Analysis of Approximation Methods for Differential and Integral Equations

✍ Scribed by H.-J. Reinhardt (auth.)

• Publisher: Springer-Verlag New York
• Year: 1985
• Tongue: English
• Leaves: 411
• Series: Applied Mathematical Sciences 57
• Edition: 1
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Front Matter....Pages N2-xi
Front Matter....Pages 1-2
Finite-Difference Methods for Boundary-Value Problems....Pages 3-19
Projection Methods for Variational Equations....Pages 20-50
Approximation Methods for Integral Equations of the Second Kind....Pages 51-73
Approximation Methods for Initial Value Problems in Partial Differential Equations....Pages 74-120
Front Matter....Pages 121-122
The Concepts of Discrete Convergence and Discrete Approximations....Pages 123-151
Discrete Convergence of Mappings and Solutions of Equations....Pages 152-180
Compactness Criteria for Discrete Convergence....Pages 181-206
Front Matter....Pages 207-208
Convergence of Finite-Difference Methods for Boundary-Value Problems....Pages 209-235
Biconvergence for Projection Methods via Variational Principles....Pages 236-250
Convergence of Perturbations of Integral Equations of the Second Kind....Pages 251-265
Front Matter....Pages 266-267
Inverse Stability and Convergence for General Discrete-Time Approximations of Linear and Nonlinear Initial Value Problems....Pages 268-305
Special Criteria for Inverse Stability....Pages 306-353
Convergence Analysis of Special Methods....Pages 354-384
Back Matter....Pages 385-399

✦ Subjects

Numerical Analysis