LINEAR EQUATIONS OF THE SECOND ORDER OF PARABOLIC TYPE

Abstract

CONTENTS Introduction § 1. The maximum principle. Uniqueness of the solutions of the basic boundary value problems § 2. A priori estimates § 3. Solution of boundary value problems by Rothe's method. The Cauchy problem § 4. The fundamental solution of a linear parabolic equation. The Green's function. The method of integral equations for the solution of boundary value problems § 5. Generalized solutions of boundary value problems. The uniqueness theorem. Some auxiliary propositions § 6. The method of finite differences § 7. Some methods of functional analysis for the solution of boundary value problems § 8. The solution of boundary value problems by the method of continuation by a parameter § 9. The application of Galerkin's method for the construction of a solution of the first boundary value problem § 10. Generalized solutions of Cauchy's problem § 11. On differentiability properties of generalized solutions § 12. The behaviour of solutions for indefinitely increasing time References