Inverse Source Problem for the Equation of Forced Vibrations of a Beam

Direct and inverse problems for the equation of forced vibrations of a finite length beam with a variable stiffness coefficient at the lowest term are investigated. The direct problem is 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 variable in the inverse problem is a multiplier in the right-hand side, which depends on the space variable x. This unknown is determined with respect to the solution of the direct problem by specifying an integral redefinition condition. The uniqueness of the solution of the direct problem is proved by the method of energy estimates. The eigenvalues and eigenfunctions of the corresponding elliptic operator are used to reduce the problems to integral equations. The method of successive approximations is used to prove existence and uniqueness theorems for solutions of these equations.