Lectures on Partial Differential Equations: (Dover Books on Mathematics)

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.

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