On the solvability of one inverse problem for a fourth-order equation

Abstract

In this paper, for a fourth-order equation in a rectangular domain, an inverse problem of finding the unknown right-hand side, which depends on one variable, is considered. Criteria for the uniqueness and existence of a solution to the inverse problem under consideration for a fourth-order equation are established. The solution to the problem is constructed as the sum of a series in eigenfunctions of the corresponding spectral problem. The uniqueness of the solution to the inverse problem follows from the completeness of the system of eigenfunctions. Sufficient conditions are established for the boundary functions that guarantee theorems of existence and stability of the solution to the problem. In a closed domain, absolute and uniform convergence of the found solution to the inverse problem in the form of a series in the class of regular solutions is shown, as well as series obtained by term-by-term differentiation with respect to t and x three and four times, respectively. The stability of the solution of the inverse problem in the norms of the space of square-summable functions and in the space of continuous functions with respect to changes in the input data has also been proven.