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 νk, for k non-negative integer and ν>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 some additional hypothesis, the solution can be expressed as a linear combination of Mittag-Leffler functions with common fractional order ν. We establish a probabilistic relationship, involving the inverse of stable subordinator, between the solutions of differential problems with order αν and ν, for α∈(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.