Fractional Models in Laser Physics: The Role of the Hilfer-Prabhakar Operator and Khalouta Integral Transform

Abstract

In this work, we seek to implement the new exponential type Kernel integral transformation called Khalouta transform to the Prabhakar integral (PFI), Prabhakar fractional derivatives (PFD), regularized Prabhakar fractional derivative (RPFD), Hilfer–Prabhakar fractional derivatives (HPFD) and its regular- ized form (RHPFD). Next, we present these results in the form of functions containing the Mittag-Leffler function and apply them to the solution of the Generalized Fractional Free-Electron Laser (FEL) equation involving the HPFD and its regularized form.