Analysis of fractional Cauchy problems with some probabilistic applications

In this paper we give an explicit solution to Dzherbashyan-Caputo-fractional Cauchy problems related to equations with derivatives of order vk, for k non -negative integer and v > 0. The solution is obtained by connecting the differential equation with the roots of the characteristic polynomial and it is expressed in terms of Mittag-Lefflertype functions. Under some additional hypothesis, the solution can be expressed as a linear combination of Mittag-Leffler functions with common fractional order v. We establish a probabilistic relationship, involving the inverse of stable subordinator, between the solutions of differential problems with order alpha v and v, for alpha is an element of (0, 1). Finally, we use the described method to solve fractional differential equations arising in the fractionalization of partial differential equations related to the probability law of planar random motions with finite velocities. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).