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

Asymptotic analysis of fundamental solution of multi-dimensional distributed-order time-fractional diffusion equation with unit density function

“…and the Laplace transform on the equation (3.29), simultaneously, and get a second order ODE with the finite solution [4] ũ (…”

“…To the best of our knowledge, for the super-slow diffusion equation any closed-form expression in one-or multi-dimensional space for the fundamental solution has not been given. Although, this solution has been presented in terms of the Mellin-Barnes integrals [4], but in fact it has not been possible yet to find any special function (existing function in the literature or a modified function of it) as a desired solution.…”

“…The asymptotic behaviors of solution was also studied using the asymptotics of the Fox H-functions. In [4], author also apply the Fourier and Laplace integral transforms, simultaneously, to extend the operational calculus of (1.1) by following equations for the whole-and half-space…”

“…The integral representations of solutions are subsequently derived after applying the Titchmarsh theorem for Bromwich's integrals. In order to answer the question about the finding a closed-form expression for the fundamental solution in the case b(σ) = 1, in contrast to the previous strategy in [4], we first study the inverse Laplace transform and the inverse Fourier transform later. This approach has not been stated in the literature and this is the main objective of our work for establishing new operational calculus of (1.1).…”