Special functions as solutions to the Euler-Poisson-Darboux equation with a fractional power of the Bessel operator

Abstract

In this paper, we consider fractional ordinary differential equations and the fractional Euler-Poisson-Darboux equation with fractional derivatives in the form of a power of the Bessel differential operator. Using the technique of the Meijer integral transform and its modification, fundamental solutions to these equations are derived in terms of the Fox-Wright function, the Fox H-function, and their particular cases. We also provide some explicit formulas for the solutions to the corresponding initial-value problems in terms of the generalized convolutions introduced in this paper