Inverse Problems for the Equation of Vibrations of a Canister Beam to Find the Source

For the beam vibration equation, inverse problems are studied to find the right side, i.e. vibration source. Solutions of the problems by methods of spectral analysis and Volterra integral equations are constructed explicitly as sums of series, and the corresponding uniqueness and existence theorems are proved. When substantiating the existence of a solution to the inverse problem by determining the factor of the right-hand side, which depends on the spatial coordinate, the problem of small denominators arises. In this regard, estimates of the denominators are established that guarantee their separation from zero, with an indication of the corresponding asymptotics. On the basis of these estimates, the convergence of the series in the class of regular solutions of the beam oscillation equation is substantiated.