# The Method of Summary Representation for Numerical Solution of Problems of Mathematical Physics

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Pure and Applied Mathematics, Volume 79: The Method of Summary Representation for Numerical Solution of Problems of Mathematical Physics presents the numerical solution of two-dimensional and three-dimensional boundary-value problems of mathematical physics. This book focuses on the second-order and fourth-order linear differential equations. Organized into two chapters, this volume begins with an overview of ordinary finite-difference equations and the general solutions of certain specific finite-difference equations. This text then examines the various methods of successive approximation that are used exclusively for solving finite-difference equations. This book discusses as well the established formula of summary representation for certain finite-difference operators that are associated with partial differential equations of mathematical physics. The final chapter deals with the formula of summary representation to enable the researcher to write the solution of the corresponding systems of linear algebraic equations in a simple form. This book is a valuable resource for mathematicians and physicists.

## Table of Contents

Author's Preface to the English Edition

Preface

Introduction

Chapter 1. General Theory of the One-Dimensional Problem of Eigenvalues and Eigenfunctions of Discrete Argument. Matrices of Type II

§ 1. Ordinary Finite-Difference Equations

§ 2. General Problem of Eigenvalues and Eigenfunctions of Discrete Argument. Matrices of Type II

§ 2.1 Formulae of Multiple Summation by Parts

§ 2.2 The Space II and the Space II' of Functions of Discrete Argument. Self-Adjoint Finite-Difference Operators

§ 2.3 Self-Adjoint Finite-Difference Boundary-Value Problems. Matrices of Type II

§ 2.4 Eigenvalues and Eigenfunctions of Discrete Argument

§ 2.5 Matrices of Simple Structure. Fundamental Properties of Matrices of Type II

§ 2.6 General Problem of Eigenvalues and Eigenfunctions for Second-Order Finite-Difference Equations

§ 3. Solution of Particular Boundary-Value Problems, and the Construction of Fundamental Matrices in Explicit Form

§ 4. On Special Functions of Discrete Argument, and Special Matrices of Type II

§ 4.1 Functions of Discrete Argument, Connected with Second-Order Finite-Difference Operators

§ 4.2 Special Functions of Discrete Argument of the First and Second Types

§ 4.3 Special Cases

Chapter 2. Numerical Solution of Two-Dimensional and Three-Dimensional Boundary-Value Problems of Mathematical Physics

§ 1. Solution of Boundary-Value Problems for Second-Order Elliptic Differential Equations with Constant Coefficients

§ 1.1 Solution of Two-Dimensional Boundary-Value Problems

§ 1.2 Extension of the Method to the Solution of Three-Dimensional Boundary-Value Problems

§ 1.3 Numerical Example

§ 1.4 Generalization of the Basic Formulae of Summary Representation for Two-Dimensional Boundary-Value Problems

§ 1.5 Generalization of the Basic Formulae of Summary Representation for Three-Dimensional Boundary-Value Problems

§ 2. Solution of Boundary-Value Problems for Fourth-Order Elliptic Differential Equations with Constant Coefficients

§ 2.1 Formula of Summary Representation for the Finite-Difference Biharmonic Operator

§ 2.2 Solution of Biharmonic Boundary-Value Problems

§ 2.3 Generalization of the Basic Formula of Summary Representation

§ 3. Formulae of Summary Representation for Finite-Difference Equations, Corresponding to Second-Order Parabolic Differential Equations with Constant Coefficients

§ 3.1 Equations with Two Independent Variables

§ 3.2 Equations with Three Independent Variables

§ 4. Solution of Finite-Difference Boundary-Value Problems, Connected with Boundary-Value Problems for Second-Order Hyperbolic Differential Equations with Constant Coefficients

§ 4.1 Equations of Hyperbolic Type with Two Independent Variables

§ 4.2 Hyperbolic Equations with Three Independent Variables

§ 5. Differential Equation for the Transverse Vibrations of Beams

§ 6. On the Numerical Solution of Two-Dimensional and Three-Dimensional Boundary-Value Problems for Differential Equations with Variable Coefficients

References

Supplement to the English Edition

§ 1. Alternating Iterative Method for the Numerical Solution of Connection Equations

§ 2. On the Approach to the Limit in Certain Formulae of Summary Representation

§ 3. On the Application of the Method of Summary Representation to the Solution of Problems of Filtration Under Pressure

§ 4. On the Solution of Bending and Torsion Problems for Prismatic Beams by the Method of Summary Representation

§ 4.1. Problems of the Bending of Prismatic Beams

§ 4.2. Problems of the Torsion of Prismatic Beams

§ 5. On the Application of the Method of Summary Representation to Biharmonic Problems of the Bending of Plates

§ 6. On Certain Formulae of Summary Representation

§ 6.1. Formulae of Summary Representation for Annular Sectors and for Annuli

§ 6.2. Formulae of Summary Representation for Sectors and Circles

§ 6.3. Formulae of Summary Representation for Angles and Planes

§ 6.4. On the Solutions of Boundary-Value Problems, and Certain Generalizations

§ 7. Bibliography of the Scientific Works of Professor G. N. Polozhii

Index

Other Titles in the Series

## Product details

- No. of pages: 304
- Language: English
- Copyright: © Pergamon 1965
- Published: January 1, 1965
- Imprint: Pergamon
- eBook ISBN: 9781483185460

## About the Editors

### I. N. Sneddon

### M. Stark

### K. A. H. Gravett

## About the Author

### G. N. Polozhii

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