In the previous post we have seen Equations of Mathematical Physics by A. V. Bitsadze. In this post we will see the associated problem book A Collection of Problems on The Equations of Mathematical Physics by A. V. Bitsadze and D. F. Kalinichenko.

About the Book:

The present book is a collection on the problems on the equations of mathematical physics studied in colleges with comprehensive mathematical programme. It consists of two parts, the first of which contains the conditions of the problems and the second the answers to the problems and detailed explanations of the solutions of the most difficult problems. The material of the first part is divided into five chapters in which the problems are grouped according to the type of partial differential equations. Each chapter begins with the necessary prerequisites taken from the corresponding division of the theory of the equations of mathematical physics which facilitates the understanding of the subject.

The book was translated from the Russian by V. M. Volosov and I. G. Volosova and was first published by Mir Publishers in 1980.

Note: Scan by the orginal uploader. We have cleaned the copy and made a new format and added a new cover. The scan quality is not very good.

You can get the book here.

Contents

Preface 5

CONDITIONS OF THE PROBLEMS 9

Chapter 1
Introduction. Classification of Partial Differential Equations and Systems of Partial Differential Equations. Normal Form of Partial Differential Equations of Second Order in Two Independent Variables. Derivation of Some Equations of Mathematical Physics. 9

1. Partial Differential Equation and Its Solution. Systems of Partial Differential Equations 9
2. Classification of Partial Differential Equations and Systems of Partial Differential Equations 12
3. Reduction to Normal Form of Linear Partial Differential Equations of Second Order in Two Independent Variables 17
4. Mathematical Models for Some Phenomena Studied in Mathematical Physics 21

Chapter 2
Elliptical Partial Differential Equations 40

1. Basic Properties of Harmonic Functions 40
2. The Dirichlet and the Neumann Boundary-Value Problems for Harmonic Functions 46
3. Potential Functions 53
4. Some Other Classes of Elliptical Partial Differential Equations 59

Chapter 3
Hyperbolic Partial Differential Equations 65

1. Wave Equation 65
2. Well-posed problems for Hyperbolic Partial Differential Equations 75
3. Some other classes of Hyperbolic Differential Equations. The Cauchy Problem for Laplace’s Equation 82

Chapter 4
Parabolic Partial Differential Equations 93

1. Heat Conduction Equation 93
2. Some other examples of Parabolic Differential Equations 97

Chapter 5
Basic Methods for the Solution of Partial Differential Equations 140

1. The Method of Separation of Variables (The Fourier Method) 101
2. Special Functions. Asymptotic Expansions. 116
3. Method of Integral Transformations 132
4. Method of Finite Differences 140
5. Variational Methods 144

ANSWERS AND SOLUTIONS 149

Appendix 319