Inverse source problem for the equation of forced vibrations of a beam

In this article, direct and inverse problems are studied for the equation of forced vibrations of a beam of finite length with a variable stiffness coefficient at the lowest term. In the direct problem, we consider the initial-boundary value problem for this equation with boundary conditions in the form of a beam fixed at one end and free at the other. The unknown inverse problem is the factor of the right side, which depends on the space variable $x$. To determine it with respect to the solution of the direct problem, an integral overdetermination condition is specified. The uniqueness of the solution of the direct problem is proved by the method of energy estimates. Using the eigenvalues and eigenfunctions of the corresponding elliptic operator, the problems are reduced to integral equations. The method of successive approximations is applied to these equations and existence and uniqueness theorems for solutions are proved.