Generalized Shehu Transform to $\Psi$-Hilfer-Prabhakar Fractional Derivative and its Regularized Version

Abstract

In this manuscript, athours interested on the generalized Shehu transform of $\Psi$-Riemann-Liouville, $\Psi$-Caputo, $\Psi$-Hilfer fractional derivatives. Moreover, $\Psi$-Prabhakar, $\Psi$-Hilfer-Prabhakar fractional derivatives and its regularized version presented in terms of the $\Psi$-Mittag-Leffler type function. They are also utilised to solve several Cauchy type problems involving $\Psi$-Hilfer-Prabhakar fractional derivatives and their regularised form, such as the space-time fractional advection-dispersion equation and the generalized fractional free-electron laser (FEL) equation.

Keywords

• $\Psi$-Prabhakar integral
• $\Psi$-Hilfer-Prabhakar derivative
• $\Psi$-Mittag-Leffler function
• $\Psi$-Shehu transform

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