79
10.3;5.3 Examples;82
10.4;5.4 Dirac Delta Function;85
10.5;5.5 Green's Function Approach;87
10.6;5.6 Examples;97
10.7;5.7 Green's Function for Nth-Order Ordinary Differential Equation;99
10.8;5.8 Modified Green's Function;101
10.9;5.9 Examples;104
10.10;Exercises;106
11;CHAPTER 6. APPLICATIONS TO PARTIAL DIFFERENTIAL EQUATIONS;109
11.1;6.1 Introduction;109
11.2;6.2 Integral Representation Formulas for the Solutions of the Laplace and Poisson Equations;111
11.3;6.3 Examples;118
11.4;6.4 Green's Function Approach;121
11.5;6.5 Examples;125
11.6;6.6 The Helmholtz Equation;131
11.7;6.7 Examples;133
11.8;Exercises;142
12;CHAPTER 7. SYMMETRIC KERNELS;147
12.1;7.1 Introduction;147
12.2;7.2 Fundamental Properties of Eigenvalues and Eigenfunctions for Symmetric Kernels;153
12.3;7.3 Expansion in Eigenfunctions and Bilinear Form;157
12.4;7.4 Hilbert-Schmidt Theorem and Some Immediate Consequences;161
12.5;7.5 Solution of a Symmetric Integral Equation;166
12.6;7.6 Examples;168
12.7;7.7 Approximation of a General L2-Kernel (Not Necessarily Symmetric) by a Separable Kernel;172
12.8;7.8 The Operator Method in the Theory of Integral Equations;173
12.9;7.9 Rayleigh-Ritz Method for Finding the First Eigenvalue;176
12.10;Exercises;179
13;CHAPTER 8. SINGULAR INTEGRAL EQUATIONS;182
13.1;8.1 The Abel Integral Equation;182
13.2;8.2 Examples;186
13.3;8.3 Cauchy Principal Value for Integrals;188
13.4;8.4 The Cauchy-Type Integrals;192
13.5;8.5 Solution of the Cauchy-Type Singular Integral Equation;195
13.6;8.6 The Hilbert Kernel;199
13.7;8.7 Solution of the Hilbert-Type Singular Integral Equation;202
13.8;8.8 Examples;205
13.9;Exercises;206
14;CHAPTER 9. INTEGRAL TRANSFORM METHODS;209
14.1;9.1 Introduction;209
14.2;9.2 Fourier Transform;211
14.3;9.3 Laplace Transform;212
14.4;9.4 Applications to Volterra Integral Equations with Convolution-Type Kernels;213
14.5;9.5 Examples;215
14.6;9.6 Hilbert Transform;222
14.7;9.7 Examples;225
14.8;Exercises;226
15;CHAPTER 10. APPLICATIONS TO MIXED BOUNDARY VALUE PROBLEMS;229
15.1;10.1 Two-Part Boundary Value Problems;229
15.2;10.2 Three-Part Boundary Value Problems;233
15.3;10.3 Generalized Two-Part Boundary Value Problems;244
15.4;10.4 Generalized Three-Part Boundary Value Problems;249
15.5;10.5 Further Examples;257
15.6;Exercises;264
16;CHAPTER 11. INTEGRAL EQUATION PERTURBATION METHODS;265
16.1;11.1 Basic Procedure;265
16.2;11.2 Applications to Electrostatics;268
16.3;11.3 Low-Reynolds-Number Hydrodynamics;272
16.4;11.4 Elasticity;287
16.5;11.5 Theory of Diffraction;294
16.6;Exercises;297
17;APPENDIX;300
18;BIBLIOGRAPHY;305
19;INDEX;308