Nonhomogeneous Boundary Value Problem With Nonlocal Conditions for a Partial Differential Equation With the Operator of the Generalized Differentiation

Abstract

The article is devoted to investigation of nonlocal boundary value problem for nonhomogeneous partial differential equation with the operator of the generalized differentiation $B=z\frac{\partial}{\partial z}$, which operate on function of scalar complex variable $z$. Problems with nonlocal conditions for partial differential equations represent an important part of the present-day theory of differential equations. Particularly, this is due with the fact that these problems are models of the propagation of heat, process of moisture transfer in capillary-porous environments, diffusion of particles in the plasma, inverse problems, and also problems of mathematical biology. One of the most important question of the general theory of partial differential equations is the establishment of conditions for the correctness of boundary value problems. However, the investigation of problems with nonlocal conditions for partial differential equations in bounded domains connected with the problem of small denominators. This problem connected with the fact, that the denominators of coefficients of the series, which represented the solutions of nonlocal problems may be arbitrary small. Specific of the present work is the investigation of a nonlocal boundary-value problem for nonhomogeneous partial differential equation with the operator of the generalized differentiation $B=z\frac{\partial}{\partial z}$, which operate on functions of one scalar complex variable $z$. The considered problem in the case of many generalized differentiation operators is incorrect in Hadamard sense, and its solvability depends on the small denominators that arise in the constructing of a solution. In the case of one scalar complex variable we showed, that the problem is Hadamard correct. The conditions of correct solvability of the nonlocal boundary value problem in Sobolev spaces are established. The uniqueness theorem and existence theorem of the solution of problem in these spaces are proved.