Analysis of Approximation Methods for Differential and Integral Equations

1. Front Matter

   Pages N2-xi

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2. Presentation of Numerical Methods

   1. Front Matter

      Pages 1-2

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   2. Finite-Difference Methods for Boundary-Value Problems

      • H.-J. Reinhardt

      Pages 3-19

   3. Projection Methods for Variational Equations

      • H.-J. Reinhardt

      Pages 20-50

   4. Approximation Methods for Integral Equations of the Second Kind

      • H.-J. Reinhardt

      Pages 51-73

   5. Approximation Methods for Initial Value Problems in Partial Differential Equations

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      Pages 74-120

3. Convergence Theory

   1. Front Matter

      Pages 121-122

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   2. The Concepts of Discrete Convergence and Discrete Approximations

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      Pages 123-151

   3. Discrete Convergence of Mappings and Solutions of Equations

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      Pages 152-180

   4. Compactness Criteria for Discrete Convergence

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      Pages 181-206

4. Convergence Analysis for Approximate Solutions to Boundary-Value Problems and Integral Equations

   1. Front Matter

      Pages 207-208

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   2. Convergence of Finite-Difference Methods for Boundary-Value Problems

      • H.-J. Reinhardt

      Pages 209-235

   3. Biconvergence for Projection Methods via Variational Principles

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      Pages 236-250

   4. Convergence of Perturbations of Integral Equations of the Second Kind

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      Pages 251-265

5. Inverse Stability, Consistency and Convergence for Initial Value Problems in Partial Differential Equations

   1. Front Matter

      Pages 266-267

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   2. Inverse Stability and Convergence for General Discrete-Time Approximations of Linear and Nonlinear Initial Value Problems

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      Pages 268-305

   3. Special Criteria for Inverse Stability

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      Pages 306-353

   4. Convergence Analysis of Special Methods

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      Pages 354-384

6. Back Matter

   Pages 385-399

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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.

• Analysis
• Equations
• Integral
• Integral equation
• numerical analysis
• ordinary differential equation
• partial differential equation
• wave equation

• Fachbereich Mathematik, Johann-Wolfgang-Goethe-Universität, 6000 Frankfurt Main, Federal Republic of Germany

  H.-J. Reinhardt

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