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.
Highlights
The role of special functions in applied mathematics and especially in differential equations and mathematical physics can hardly be overestimated
In the rest of this section, we introduce a generalized translation operator and a generalized convolution that we employ for analytical treatment of the fractional differential equations with the fractional Bessel derivative
To shorten the formulations of the results, each time the Meijer integral transform is applied to a function, we suppose that this function is from the space Kγ introduced in the previous section
Summary
The role of special functions in applied mathematics and especially in differential equations and mathematical physics can hardly be overestimated. We discuss some new applications of the Fox–Wright function p Ψq (z) and the Fox H-function Hm,n p,q ( z ) in the theory of fractional differential equations with the fractional Bessel operator, which is defined as a power of the conventional Bessel γ ∂. They are our main tools for solving fractional differential equations with the fractional powers of the Bessel operator.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
