Nonlocal boundary value problem for a nonhomogeneous pseudoparabolic-type integro-differential equation with degenerate kernel

Mathematical modeling of many processes occurring in the real world leads to the study of direct and inverse problems for equations of mathematical physics. Direct problems for partial differential and integro-differential equations by virtue of their importance in the application are one of the most important parts of the theory of differential equations. In the case, when the boundary of the flow of physical process is not applicable for measurements, an additional information can be used in the nonlocal conditions in the integral form.
We propose a method of studying the one-value solvability of the nonlocal problem for a non-homogeneous third-order pseudoparabolic-type integro-differential equation with degenerate kernel. Such type of integro-differential equations models many natural phenomena and appears in many fields of sciences. For this reason, a great importance in the works of many researchers was given to this type of equations.
We use the spectral method based on Fourier series and separation of variables. Application of this method of separation of variables can improve the quality of formulation of the considered problem and facilitate the processing procedure.
Thus, in this article we consider the questions of solvability and constructing the solution of nonlocal boundary value problem for a three dimensional non-homogeneous third-order pseudoparabolic-type integro-differential equation with degenerate kernel. The criterion of one-value solvability of the considered problems is installed. Under this criterion the theorems of one-valued solvability of the considered problems are proved. It is also checked that the solutions of considering problems are smooth. Every estimate was obtained with the aid of the Hölder inequality and Minkovsky inequality. This paper advances the theory of partial integro-differential equations with degenerate kernel.