Integral Equations

The theory of integral equations has been an active research field for many years and is based on analysis, function theory, and functional analysis. On the other hand, integral equations are of practical interest because of the «boundary integral equation method», which transforms partial differential equations on a domain into integral equations over its boundary. This book grew out of a series of lectures given by the author at the Ruhr-Universitat Bochum and the Christian-Albrecht-Universitat zu Kiel to students of mathematics. The contents of the first six chapters correspond to an intensive lecture course of four hours per week for a semester. Readers of the book require background from analysis and the foundations of numeri­ cal mathematics. Knowledge of functional analysis is helpful, but to begin with some basic facts about Banach and Hilbert spaces are sufficient. The theoretical part of this book is reduced to a minimum; in Chapters 2, 4, and 5 more importance is attached to the numerical treatment of the integral equations than to their theory. Important parts of functional analysis (e. g. , the Riesz-Schauder theory) are presented without proof. We expect the reader either to be already familiar with functional analysis or to become motivated by the practical examples given here to read a book about this topic. We recall that also from a historical point of view, functional analysis was initially stimulated by the investigation of integral equations.

1 Introduction.- 1.1 Integral Equations.- 1.2 Basics from Analysis.- 1.2.1 Continuous Functions.- 1.2.2 Lipschitz Continuous Functions.- 1.2.3 Hölder Continuous Functions.- 1.3 Basics from Functional Analysis.- 1.3.1 Banach Spaces.- 1.3.2 Banach Spaces CLIEN(D), CLk(D), ?LIEN (D).- 1.3.3 Banach Spaces L1(D), L2(D), L?(D).- 1.3.4 Dense Subspaces.- 1.3.5 Banach’s Fixed Point Theorem.- 1.3.6 Linear Operators.- 1.3.7 Theorem of Uniform Boundedness.- 1.3.8 Compact Sets and Compact Mappings.- 1.3.9 Riesz-Schauder Theory.- 1.3.10 Hilbert Spaces, Orthogonal Complements, Projections.- 1.4 Basics from Numerical Mathematics.- 1.4.1 Interpolation.- 1.4.2 Quadrature.- 1.4.3 Condition Number of a System of Equations.- 2 Volterra Integral Equations.- 2.1 Theory of Volterra Integral Equations of the Second Kind.- 2.1.1 Existence und Uniqueness of the Solution.- 2.1.2 Regularity of the Solution.- 2.2 Numerical Solution by Quadrature Methods.- 2.2.1 Derivation of the Discretisation.- 2.2.2 Error Estimate.- 2.3 Further Numerical Methods.- 2.4 Linear Volterra Integral Equations of Convolution Type.- 2.5 The Volterra Integral Equations of the First Kind.- 3 Theory of Fredholm Integral Equations of the Second Kind.- 3.1 The Fredholm Integral Equation of the Second Kind.- 3.2 Compactness of the Integral Operator K.- 3.2.1 General Considerations.- 3.2.2 The Case X = C(D).- 3.2.3 The Case X = L2(D).- 3.2.4 The Case of an Unbounded Interval I.- 3.3 Finite Approximability of the Integral Operator K.- 3.3.1 Convergence with Respect to the Operator Norm.- 3.3.2 Degenerate Kernels.- 3.4 The Image Space of K.- 3.4.1 Smooth Kernels k(x, y).- 3.4.2 The Image Kf for f?C?(I).- 3.4.3 Kernels with Integrable Singularity.- 3.4.4 Compactness.- 3.4.5 Volterra Integral Equation.- 3.4.6 K as Mapping Defined on L?(D).- 3.5 Solution of the Fredholm Integral Equation of the Second Kind.- 3.5.1 Existence and Uniqueness.- 3.5.2 Regularity.- 4 Numerical Treatment of Fredholm Integral Equations of the Second Kind.- 4.1 General Considerations.- 4.1.1 Notation of the Semidiscrete Problem.- 4.1.2 Consistency and Stability.- 4.1.3 Convergence.- 4.1.4 Stability and Convergence Theorem.- 4.1.5 Error Estimates.- 4.1.6 Condition Numbers.- 4.2 Discretisation by Kernel Approximation.- 4.2.1 Degenerate Kernels.- 4.2.2 Setting Up the System of Equations.- 4.2.3 Kernel Approximation by Interpolation.- 4.2.4 Tensor Approximation of k.- 4.2.5 Examples of Kernel Approximations.- 4.2.6 A Variant of the Kernel Approximation.- 4.2.7 Analysis of the System of Equations.- 4.2.8 Numerical Examples.- 4.3 Projection Methods in General.- 4.3.1 Subspaces.- 4.3.2 Projections.- 4.3.3 Lemmata.- 4.3.4 Discretisation by means of a Projection.- 4.3.5 Convergence Analysis.- 4.3.6 Error Estimate.- 4.4 Collocation Method.- 4.4.1 Definition of the Projection by Interpolation.- 4.4.2 Setting up the System of Linear Equations.- 4.4.3 Examples for Interpolations.- 4.4.4 Condition Number of the System of Equations.- 4.4.5 Numerical Examples.- 4.5 Galerkin Method.- 4.5.1 Subspace, Orthogonal Projection.- 4.5.2 Derivation of the System of Equations.- 4.5.3 Convergence in L2(D) and L?(D).- 4.5.4 Error Estimates.- 4.5.5 Condition Number of the System of Equations.- 4.5.6 Example: Piecewise Constant Functions.- 4.5.7 Example: Piecewise Linear Functions.- 4.5.8 General Analysis of Projection Errors.- 4.5.9 Revisited: Piecewise Linear Functions.- 4.5.10 Numerical Examples.- 4.6 Additional Comments Concerning Projection Methods.- 4.6.1 Regularisation Method.- 4.6.2 Estimates with Respect to Weaker Norms.- 4.6.3 The Iterated Approximation.- 4.6.4 Superconvergence.- 4.6.5 More General Formulations of the Projection Method.- 4.6.6 Numerical Quadrature.- 4.6.7 Product Integration.- 4.7 Discretisation by Quadrature: The Nyström Method.- 4.7.1 Description of the Method.- 4.7.2 Convergence Analysis.- 4.7.3 Stability.- 4.7.4 Consistency Order.- 4.7.5 Condition Number of the System of Equations.- 4.7.6 Regularisation.- 4.7.7 Numerical Examples.- 4.7.8 Product Integration.- 4.8 Supplements.- 4.8.1 Connection between the Discretisation Methods.- 4.8.1.1 The Kernel Approximation and the Galerkin Method.- 4.8.1.2 From the Galerkin Method to the Collocation and Nyström-Method.- 4.8.1.3 From the Collocation to the Nyström Method.- 4.8.1.4 From the Collocation to the Galerkin Method.- 4.8.2 Method of the Defect Correction.- 4.8.3 Extrapolation Method.- 4.8.4 Eigenvalue Problems.- 4.8.5 Complementary Integral Equations.- 4.8.6 Supplement: Perturbation Theorem for Stability.- 5 Multi-Grid Methods for Solving Systems Arising from Integral Equations of the Second Kind.- 5.1 Preliminaries.- 5.1.1 Notation.- 5.1.2 Direct Solution of the System of Equations.- 5.1.3 Picard Iteration.- 5.1.4 Conjugate Gradient Method.- 5.2 Stability and Convergence (Discrete Formulation).- 5.2.1 Prolongations and Restrictions.- 5.2.2 The Banach Space Y and the Discrete Spaces Yn.- 5.2.3 The Interpolation Error or Projection Error.- 5.2.4 Consistency.- 5.2.5 Stability.- 5.2.6 Convergence.- 5.3 The Hierarchy of Discrete Problems.- 5.3.1 Levels of Discretisations.- 5.3.2 Prolongations and Restrictions.- 5.3.3 Relative Consistency.- 5.3.4 Convergence.- 5.4 Two-Grid Iteration.- 5.4.1 The Two-Grid Algorithm.- 5.4.2 Convergence Analysis.- 5.4.3 Amount of Computational Work.- 5.4.4 Variant for A? ? I.- 5.4.5 Numerical Examples.- 5.5 Multi-Grid Iteration.- 5.5.1 Algorithm (Basic Version).- 5.5.2 Amount of Computational Work.- 5.5.3 Convergence.- 5.5.4 Numerical Examples.- 5.5.5 Variants of the Multi-Grid Methods.- 5.6 Nested Iteration.- 5.6.1 Algorithm.- 5.6.2 Amount of Computational Work.- 5.6.3 Convergence.- 5.6.4 Numerical Examples.- 5.6.5 Nested Iteration with Nyström Interpolation.- 6 Abel’s Integral Equation.- 6.1 Notations and Examples.- 6.1.1 Abel’s Integral Equation and its Generalisations.- 6.1.2 Examples from Applications.- 6.1.3 Improper Integrals.- 6.2 A Necessary Condition for a Bounded Solution.- 6.3 Euler’s Integrals.- 6.4 Inversion of Abel’s Integral Equation.- 6.5 Reformulation for Kernels k(x,y)/(x-y)?.- 6.6 Numerical Methods for Abel’s Integral Equation.- 7 Singular Integral Equations.- 7.1 The Cauchy Principal Value.- 7.1.1 Definition and Properties.- 7.1.2 Curvilinear Integrals.- 7.1.3 Cauchy’s Principal Value for Curvilinear Integrals.- 7.1.4 The Example f (?)=1/(?-z).- 7.2 The Cauchy Kernel.- 7.2.1 Definition and Properties.- 7.2.2 Regularity Properties.- 7.2.3 Properties of the Generated Holomorphic Function.- 7.2.4 Representation of K2.- 7.2.5 The Cauchy Integral on the Unit Circle.- 7.3 The Singular Integral Equation.- 7.3.1 The Case of Constant Coefficients.- 7.3.2 The Case of Variable Coefficients.- 7.3.3 General Singular Integral Equations.- 7.3.4 Approximation of the Cauchy Integral on the Unit Circle.- 7.3.5 Approximation of the Cauchy Integral on an Arbitrary Curve ?.- 7.3.6 Multi-Grid Methods for Equations of a Special Form.- 7.4 Application to the Dirichlet Problem for Laplace’s Equation.- 7.4.1 The Problem in the Interior Domain.- 7.4.2 The Double-Layer Potential.- 7.4.3 Uniqueness and Representation Theorem.- 7.4.4 The Case of a Smooth Boundary ?.- 7.4.5 The Double-Layer Potential for Solving the Exterior Problem.- 7.4.6 The Tangential Derivative of the Single-Layer Potential.- 7.5 Hypersingular Integrals.- 8 The Integral Equation Method.- 8.1 The Single-Layer Potential.- 8.1.1 The Singularity Function.- 8.1.2 Continuity of the Single-Layer Potential.- 8.1.2.1 Definition.- 8.1.2.2 Surface Integrals.- 8.1.2.3 Improper Integrals on Surfaces.- 8.1.2.4 Properties of the Single-Layer Potential.- 8.1.3 Derivatives of the Single-Layer Potential.- 8.1.3.1 The Normal Derivative.- 8.1.3.2 The Cauchy Principal Value for Surface Integrals.- 8.1.3.3 Other Directional Derivatives.- 8.1.4 Formulation of the Dirichlet Boundary Value Problem as First Kind Integral Equation for the Single-Layer Potential.- 8.1.4.1 Concerning the Interior and Exterior Problem of the Laplace Equation.- 8.1.4.2 The Integral Equation of the First Kind.- 8.1.5 Formulation of the Neumann Boundary Value Problem as Second Kind Integral Equation for the Single-Layer Potential.- 8.2 The Double-Layer Potential.- 8.2.1 Definition.- 8.2.2 Regularity Properties of the Double-Layer Integral Operator.- 8.2.3 Jump Properties of the Double-Layer Potential.- 8.2.4 Further Properties of the Double-Layer Potential.- 8.2.4.1 Hölder Continuity.- 8.2.4.2 The Potential close to a Jump Discontinuity of the Density.- 8.2.4.3 The Double-Layer Potential of the Density f = 1.- 8.2.5 Derivatives of the Double-Layer Potential.- 8.2.6 Integral Equations with the Double-Layer Operator.- 8.2.6.1 Formulation of the Dirichlet Boundary Value Problem as Integral Equation of the Second Kind with the Double-Layer Operator.- 8.2.6.2 Formulation of the Neumann Boundary Value Problem as Integral Equation of the Second Kind with the Double-Layer Operator.- 8.2.7 Non-smooth Curves or Surfaces.- 8.3 The Hypersingular Integral Equation.- 8.4 Synopsis: Integral Equations for the Laplace Equation.- 8.5 The Integral Equation Method for Other Differential Equations.- 8.5.1 Differential Equations of Second Order.- 8.5.2 Equations of Higher Order.- 8.5.3 Systems of Differential Equations.- 9 The Boundary Element Method.- 9.1 Construction of the Boundary Element Method.- 9.1.1 Definition of the Boundary Element Method.- 9.1.2 Galerkin Method.- 9.1.3 Collocation Method.- 9.1.4 Convergence in the Compact Case.- 9.1.5 Convergence in the Case of Elliptic Bilinear Forms.- 9.2 The Boundary Elements.- 9.2.1 Elements in the Two-Dimensional Case.- 9.2.2 Geometric Discretisation.- 9.2.3 Elements in the Three-Dimensional Case.- 9.2.4 Error Considerations.- 9.3 Multi-Grid Methods.- 9.3.1 Equations of the Second Kind.- 9.3.2 Equations of the First Kind.- 9.4 Integration and Numerical Quadrature.- 9.4.1 General Considerations.- 9.4.2 Weakly Singular Integrals.- 9.4.3 Nearly Singular Integrals.- 9.4.4 Strongly Singular Integrals.- 9.4.5 Treatment of Double Integrals Arising from the Galerkin Method.- 9.5 Solution of Inhomogeneous Equations.- 9.6 Computation of the Potential.- 9.6.1 Evaluation of the Potential.- 9.6.2 Evaluation of the Derivatives.- 9.6.3 Error Considerations.- 9.6.4 Extrapolation.- 9.7 The Panel Clustering Algorithm.- 9.7.1 Introduction.- 9.7.2 Panels.- 9.7.3 The Panel Clustering Method.- 9.7.3.1 The Far Field Expansion.- 9.7.3.2 Panel Clustering.- 9.7.3.3 Admissible Clusters and Admissible Coverings.- 9.7.3.4 The Algorithm for Matrix Multiplication.- 9.7.4 The Additional Quadrature Error.- 9.7.5 Complexity of the Algorithm.