Analysis of Approximation Methods for Differential and Integral Equations

by Hans-Jürgen Reinhardt

Overview

This book is primarily based on the research done by the Numerical Analysis Group at the Goethe-Universitat in Frankfurt/Main, and on material presented in several graduate courses by the author between 1977 and 1981. It is hoped that the text will be useful for graduate students and for scientists interested in studying a fundamental theoretical analysis of numerical methods along with its application to the most diverse classes of differential and integral equations. The text treats numerous methods for approximating solutions of three classes of problems: (elliptic) boundary-value problems, (hyperbolic and parabolic) initial value problems in partial differential equations, and integral equations of the second kind. The aim is to develop a unifying convergence theory, and thereby prove the convergence of, as well as provide error estimates for, the approximations generated by specific numerical methods. The schemes for numerically solving boundary-value problems are additionally divided into the two categories of finite­ difference methods and of projection methods for approximating their variational formulations.

Product Details

ISBN-13:	9780387962146
Publisher:	Springer New York
Publication date:	10/07/1985
Series:	Applied Mathematical Sciences , #57
Edition description:	Softcover reprint of the original 1st ed. 1985
Pages:	398
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

I: Presentation of Numerical Methods.- 1. Finite-Difference Methods for Boundary-Value Problems.- 2. Projection Methods for Variational Equations.- 3. Approximation Methods for Integral Equations of the Second Kind.- 4. Approximation Methods for Initial Value Problems in Partial Differential Equations 74.- II: Convergence Theory.- 5. The Concepts of Discrete Convergence and Discrete Approximations.- 6. Discrete Convergence of Mappings and Solutions of Equations.- 7. Compactness Criteria for Discrete Convergence.- III: Convergence Analysis for Approximate Solutions of Boundary-Value Problems and Integral Equations.- 8. Convergence of Finite-Difference Methods for Boundary-Value Problems.- 9. Biconvergence for Projection Methods Via Variational Principles.- 10. Convergence of Perturbations of Integral Equations of The Second Kind.- IV: Inverse Stability, Consistency and Convergence for Initial Value Problems in Partial Differential Equations.- 11. Inverse Stability and Convergence for General Discrete-Time Approximations of Linear and Nonlinear Initial Value Problems.- 12. Special Criteria for Inverse Stability.- 13. Convergence Analysis of Special Methods.- Glossary of Symbols.