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.