Linear Integral Equations

PrefaceChapter 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 NotationChapter 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 ExercisesChapter 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 ExercisesChapter 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 ExercisesChapter 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 ExercisesChapter 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 ExercisesChapter 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 ExercisesChapter 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 ExercisesChapter 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 ExercisesChapter 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 ExercisesChapter 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 ExercisesAppendixBibliographyIndex