Synopses & Reviews
Graduate-level exposition by noted Russian mathematician offers rigorous, transparent, highly readable coverage of classification of equations, hyperbolic equations, elliptic equations and parabolic equations. Wealth of commentary and insight invaluable for deepening understanding of problems considered in text. Translated from the Russian by A. Shenitzer.
Synopsis
The field of partial differential equations is an extremely important component of modern mathematics. It has great intrinsic beauty and virtually unlimited applications. This book, written for graduate-level students, grew out of a series of lectures the late Professor Petrovsky gave at Moscow State University. The first chapter uses physical problems to introduce the subjects and explains its division into hyperbolic, elliptic, and parabolic partial differential equations. Each of these three classes of equations is dealt with in one of the remaining three chapters of the book in a manner that is at once rigorous, transparent, and highly readable.
Petrovsky was a leading figure in Russian mathematics responsible for many advances in the field of partial differential equations. In these masterly lectures, his commentary and discussion of various aspects of the problems under consideration will prove valuable in deepening students' understanding and appreciation of these problems.
Synopsis
Graduate-level exposition by noted Russian mathematician offers rigorous, readable coverage of classification of equations, hyperbolic equations, elliptic equations, and parabolic equations. Translated from the Russian by A. Shenitzer.
1954 edition.
Table of Contents
Foreword, by R. Courant; Translator's Note, by Abe Shenitzer; Preface
Chapter I. Introduction. Classification of equations
1. Definitions. Examples
2. The Cauchy problem. The Cauchy-Kowalewski theorem
3. The generalized Cauchy problem. Characteristics
4. Uniqueness of the solution of the Cauchy problem in the class of non-analytic functions
5. Reduction to canonical form at a point and classification of equations of the second order in one unknown function
6. Reduction to canonical form in a region of a partial differential equation of the second order in two independent variables
7. Reduction to canonical form of a system of linear partial differential equations of the first order in two independent variables
Chapter II. Hyperbolic equations
The Cauchy problem for non-analytic functions
8. The reasonableness of the Cauchy problem
9. The notion of generalized solutions
10. The Cauchy problem for hyperbolic systems in two independent variables
11. The Cauchy problem for the wave equation. Uniqueness of the solution
12. Formulas giving the solution of the Cauchy problem for the wave equation
13. Examination of the formulas which give the solution of the Cauchy problem
14. The Lorentz transformation
15. The mathematical foundations of the special principle of relativity
16. Survey of the fundamental facts of the theory of the Cauchy problem for general hyperbolic systems
II. Vibrations of bounded bodies
17. Introduction
18. Uniqueness of the mixed initial and boundary-value problem
19. Continuous dependence of the solution on the initial data
20. The Fourier method for the equation of a vibrating string
21. The general Fourier method (introductory considerations)
22. General properties of eigenfunctions and eigenvalues
23. Justification of the Fourier method
24. Another justification of the Fourier method
25. Investigation of the vibration of a membrane
26. Supplementary information concerning eigenfunctions
Chapter III. Elliptic equations
27. Introduction
28. The minimum-maximum property and its consequences
29. Solution of the Dirichlet problem for a circle
30. Theorems on the fundamental properties of harmonic functions
31. Proof of the existence of a solution of Dirichlet's problem
32. The exterior Dirichlet problem
33. The Neumann problem (the second boundary-value problem)
34. Potential theory
35. Application of potential theory to the solution of boundary-value problems
36. Approximate solution of the Dirichlet problem by the method of finite differences
37. Survey of the most important results for general elliptic equations
Chapter IV. Parabolic equations
38. Conduction of heat in a bounded strip (the first boundary-value problem)
39. Conduction of heat in an infinite strip (the Cauchy problem)
40. Survey of some further investigations of equations of the parabolic type