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.

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Volterra and Fredholm integral equations form the domain of this book. Special chapters are devoted to Abel's integral equations and the singular integral equation with the Cauchy kernel; others focus on the integral equation method and the boundary element method (BEM). While a small section affords some theoretical grounding in integral equations (covering existence, regularity, etc.), the larger part of the book is devoted to a description and analysis of the discretisation methods (Galerkin / collocation / Nyström). Also the multigrid method for the solution of discrete equations is analysed. The most prominent application of integral equations occurs in the use of the boundary element method, which here is discussed from the numerical point of view in particular. New results about numerical integration and the panel clustering technique are included. Many chapters have an introductory character, while special subsections give more advanced information. Intended readers are students of mathematics as well as postgraduates.

Contenu
1 Introduction.- 1.1 Integral Equations.- 1.2 Basics from Analysis.- 1.3 Basics from Functional Analysis.- 1.4 Basics from Numerical Mathematics.- 2 Volterra Integral Equations.- 2.1 Theory of Volterra Integral Equations of the Second Kind.- 2.2 Numerical Solution by Quadrature Methods.- 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.- 3.3 Finite Approximability of the Integral Operator K.- 3.4 The Image Space of K.- 3.5 Solution of the Fredholm Integral Equation of the Second Kind.- 4 Numerical Treatment of Fredholm Integral Equations of the Second Kind.- 4.1 General Considerations.- 4.2 Discretisation by Kernel Approximation.- 4.3 Projection Methods in General.- 4.4 Collocation Method.- 4.5 Galerkin Method.- 4.6 Additional Comments Concerning Projection Methods.- 4.7 Discretisation by Quadrature: The Nyström Method.- 4.8 Supplements.- 5 Multi-Grid Methods for Solving Systems Arising from Integral Equations of the Second Kind.- 5.1 Preliminaries.- 5.2 Stability and Convergence (Discrete Formulation).- 5.3 The Hierarchy of Discrete Problems.- 5.4 Two-Grid Iteration.- 5.5 Multi-Grid Iteration.- 5.6 Nested Iteration.- 6 Abel's Integral Equation.- 6.1 Notations and Examples.- 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.- 7.2 The Cauchy Kernel.- 7.3 The Singular IntegralEquation.- 7.4 Application to the Dirichlet Problem for Laplace's Equation.- 7.5 Hypersingular Integrals.- 8 The Integral Equation Method.- 8.1 The Single-Layer Potential.- 8.2 The Double-Layer Potential.- 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.- 9 The Boundary Element Method.- 9.1 Construction of the Boundary Element Method.- 9.2 The Boundary Elements.- 9.3 Multi-Grid Methods.- 9.4 Integration and Numerical Quadrature.- 9.5 Solution of Inhomogeneous Equations.- 9.6 Computation of the Potential.- 9.7 The Panel Clustering Algorithm.