Forward and inverse source reconstruction problems for the equations of vibrations of a rectangular plate

For the equation of vibrations of a rectangular plate, the initial-boundary value and inverse problems of finding the right-hand side (the source of vibrations) are studied. Solutions of the problems 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 of determining the factor on the right-hand side, which depends on spatial coordinates, the problem of small denominators of two natural variables arises, for which estimates of the separation from zero with the corresponding asymptotics are established. These estimates made it possible to prove the existence theorem for this problem in the class of regular solutions by imposing certain smoothness conditions on the given boundary functions.