The inverse problem of determining the right-hand side of afourth-order differential equation

Abstract

This article studies the inverse problem of finding a multiplier on the right-hand side, depending on the spatial variable $x$. In the direct problem, an initial-boundary value problem for a fourth-order differential equation is considered. Using the Fourier method, the solution to the initial-boundary value problem is constructed, and its properties are investigated. Sufficient conditions for the existence of a solution to the direct problem are obtained, which will be used in the study of the inverse problem. Theorems on local existence and global uniqueness are proven, and an estimate of the conditional stability of the solution to both the direct and inverse problems is provided.