Shehu Transform of Hilfer-Prabhakar Fractional Derivatives and Applications on some Cauchy Type Problems

Abstract

In this paper, we are interested on the Shehu transform of both Prabhakar and Hilfer–Prabhakar fractional derivative and its regularized version. These results are presented in terms of Mittag-Leffler type function and also utilized to obtain the solutions of some Cauchy type problems, such as Space-time Fractional Advection-Dispersion equation and Generalized fractional Free Electron Laser (FEL) equation, at which Hilfer-Prabhakar fractional derivative of fractional order and its regularized version are involved.