Integration of Equations of Parabolic Type by the Method of Nets - 1st Edition - ISBN: 9780080101958, 9781483155326

Integration of Equations of Parabolic Type by the Method of Nets, Volume 54

1st Edition

Editors: I. N. Sneddon M. Stark S. Ulam
Authors: V. K. Saul'Yev
eBook ISBN: 9781483155326
Imprint: Pergamon
Published Date: 1st January 1964
Page Count: 364
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Table of Contents


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



Description

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.


Details

No. of pages:
364
Language:
English
Copyright:
© Pergamon 1964
Published:
Imprint:
Pergamon
eBook ISBN:
9781483155326

About the Editors

I. N. Sneddon Editor

M. Stark Editor

S. Ulam Editor

About the Authors

V. K. Saul'Yev Author