On the inverse problem of identifying the source term in a pseudoparabolic equation with a final time overdetermination condition

Abstract

In this paper, we consider the inverse problem for a linear pseudoparabolic equation describing the temperature distribution taking into account external forces that depend only on the spatial variable. The classical solution to the inverse problem under consideration satisfies the usual pseudoparabolic equation, initial and nonlocal boundary conditions, and a final additional condition. The issues of existence and uniqueness of the solution to the inverse problem are the subject of study in the work presented by the author. As the main result, theorems on the existence and uniqueness of the classical solution to the problem under study are formulated and rigorously proven. These theorems are completely proven in a mathematically rigorous language using the method of separation of variables. In the course of the proof, a system of orthogonal and biorthogonal basis functions of a special type was chosen in accordance with the nonlocal boundary conditions. First of all, to prove the theorem on the existence of a solution, an analytical formula for the solution was derived in the form of a series in the system of these functions, their uniform convergence was analyzed according to the Weierstrass theorem, and the convergence to the classical solution of the inverse problem under consideration was investigated. The proof of the theorem on uniqueness was carried out by the method of the opposite assumption.

Keywords: inverse problem, pseudoparabolic equation, existence of solution, uniqueness of solution, nonlocal boundary condition.