A Collection of Problems on the Equations of Mathematical Physics – Vladimirov

In this post, we will see the book A Collection of Problems on the Equations of Mathematical Physics edited by V. S. Vladimirov. The contributors to the book include V .S. Vladimirov, V .P . Mikhailov, A. A. Vasharin, Kh. Kh.Karimova, Yu. V. Sidorov, and M.I. Shabunin. This is the associated problem book for Equations of Mathematical Physics and Partial Differential Equations which we have seen earlier.

About the book

The extensive application of modern mathematical techniques to
theoretical and mathematical physics requires a fresh approach to the course of equations of mathematical physics. This is especially true with regards to such a fundamental concept as the solution of a boundary value problem. The concept of a generalized solution considerably broadens the field of problems and enables solving from a unified position the most interesting problems that cannot be solved by applying classical methods. To this end two new courses have been written at the Department of Higher Mathematics at the Moscow Physics and Technology Institute, namely, “Equations of Mathematical Physics” by V.S. Vladimirov and “Partial Differential Equations” by V.P. Mikhailov (both books have been translated into English by Mir Publishers, the first in 1984 and the second
in 1978).

The present collection of problems is based on these courses and
amplifies them considerably. Besides the classical boundary value problems, we have included a large number of boundary value problems that have only generalized solutions. Solution of these requires using the methods and results of various branches of modern analysis. For this reason we have included problems in Lebesgue integration, problems involving function spaces (especially spaces of generalized differentiable functions) and generalized functions (with Fourier and Laplace transforms), and integral equations.

The book is aimed at undergraduate and graduate students in the physical sciences, engineering, and applied mathematics who have taken the typical “methods” course that includes vector analysis, elementary complex variables, and an introduction to Fourier series and boundary value problems. Asterisks denote the more difficult problems.

The book was translated from Russian by Eugene Yankovsky and was published in 1986.

Credits to the original uploader.

You can get the book here.

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Contents

Preface 5

Symbols and Definitions 9

Chapter I Statement of Boundary Value Problems in Mathematical Physics 12

1 Deriving Equations of Mathematical Physics 12
2 Classification of Second-order Equations 35

Chapter II Function Spaces and Integral Equations 41

3 Measurable Functions. The Lebesgue Integral 41
4 Function Spaces 48
5 Integral Equations 67

Chapter III Generalized Functions 88

6 Test and Generalized Functions 88
7 Differentiation of Generalized Functions 95
8 The Direct Product and Convolution of Generalized Functions 104
9 The Fourier Transform of Generalized Functions of Slow Growth 114
10 The Laplace Transform of Generalized Functions 121
11 Fundamental Solutions of Linear Differential Operators 125

Chapter IV The Cauchy Problem 134

12 The Cauchy Problem for Second-order Equations of Hyperbolic Type 134
13 The Cauchy Problem for the Heat Conduction Equation 157
14 The Cauchy Problem for Other Equations and Goursat’s Problem 167

Chapter V Boundary Value Problems for Equations of Elliptic Type 180

15 The Sturm-Liouville Problem 181
16 Fourier’s Method for Laplace’s and Poisson’s Equations 190
17 Green’s Functions of the Dirichlet Problem 205
18 The Method of Potentials 211
19 Variational Methods 230

Chapter VI Mixed Problems 239

20 Fourier’s Method 239
21 Other Methods 269

Appendix Examples of Solution Techniques for Some Typical Problems 277

A1 Method of Characteristics 277
A2 Fourier’s Method 279
A3 Integral Equations with a Degenerate Kernel 281
A4 Variational Problems 283

References 284

Subject Index 287

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