Inverse Problems of Determining the Right-Hand Side and the Initial Conditions for the Beam Vibration Equation

Abstract

For the beam vibration equation, we study the inverse problems of finding the right-hand side (the source of vibrations) and the initial conditions. The solutions of the problems are constructed in closed form as sums of series, and the corresponding existence and uniqueness theorems are proved. The problem of small denominators arises when justifying the convergence of the series. In this connection, we establish estimates for the denominators that guarantee their boundedness away from zero and indicate the corresponding asymptotics. Based on these estimates, we prove the convergence of the series in the class of regular solutions of the beam vibration equation.