Integral Equations

Title
Integral Equations
Copyright
Contents
Preface
Acknowledgements
1. Basic Concepts
	1.1 Introduction
	1.2 Abel’s Problem
	1.3 Initial Value Problem and Boundary Value Problem
	1.4 Integral Equation
	1.5 Special Kinds of Kernels
	1.6 Classification of Integral Equation
	1.7 Iterated Kernels
	1.8 Reciprocal Kernal or Resolvent Kernel
	1.9 Eigenvalues and Eigenfunctions
	1.10 Solution of an Integral Equation
	Exercise 1.1
2. Applications to Ordinary Differential Equations
	2.1 Introduction
	2.2 Method of Conversion of an Initial Value Problem to a
 Volterra Integral Equation
	Exercise 2.1
	2.3 Alternate Method of Transforming the Initial Value
 Problem into a Volterra Integral Equation
	Exercise 2.2
	2.4 Boundary Value Problem and its Conversion to Fredholm
 Integral Equation
	Exercise 2.3
3. Solution of Homogeneous Fredholm Integral
 Equations of the Second Kind
	3.1 Introduction
	3.2 Characteristic Value (or Eigenvalue) and Characteristic
 Function (or Eigenfunction
	3.3 Solution of Homogeneous Fredholm Integral Equation
 of the Second Kind with Separable (or Degenerate) Kernel
	3.4 Orthogonality of Two Functions
	3.5 Orthogonality of Eigenfunctions
	3.6 Real Eigenvalues
	Exercise 3.1
4. Fredholm Integral Equations with
 Separable Kernels
	4.1 Introduction
	4.2 Solution of Fredholm Integral Equation of the Second Kind with
 Separable (or Degenerate) Kernel
	Exercise 4.1
5. Integral Equations with Symmetric Kernels
	5.1 Introduction
	5.2 Symmetric Kernal
	5.3 Regularity Condition
	5.4 Inner or Scalar Product of Two Functions
	5.5 Orthogonal System of Functions
	5.6 Fundamental Properties of Eigenvalues and
 Eigenfunctions of Symmetric Kernels
	5.7 Hilbert–Schmidt Theorem
	5.8 Schmidt’s Solution of Non-homogeneous Fredholm
 Integral Equation of the Second Kind
	Practice Questions with Intermediate Results
	Exercise 5.1
6. Solution of Integral Equations of the Second
 Kind by Successive Approximation
	6.1 Introduction
	6.2 Iterated Kernel or Function
	6.3 Resolvent Kernel or Reciprocal Kernel
	6.4 Solution of Fredholm Integral Equation of the Second
 Kind by Successive Substitution
	6.5 Solution of Volterra Integral Equation of the Second
 Kind by Successive Substitutions
	6.6 Solution of Fredholm Integral Equation of the Second
 Kind by Successive Approximations: Iterative Method
 (Iterative Scheme) Neumann Series
	6.7 Resolvent Kernel of a Fredholm Integral Equation
	6.8 Illustrations Based on the Solution of Fredholm Integral
 Equation by Successive Approximations (Iterative Method
	Exercise 6.1
	Exercise 6.2
	6.9 Reciprocal Functions
	6.10 Another Approach to Solve Fredholm Integral
 Equation of the Second Kind (Volterra Solution
	6.11 Solution of Volterra Integral Equation of the Second
 Kind by Successive Approximations: Iterative Method
 (Neumann Series
	6.12 Resolvent Kernel and Volterra Integral Equation
	6.13 Illustrations to Explain the Solution of Volterra Integral
 Equation by Successive Approximations (or Iterative
 Method
	Exercise 6.3
7. Classical Fredholm Theory
	7.1 Introduction
	7.2 Fredholm’s First Theorem
	7.3 Working Rule for Evaluating the Resolvent Kernel and
 Solution of Fredholm Integral Equation of the Second
 Kind by Using Fredholm’s First Theorem
	7.4 Fredholm’s Second Fundamental Theorem
	7.5 Fredholm’s Third Theorem
	Exercise 7.1
8. Integral Transform Methods
	8.1 Introduction
	8.2 Singular Integral Equation
	8.3 Laplace Transform
	8.4 Some Important Properties of Laplace Transform
	8.5 Inverse Laplace Transform
	8.6 Some Important Properties of Inverse Laplace Transform
	8.7 Convolution of Two Functions
	8.8 The Heaviside Expansion Formula
	8.9 The Complex Inversion Formula
	8.10 Integral Equations in Special Forms
	8.11 Application of Laplace Transform to Find the Solutions of Volterra
 Integral Equation
	8.11.1 Convolution Type Kernels of Volterra Integral Equation:
 Working Procedure
	8.11.2 Resolvent Kernel of Volterra Integral Equation by Using
 Laplace Transform
	8.11.3 Solution of Integral Equations of the Type
 by Using Laplace
 Transform: Working Procedure
	8.12 Fourier Transforms and Their Important Properties
	8.13 Application of Fourier Transform to Determine the Solution of
 Singular Integral Equations
Exercise 8.1
Index
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