This book has developed from courses of lectures given by the
author over a period of years to the students of the Moscow PhysicoTechnical Institute. It is intended for the students having basic knowledge of mathematical analysis, algebra and the theory of ordinary differential equations to the extent of a university course.
Except Chapter I, where some general questions regarding partial differential equations have been examined, the material has been arranged so as to correspond to the basic types of equations. The central role in the book is played by Chapter IV, the largest of all, which discusses elliptic equations. Chapters V and VI are devoted to the hyperbolic and parabolic equations.
The method used in this book for investigating the boundary value problems and, partly, the Cauchy problem is based on the notion of generalized solution which enables us to examine equations with variable coefficients with the same ease as the simplest equations: Poisson's equation, wave equation and heat equation. Apart from discussing the questions of existence and uniqueness of solutions of the basic boundary value problems, considerable space has been devoted to the approximate methods of solving these equations:
Ritz's method in the case of elliptic equations and Galerkin's
method for hyperbolic and parabolic equations.
Author(s): V. P. Mikhailov
Publisher: MIR
Year: 1978
Language: English
Pages: 408
Preface 7
CHAPTER I
INTRODUCTION. CLASSIFICATION OF EQUATIONS. FORMULATION OF SOME PROBLEMS
§1. The Cauchy Problem. Kovalevskaya's Theorem 12
§2. Classification of Linear Differential Equations of the Second Order 31
§3. Formulation of Some Problems 34
Problems on Chapter I 41
Suggested Reading on Chapter I 41
CHAPTER II
THE LEBESGUE INTEGRAL AND SOME QUESTIONS OF FUNCTIONAL ANALYSIS
§1. The Lebesgue Integral 42
§2. Normed Linear Spaces. Hilbert Space 64
§3. Linear Operators. Compact Sets. Completely Continuous Operators 72
§4. Linear Equations in a Hilbert Space 85
§5. Selfadjoint Completely Continuous Operators 94
CHAPTER III
FUNCTION SPACES
§1. Spaces of Continuous and Continuously Differentiable Functions 101
§2. Spaces of Integrable Functions 104
§3. Generalized Derivatives 111
§4. Spaces Hk(Q) 121
§5. Properties of Functions Belonging to H1(Q) and H1(Q) 135
§6. Properties of Functions Belonging to Hk(Q) 149
§7. Spaces cr,0andC2s,8• SpacesHT,0andH28,8 155
§8. Examples of Operators in Function Spaces 161
Problems on Chapter III 166
Suggested Reading on Chapter III 168
CHAPTER IV
ELLIPTIC EQUATIONS
§1. Generalized Solutions of Boundary-Value Problems. Eigenvalue Problems 169
§2. Smoothness of Generalized Solutions. Classical Solutions 208
§3. Classical Solutions of Laplace's and Poisson's Equations 232
Problems on Chapter IV 261
Suggested Reading on Chapter IV 264
CHAPTER V
HYPERBOLIC EQUATIONS
§1. Properties of Solutions of Wave Equation. The Cauchy Problem for Wave Equation 266
§2. Mixed Problems 284
§3. Generalized Solution of the Cauchy Problem 328
Problems on Chapter V 339
Suggested Reading on Chapter V 341
CHAPTER VI
PARABOLIC EQUATIONS
§1. Properties of Solutions of Heat Equation. The Cauchy Problem for Heat Equation 342
§2. Mixed Problems 362
Problems on Chapter VI 388
Suggested Reading on Chapter VI 391
Index 392
