Analyticity of solutions to a distributed order time-fractional diffusion equation and its application to an inverse problem.

This paper deals with an initial–boundary value problem for a distributed order time-fractional diffusion equation with the fractional derivative in the Caputo sense. The method of the eigenfunctions expansion in combination with the Laplace transform is first employed to prove the uniqueness and existence of the solution to the initial–boundary value problem and then to show its analyticity in time. As an application of the analyticity of the solution, a uniqueness result for an important inverse problem of determination of the weight function in the distributed order derivative contained in the time-fractional diffusion equation from one interior point observation of its solution is obtained.