Forward and inverse problems for a mixed type equation with Caputo fractional derivative and Bessel operator

This work is devoted to the investigation of initial-boundary value and inverse problems for mixed-type equations involving a time-fractional derivative of order ρ in the sense of Caputo, together with a Bessel operator. By applying the method of separation of variables, the problem is reduced to solving a system of ordinary differential equations for the coefficients of the series expansions of the unknown functions. These expansions are represented as FourierBessel series in terms of orthonormal Bessel functions of the first kind and zero order. The conditions for the existence and uniqueness of the solution are established. Further, it is assumed that the order of the derivative ρ is unknown, and the inverse problem of determining this parameter is considered. An additional condition is found that guarantees the existence and uniqueness of a solution to this inverse problem.