Saul'Yev / Sneddon / Stark Integration of Equations of Parabolic Type by the Method of Nets

1. Auflage 2014
ISBN: 978-1-4831-5532-6
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 364 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

International Series of Monographs in Pure and Applied Mathematics, Volume 54: Integration of Equations of Parabolic Type by the Method of Nets deals with solving parabolic partial differential equations using the method of nets. The first part of this volume focuses on the construction of net equations, with emphasis on the stability and accuracy of the approximating net equations. The method of nets or method of finite differences (used to define the corresponding numerical method in ordinary differential equations) is one of many different approximate methods of integration of partial differential equations. The other methods, and some based on newer equations, are described. By analyzing these newer methods, older and existing methods are evaluated. For example, the asymmetric net equations; the alternating method of using certain equations; and the method of mean arithmetic and multi-nodal symmetric method point out that when the accuracy needs to be high, the requirements for stability become more defined. The methods discussed are very theoretical and methodological. The second part of the book concerns the practical numerical solution of the equations posed in Part I. Emphasis is on the commonly used iterative methods that are programmable on computers. This book is suitable for statisticians and numerical analysts and is also recommended for scientists and engineers with general mathematical knowledge.

Autoren/Hrsg.

Saul'Yev, V. K.

Sneddon, I. N.

Stark, M.

Ulam, S.

Inhaltsverzeichnis

1;Front Cover;1
2;Integration of Equations of Parabolic Type by the Method of Nets;4
3;Copyright Page;5
4;Table of Contents;6
5;EDITORIAL PREFACE;8
6;FOREWORD;10
7;AUTHOR'S PREFACE;14
8;PART I: CONSTRUCTION OF NET EQUATIONS;20
8.1;Introduction;22
8.2;Chapter § 1. Absolutely Unstable Net Equations;37
8.3;Chapter § 2. Six-point Symmetric Equation;41
8.4;Chapter § 3. Asymmetric Net Equations;48
8.5;Chapter § 4. Alternating Method;62
8.6;Chapter § 5. Method of Mean Arithmetic, and Multi-nodal Symmetric Method;71
8.7;Chapter § 6. Comparison between Explicit and Implicit Equations, and the "Implicitly-explicit" Methods;83
8.8;Chapter § 7. Spherical and Cylindrical Regions;92
8.9;Chapter § 8. Equations of Increased Accuracy;102
8.10;Chapter § 9. Net Equations with Fictitious Nodes;120
8.11;Chapter §10. On Bilateral Approximations;127
8.12;Chapter §11. Two-dimensional and Three-dimensional Equations;135
8.13;Chapter §12. Two-dimensional and Multi-dimensional Net Equations of Increased Accuracy;149
8.14;Chapter §13. Non-uniform Nets;166
8.15;Chapter § 14. Multi-step Equations;172
8.16;Chapter § 15. General Case of Variable and Discontinuous Coefficients;181
8.17;Chapter § 16. Parabolic Equations of Higher than the Second Order;189
8.18;Chapter § 17. Non-linear Equations;203
8.19;Conclusions;214
9;PART II: THE SOLUTION OF NET EQUATIONS;220
9.1;Introduction;222
9.2;Chapter § 1. "One-dimensional" Elliptic Net Equations;225
9.3;Chapter § 2. Direct Methods;231
9.4;Chapter § 3. Ill-conditioned Net Matrices;243
9.5;Chapter § 4. Simplest Iterative Method;249
9.6;Chapter § 5. Variational Methods;260
9.7;Chapter § 6. Methods Using Chebyshev Polynomials;269
9.8;Chapter § 7. Iterative Methods of the Second Degree;279
9.9;Chapter § 8. Iterative Methods of the nth Degree;287
9.10;Chapter § 9. Methods of Successive Displacements;299
9.11;Chapter § 10. Methods of Block Iteration;318
10;APPENDIX;334
10.1;On the Application of Chebyshev Polynomials to Parabolic Net Equations;334
11;REFERENCES;340
12;INDEX;358
13;VOLUMES PUBLISHED IN THIS SERIES;364