Boundary Value Problem for a Loaded Pseudoparabolic Equation with a Fractional Caputo Operator

Abstract

Differential equations containing fractional derivatives, for both time and spatial variables, have now begun to attract the attention of mathematicians and physicists; they are used in connection with these equations as mathematical models of various processes. The fractional derivative equation tool plays a crucial role in describing plenty of natural processes concerning physics, biology, geology, and so on. In this paper, we studied a loaded equation in relation to a spatial variable for a linear pseudoparabolic equation, with an initial and second boundary value condition (the Neumann condition), and a fractional Caputo derivative. A distinctive feature of the considered problem is that the load at the point is in the higher partial derivatives of the solution. The problem is reduced to a loaded equation with a nonlocal boundary value condition. A way to solve the considered problem is by using the method of energy inequalities, so that a priori estimates of solutions for non-local boundary value problems are obtained. To prove that this nonlocal problem is solvable, we used the method of continuation with parameters. The existence and uniqueness theorems for regular solutions are proven.