Abstract. For a time fractional diffusion equation and diffusion-wave equation with Caputo partial derivatives we prove the correctness of an inverse problem. This problem is to find a solution of direct problem, which is classical in time with values in the space of periodic spatial distributions, and a source term of the equation. A time integral over-determination condition is used.Mathematics subject classification (2010): 35S10.

We study the inverse Cauchy problem to a time fractional diffusion-wave equation with distributions in right-hand sides. This problem is to find a generalized solution of direct problem and an unknown time-dependent part of a source from the space of distributions. The unique solvability of the problem is established.