Analysis of fractional Cauchy problems with some probabilistic applications

TL;DR

This paper provides explicit solutions to fractional Cauchy problems involving derivatives of order multiples of a fractional parameter, linking them to Mittag-Leffler functions and probabilistic models, with applications to random motions.

Contribution

It introduces a method to solve fractional differential equations using characteristic roots and Mittag-Leffler functions, establishing probabilistic links between solutions of different fractional orders.

Findings

Explicit solutions expressed via Mittag-Leffler functions.

Probabilistic relationships between solutions of different fractional orders.

Applications to fractionalization of PDEs and random motion models.

Abstract

In this paper we give an explicit solution of Dzherbashyan-Caputo-fractional Cauchy problems related to equations with derivatives of order $\nu k$, for $k$ non-negative integer and $\nu>0$. The solution is obtained by connecting the differential equation with the roots of the characteristic polynomial and it is expressed in terms of Mittag-Leffler-type functions. Under the some stricter hypothesis the solution can be expressed as a linear combination of Mittag-Leffler functions with common fractional order $\nu$. We establish a probabilistic relationship between the solutions of differential problems with order $\nu/m$ and $\nu$, for natural $m$. 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.