| Preface |
| Acknowledgements |
| 1 Integral Equations, Origin, and Basic Tools |
| 1.1 Various Problems as Integral Equations |
| Exercises 1.1 |
| 1.2 Classification of Integral Equations |
| Exercises 1.2 |
| 1.3 Some Important Identities and Basic Definitions |
| 1.3.1 Multiple Integrals Reduced to Single Integrals |
| 1.3.2 Generalized Leibnitz Formula |
| 1.3.3 Convergence of Integrals and Basic Definitions |
| Exercises 1.3 |
| 1.4 Laplace, Fourier, and Other Transforms |
| 1.4.1 The Laplace Transform |
| 1.4.2 Fourier Transforms |
| 1.4.3 Other Transforms |
| Exercises 1.4 |
| 1.5 Basic Numerical Integration Formulas |
| 1.5.1 Basic (Elementary) Integration Formulas |
| 1.5.2 The Smoothing Effect of Integration |
| 1.5.3 Interpolation of the Numerical Solutions of Integral Equations |
| 1.5.4 Review of Cramer's Rule |
| Exercises 1.5 |
| 2 Modeling of Problems as Integral Equations |
| 2.1 Population Dynamics |
| 2.1.1 Human Population |
| 2.1.2 Biological Species Living Together |
| Exercises 2.1 |
| 2.2 Control and Other Problems |
| 2.2.1 Mortality of Equipment and Rate of Replacement |
| Exercises 2.2 |
| 2.2 Mechanics Problems |
| 2.3.1 Hanging Chain |
| 2.3.2 Sliding a Bead Along a Wire: Abel's Problem |
| Exercises 2.3 |
| 2.4 Initial Value Problems Reduced to Volterra Integral Equations |
| Exercises 2.4 |
| 2.5 Boundary Value Problems Reduced to Fredholm Integral Equations |
| Exercises 2.5 |
| 2.6 Mixed Boundary Conditions: Dual Integral Equations |
| 2.6.1 Electrified Infinite Plane |
| 2.6.2 Electrified Disc |
| Exercises 2.6 |
| 2.7 Integral Equaitons in Higher Dimensions |
| 2.7.1 Schrodinger Equations as an Integral Equation in the Three-Dimensional Momentum Space |
| 3 Volterra Integral Equations |
| 3.1 Volterra Equations of the Second Kind |
| 3.1.1 Resolvent Kernel Method: Neumann Series |
| 3.1.2 Method of Successive Approximations(Iterations) |
| 3.1.3 Laplace Transform Method: Difference Kernel |
| Exercises 3.1 |
| 3.2 Volterra Integral Equation of the First Kind with a Difference Kernel-Laplace Transform Method |
| Exercises 3.2 |
| 3.3 Numerical Solution of Volterra Integral Equations |
| Exercises 3.3 |
| 4 Green's Function |
| 4.1 Construction of the Green's Function |
| 4.1.1 Nonhomogeneous Differential Equations |
| 4.1.2 Construction of the Green's Function-Variation of Parameters Method |
| 4.1.3 Orthogonal Series Representation of Green's Function |
| 4.1.4 Green's Function in Two Dimensions |
| Exercises 4.1 |
| 4.2 Fredholm Integral Equations and the Green's Function |
| Exercises 4.2 |
| 5 Fredholm Interal Equations |
| 5.1 Fredholm Integral Equations with Degenerate Kernel |
| 5.1.1 Nonhomogeneous Fredholm Equations with Degenerate Kernel |
| 5.1.2 Fredholm Alternative |
| 5.1.3 Approximating a Kernel by a Degenerative One |
| Exercises 5.1 |
| 5.2 Fredholm Integral Equations with Symmetric Kernel |
| 5.2.1 Homogeneous Fredholm Equations with Symmetric Kernel |
| 5.2.2 Solution of Fredholm Equations of the Second Kind with Symmetric Kernel |
| Exercises 5.2 |
| 5.3 Fredholm Integral Equations of the Second Kind |
| 5.3.1 Method of Fredholm Resolvent Kernel |
| 5.3.2 Method of Iterated Kernels |
| 5.3.3 Some Basic Approximate Methods |
| Exercises 5.3 |
| 5.4 Fredholm Integral Equations of the First Kind |
| 5.4.1 Fredholm Equations of the First Kind with Symmetric Kernels |
| 5.4.2 Ill-Posed Problems and the Fredholm Equation of the First Kind |
| Exercises 5.4 |
| 5.5 Numerical Solution of Fredholm Integral Equations |
| 5.5.1 Numerical Approximation Setting of Fredholm |
| 5.5.2 Homogeneous Fredholm Equations |
| Exercises 5.5 |
| 6 Existence of the Solutions: Basic Fixed Point Theorms |
| 6.1 Preliminaries: Toward a Contractive Mapping |
| 6.1.1 Basic Definitions: Complete Metric Spaces |
| 6.1.2 Contractive Mapping for Linear Fredholm Equations |
| 6.1.3 Contractive Mapping for Linear Volterra Equations |
| 6.2 Fixed Point Theorm of Banach |
| 6.2.1 Existence of the Solution for Linear Integral Equations |
| 6.2.2 Existence of the Solution for Nonlinear Integral Equations |
| 6.2.3 Existence of the Solution for Nonlinear Differential Equations |
| 7 Higher Quadrature Rules for the Numerical Solution |
| 7.1 Higher Quadrature Rules of Integration with Tables |
| Exercises 7.1 |
| 7.2 Higher Quadrature Rules for Volterra Equations |
| Exercises 7.2 |
| 7.3 Higher Quadrature Rules for Fredholm Equations |
| 7.3.1 Comments on Higher Quadrature Rules for Some Singular Fredholm Equations |
| Exercises 7.3 |
| Appendix A The Hankel Transforms |
| A.1 The Hankel Transform for the Electrified Disc |
| A.2 The Finite Hankel Transform |
| Exercises: Appendix A |
| Appendix B Green's Function for Various Boundary Value Problems |
| B.1 Green's Functions in Terms of Simple Functions |
| B.2 Green's Function in Terms of Special Functions |
| Answers to Exercises |
| Chapter 1 |
| Chapter 2 |
| Chapter 3 |
| Chapter 4 |
| Chapter 5 |
| Chapter 7 |
| Appendix A |
| References |
| Index |