Linear Integral Equations
1st Edition
Theory and Technique
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Description
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.
Table of Contents
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
Details
- No. of pages:
- 310
- Language:
- English
- Copyright:
- © Academic Press 1971
- Published:
- 28th June 1971
- Imprint:
- Academic Press
- eBook ISBN:
- 9781483262505
About the Author
Ram P. Kanwal
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