## Equations of Mathematical Physics – Vladimirov

In this post, we will see the book Equations of Mathematical Physics by V. S. Vladimirov. This book examines classical boundary value problems for differentia equations of mathematical physics. Instead of the traditional means of presentation, we use the concept of the generalized solution. Generalized solutions arise in solving integral equations of the local balance type and the calculation of these solutions leads to generalized formulations of the problems of mathematical physics.

Many sections contain problems for exercises. A number of problems are given in the form of theorems which are an important addition to the basic material. For further exercises we recommend the books of B. M. Budak et al. (1) and M. M. Smirnov (1).
This book is a fuller version of the lectures which I have given over the years to the students of the Moscow Physical and Technical Institute. It is intended for students, physicists, and mathematicians who have already mastered the basis of mathematical analysis during the first two courses at university.

The book was translated from the Russian by Audrey Littlewood and edited by ALan Jeffrey and was published in 1971.

You can get the book here.

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Contents

Preface

CHAPTER 1. FORMULATION OF BOUNDARY VALUE PROBLEMS IN MATHEMATICAL PHYSICS 1

1. Some Concepts and Propositions Concerning the Theory of Sets, Theory of Functions, and the Theory of Operators 1
2. Basic Equations of Mathematical Physics 27
3. Classification of Linear (Second-Order) Differential Equations 38
4. Formulation of Boundary Value Problems for Linear Second-Order Differential Equations 52

CHAPTER 2. GENERALIZED FUNCTIONS 63

5. Test and Generalized Functions 63
6. Differentiation of Generalized Functions 79
7. The Direct Product and Convolution of Generalized Functions 95
8. Generalized Functions of Slow Growth (Tempered Distributions) 113
9. Fourier Transform of Generalized Functions of Slow Growth 121

CHAPTER 3. FUNDAMENTAL SOLUTIONS AND THE CAUCHY PROBLEM 139

10. Fundamental Solutions of Linear Differential Operators 139
11. Retarded Potential 157
12. The Cauchy Problem for the Wave Equation 171
13. Wave Propagation 178
14. The Cauchy Problem for the Equation of Heat Conduction 193

CHAPTER 4. INTEGRAL EQUATIONS 201

15. The Method of Successive Approximations 202
16. Fredholm’s Theorems 217
17. Integral Equations with an Hermitian Kernel 231
18. The Hilbert-Schmidt Theorem and its Corollaries 237

CHAPTER 5. BOUNDARY VALUE PROBLEMS FOR ELLIPTIC EQUATIONS 255

19. The Eigenvalue Problem 255
20. The Sturm-Liouville Problem 270
21. Harmonic Functions 278
22. Newtonian Potential 291
23. Boundary Value Problems for Laplace and Poisson Equations in Space 307
24. The Green’s Function for the Dirichlet Problem 318
25. Spherical Functions 333
26. Boundary Value Problems for Laplace’s Equation in a Plane 347
27. Helmholtz’s Equation 362

CHAPTER 6. THE MIXED PROBLEM 373

28. Fourier’s Method 373
29. The Mixed Problem for an Equation of Hyperbolic Type 388
30. The Mixed Problem for an Equation of Parabolic Type 404

BIBLIOGRAPHY 411 