The Two Forms of Fractional Relaxation of Distributed Order

The first-order differential equation of exponential relaxation can be generalized by using either the 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, in particular on the asymptotic behaviour of the fundamental solution at small and large times. We give an outline of the theory providing the general form of the solution in terms of an integral of Laplace type over a positive measure depending on the orderdistribution. We consider in some detail two cases of fractional relaxation of distribution order: the doubleorder and the uniformly distributed order discussing the differences between the R-L and C approaches. For all the cases considered we give plots of the solutions for moderate and large times.