An inverse source problem for distributed order time-fractional diffusion equation

We consider an inverse time-dependent source problem governed by a distributed time-fractional diffusion equation using interior measurement data. Such a problem arises in some ultra-slow diffusion phenomena in many applied areas. Based on the regularity result of the solution to the direct problem, we establish the solvability of this inverse problem as well as the conditional stability in suitable function space with a weak norm. By a variational identity connecting the unknown time-dependent source and the interior measurement data, the conjugate gradient method is also introduced to construct the inversion algorithm under the framework of regularizing scheme. We show the validity of the proposed scheme by several numerical examples.