Sufficient conditions for the reduction of a BVP for PDE with non-local and global boundary conditions to Fredholm integral equations (on a rectangular domain)

摘要

With the expanding domain of application of mathematical methods at present time there often arise problems connected with the study of partial differential equations that do not belong to any of the classical type. The elucidation of the correct formulation of these problems and the study of the specific properties of the solutions of similar equations are closely related to the study of questions of a general nature. In this respect, a time dependent PDE on a bounded rectangular region D⊂R2 with a more general non-local and global boundary conditions is considered. Its equivalent boundary value problem through an application of the method of contour integral of Rasulov [Methods of Contour Integration, North-Holland, 1967] is obtained. First, by using the theory of fundamental solutions, the essential (or necessary) conditions are obtained. Using these conditions one is able to give sufficient conditions to reduce the problem to system of Fredholm integral equations of second kind with at most weak singularities.