On solvability of one inverse problem for a fourth order equation in the rectangular domain

In this paper, for the fourth order equation in the rectangular domain, the inverse problem of finding the unknown right-hand side is considered. The solution of the problem is constructed as a sum of series of eigenfunctions and its associated functions of the corresponding spectral problem. The eigenfunctions of the corresponding spectral problem and its associated functions are complete system and form a Riesz basis in the space $L_2(0,1)$. The uniqueness of the solution to the inverse problem follows from the completeness of the system of eigen and associated functions. Sufficient conditions are established for the given initial functions, which guarantee existence and stability theorems for the solution of the problem. In a closed region, the absolute and uniform convergence of the found solution to the inverse problem is shown in the form of a series, as well as series obtained by term-by-term differentiation with respect to $t$ and $x$, three and four times, respectively. It has also been proven that the solution to the inverse problem is stable according to the norms of spaces $L_2(0,1), W^2_n\left( 0,1 \right)$, and $C(\Omega)$, with respect to changes in input data.