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Integration of Equations of Parabolic Type by the Method of Nets
- 1st Edition - January 1, 1964
- Author: V. K. Saul'Yev
- Editors: I. N. Sneddon, M. Stark, S. Ulam
- Language: English
- eBook ISBN:9 7 8 - 1 - 4 8 3 1 - 5 5 3 2 - 6
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… Read more
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Request a sales quoteInternational 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.
Editorial Preface
Foreword
Author's Preface
Part I. Construction of Net Equations
Introduction
1. Absolutely Unstable Net Equations
2. Six-point Symmetric Equation
3. Asymmetric Net Equations
4. Alternating Method
5. Method of Mean Arithmetic, and Multi-nodal Symmetric Method
6. Comparison between Explicit and Implicit Equations, and the "Implicitly-explicit" Methods
7. Spherical and Cylindrical Regions
8. Equations of Increased Accuracy
9. Net Equations with Fictitious Nodes
10. On Bilateral Approximations
11. Two-dimensional and Three-dimensional Equations
12. Two-dimensional and Multi-dimensional Net Equations of Increased Accuracy
13. Non-uniform Nets
14. Multi-step Equations
15. General Case of Variable and Discontinuous Coefficients
16. Parabolic Equations of Higher than the Second Order
17. Non-linear Equations
Conclusions
Part II. Solution of Net Equations
Introduction
1. "One-dimensional" Elliptic Net Equations
2. Direct Methods
3. Ill-conditioned Net Matrices
4. Simplest Iterative Method
5. Variational Methods
6. Methods using Chebyshev Polynomials
7. Iterative Methods of the Second Degree
8. Iterative Methods of the nth Degree
9. Methods of Successive Displacements
10. Methods of Block Iteration
Appendix
On the Application of Chebyshev Polynomials to Parabolic Net Equations
References
Index
Volumes Published in this Series
- No. of pages: 364
- Language: English
- Edition: 1
- Published: January 1, 1964
- Imprint: Pergamon
- eBook ISBN: 9781483155326
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