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In the present paper we consider an inverse source problem for a time-fractional mixed parabolic-hyperbolic equation with Caputo derivatives. In the case when the hyperbolic part of the considered mixed-type equation is the wave equation, the uniqueness of the source and the solution are strongly influenced by the initial time and the problem is generally ill-posed. However, when the hyperbolic part is time-fractional, the problem is well-posed if the end time is large. Our method relies on the orthonormal system of eigenfunctions of the operator with respect to the space variables. Finally, we prove uniqueness and stability of certain weak solutions for the problems under consideration.

Inverse source problems for time-fractional mixed parabolic-hyperbolic-type equations