Time-Fractional Diffusion of Distributed Order

The partial differential equation of Gaussian diffusion is generalized by using the time-fractional derivative of distributed order between 0 and 1, in both the Riemann-Liouville and the Caputo sense. For a general distribution of time orders we provide the fundamental solution, which is a probability density, in terms of an integral of Laplace type. The kernel depends on the type of the assumed fractional derivative, except for the single order case where the two approaches turn out to be equivalent. We consider in some detail two cases of order distribution: Double-order, and uniformly distributed order. Plots of the corresponding fundamental solutions and their variance are provided for these cases, pointing out the remarkable difference between the two approaches for small and large times.