Overview

This volume explores integral equations, devoting special chapters to Abel's integral equations and the singular integral equation with the Cauchy kernel. Other chapters focus on the integral equation method and the boundary element method. While a small section affords some theoretical grounding in integral equations (covering such topics as existence and regularity), the larger part of the book is devoted to a description and analysis of the discretization methods (such as Galerkin, collocation and Nystrom). Also, the multi-grid method for the solution of discrete equations is analyzed. The most prominent application of integral equations occurs in the use of the boundary element method, which 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. The book is intended for students of mathematics, as well as postgraduates.

Full Product Details

Author:   Wolfgang Hackbusch
Publisher:   Birkhauser Verlag AG
Imprint:   Birkhauser Verlag AG
Edition:   1995 ed.
Volume:   120
Dimensions:   Width: 15.60cm , Height: 2.20cm , Length: 23.40cm
Weight:   1.570kg
ISBN:  

9783764328719

ISBN 10:   3764328711
Pages:   362
Publication Date:   01 June 1995
Audience:   College/higher education ,  Undergraduate ,  Postgraduate, Research & Scholarly
Format:   Hardback
Publisher's Status:   Active
Availability:   Out of print, replaced by POD  
We will order this item for you from a manufatured on demand supplier.

Table of Contents

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.

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