The inverse problem for a mixed type equation with a fractional order operator in a rectangular domain

We study the inverse problem for a mixed type equation with the Riemann-Liouville and Caputo operator in a rectangular domain. A criterion for the uniqueness and existence of a solution to the inverse problem is established. The solution of the problem is constructed in the form of the sum of a series of eigenfunctions of the corresponding one-dimensional spectral problem. It is proved that the unique solvability of the inverse problem substantially depends on the choice of the boundary of a rectangular region. An example is constructed in which the inverse problem with homogeneous conditions has a nontrivial solution. Estimates are obtained that allow substantiating the convergence of series in the class of regular solutions of this equation and the stability of the solution of the inverse problem from boundary data.