FRACTIONAL RELAXATION AND TIME-FRACTIONAL DIFFUSION OF DISTRIBUTED ORDER

The basic differential equations of exponential relaxation and Gaussian diffusion can be generalized by replacing the first-order time derivative with a fractional derivative in the Riemann-Liouville (R-L) sense and in the Caputo (C) sense, both of a single order less than 1. The two forms turn out to be equivalent. When, however we use fractional derivatives of distributed order (between zero and 1), the equivalence is lost: then the analysis of asymptotic behaviour of the solutions at small and large times becomes relevant, over all in the absence of closed-form solutions. For the general case of a distribution of orders we give an outline of the theory providing the solutions in terms of an integral of Laplace type. We consider with some detail two cases of relaxation and diffusion of distribution order: the double-order and the uniformly distributed order discussing the differences between the R-L and C approaches. For all the cases considered we exhibit plots of the solutions for moderate and large times.