Integration of Equations of Parabolic Type by the Method of Nets

Integration of Equations of Parabolic Type by the Method of Nets

1st Edition - January 1, 1964

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  • Editors: I. N. Sneddon, M. Stark, S. Ulam
  • eBook ISBN: 9781483155326

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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.

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


Product details

  • No. of pages: 364
  • Language: English
  • Copyright: © Pergamon 1964
  • Published: January 1, 1964
  • Imprint: Pergamon
  • eBook ISBN: 9781483155326

About the Editors

I. N. Sneddon

M. Stark

S. Ulam

About the Author

V. K. Saul'Yev

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