Fundamental solution of a multi-dimensional distributed order fractional diffusion equation

In this paper, we consider the multi-dimensional distributed order fractional diffusion equation in the whole space, the half space and the corridor space. We use the Titchmarsh theorem to show that the fundamental solutions are expressed in terms of the Laplace-type integrals and can be applied for the Gauss–Laguerre quadrature. The mathematical identities and their relationships for the fundamental solutions in different dimensions are also discussed, and the alternative representation is given in terms of the Mellin–Barnes integrals for the fundamental solution in the whole space.