Random-time processes governed by differential equations of fractional distributed order

We analyze here different types of fractional differential equations, under the assumption that their fractional order nu is an element of (0,1] is random with probability density n(nu). We start by considering the fractional extension of the recursive equation governing the homogeneous Poisson process N(t), t > 0. We prove that, for a particular (discrete) choice of n(nu), it leads to a process with random time, defined as N((T) over tilde (nu 1,nu 2)(t)), t > 0. The distribution of the random time argument (T) over tilde (nu 1,nu 2)(t) can be expressed, for any fixed t, in terms of convolutions of stable-laws. The new process N((T) over tilde (nu 1,nu 2)) is itself a renewal and can be shown to be a Cox process. Moreover we prove that the survival probability of N((t) over tilde (nu 1,nu 2)) as well as its probability generating function, are solution to the so-called fractional relaxation equation of distributed order (see [16]). In view of the previous results it is natural to consider diffusion-type fractional equations of distributed order. We present here an approach to their solutions in terms of composition of the Brownian motion B(t),t > 0 with the random time (T) over tilde (nu 1,nu 2). We thus provide an alternative to the constructions presented in Mainardi and Pagnini [19] and in Chechkin et al. [6], at least in the double-order case. (C) 2012 Elsevier Ltd. All rights reserved.