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 Back cover