Linear Integral Equations

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.

Preface

Chapter 1. Introduction

1.1 Definition

1.2 Regularity Conditions

1.3 Special Kinds of Kernels

1.4 Eigenvalues and Eigenfunctions

1.5 Convolution Integral

1.6 The Inner or Scalar Product of Two Functions

1.7 Notation

Chapter 2. Integral Equations with Separable Kernels

2.1 Reduction to a System of Algebraic Equations

2.2 Examples

2.3 Fredholm Alternative

2.4 Examples

2.5 An Approximate Method

Exercises

Chapter 3. Method of Successive Approximations

3.1 Iterative Scheme

3.2 Examples

3.3 Volterra Integral Equation

3.4 Examples

3.5 Some Results about the Resolvent Kernel

Exercises

Chapter 4. Classical Fredhold Theory

4.1 The Method of Solution of Fredholm

4.2 Fredholm's First Theorem

4.3 Examples

4.4 Fredholm's Second Theorem

4.5 Fredholm's Third Theorem

Exercises

Chapter 5. Applications to Ordinary Differential Equations

5.1 Initial Value Problems

5.2 Boundary Value Problems

5.3 Examples

5.4 Dirac Delta Function

5.5 Green's Function Approach

5.6 Examples

5.7 Green's Function for Nth-Order Ordinary Differential Equation

5.8 Modified Green's Function

5.9 Examples

Exercises

Chapter 6. Applications to Partial Differential Equations

6.1 Introduction

6.2 Integral Representation Formulas for the Solutions of the Laplace and Poisson Equations

6.3 Examples

6.4 Green's Function Approach

6.5 Examples

6.6 The Helmholtz Equation

6.7 Examples

Exercises

Chapter 7. Symmetric Kernels

7.1 Introduction

7.2 Fundamental Properties of Eigenvalues and Eigenfunctions for Symmetric Kernels

7.3 Expansion in Eigenfunctions and Bilinear Form

7.4 Hilbert-Schmidt Theorem and Some Immediate Consequences

7.5 Solution of a Symmetric Integral Equation

7.6 Examples

7.7 Approximation of a General L2-Kernel (Not Necessarily Symmetric) by a Separable Kernel

7.8 The Operator Method in the Theory of Integral Equations

7.9 Rayleigh-Ritz Method for Finding the First Eigenvalue Exercises

Chapter 8. Singular Integral Equations

8.1 The Abel Integral Equation

8.2 Examples

8.3 Cauchy Principal Value for Integrals

8.4 The Cauchy-Type Integrals

8.5 Solution of the Cauchy-Type Singular Integral Equation

8.6 The Hilbert Kernel

8.7 Solution of the Hilbert-Type Singular Integral Equation

8.8 Examples

Exercises

Chapter 9. Integral Transform Methods

9.1 Introduction

9.2 Fourier Transform

9.3 Laplace Transform

9.4 Applications to Volterra Integral Equations with Convolution-Type Kernels

9.5 Examples

9.6 Hilbert Transform

9.7 Examples

Exercises

Chapter 10. Applications to Mixed Boundary Value Problems

10.1 Two-Part Boundary Value Problems

10.2 Three-Part Boundary Value Problems

10.3 Generalized Two-Part Boundary Value Problems

10.4 Generalized Three-Part Boundary Value Problems

10.5 Further Examples

Exercises

Chapter 11. Integral Equation Perturbation Methods

11.1 Basic Procedure

11.2 Applications to Electrostatics

11.3 Low-Reynolds-Number Hydrodynamics

11.4 Elasticity

11.5 Theory of Diffraction

Exercises

Appendix

Bibliography

Index