A MIXED PARABOLIC WITH A NON-LOCAL AND GLOBAL LINEAR CONDITIONS

Abstract

Krein [1] mentioned that for each PD equation we have two extreme operators, one is the minimal in which solution and its derivatives on the boundary are zero, the other one is the maximal operator in which there is no prescribed boundary conditions. They claim it is not possible to have a related boundary value problem for an arbitrarily chosen operator in between. They have only considered local conditions and so their claim is justified, particularly, for partial differential boundary value problems of odd orders. By considering more general (non-local and global) conditions, we showed this is not necessarily true. With similarly general conditions as considered in this paper one can define a boundary value problem for PDEs of odd order and also problems with local conditions will be a particular case of this general form. In this paper, a mixed problem for a parabolic equation with general conditions is analytically investigated and, in a closed from, its unique solution is shown.