Lectures on Partial Differential Equations

by I. G. Petrovsky, Abe Shenitzer (Other)

Overview

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.

Product Details

ISBN-13:	9780486669021
Publisher:	Dover Publications
Publication date:	01/14/1992
Series:	Dover Books on Mathematics Series
Pages:	272
Product dimensions:	5.41(w) x 8.52(h) x 0.55(d)

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