Direct and inverse problem for fractional diffusion wave equation with Bessel operator

The problem of determining the source function of a time fractional differential equation involving the Bessel operator, depending on the inverse spatial variable, is considered. We analyze the initial-boundary value problem for the diffusion equation involving the Bessel operator in a rectangular domain with respect to the inverse coefficient. The solution of the direct problem was expressed through the Fourier-Bessel series. Sufficient smoothness conditions were given to the given functions for the existence and uniqueness of the solution of the direct problem. In addition, to prove the existence of a solution, estimates for the coefficients of the series and eigenfunctions were derived based on asymptotic formulas for the Bessel function and its zeros. In the study of the inverse problem, a solution to the inverse problem was constructed using the corresponding additional condition. The existence and uniqueness of the solution to the inverse problem have been proven.