In this paper, we obtain the new generalized integral transform on Prabhakar integral, Hilfer-Prabhakar derivatives and regularized Hilfer-Prabhakar fractional derivatives. Next, we evaluate the solution of some Cauchy type fractional differential equation with Hilfer-Prabhakar fractional derivatives by applying the new integral transform and Fourier transforms which involves three parameter Mittage-Leffler function.

In this paper, we will derive the Formable integral transform of the Hilfer-Prabhakar and its regularized version of the Hilfer-Prabhakar fractional derivative. Then, we will use the Formable and Fourier transforms, which involve the three-parameter Mittag-Leffler function, to find the solution of some Cauchy type fractional differential equations with Hilfer-Prabhakar fractional derivatives.

This work is focused on the study of the generalized forms of the fractional derivatives of Reimann-Liouville, Caputo and Hilfer, in terms of Ψ functions. The fractional derivatives of Ψ-Prabhakar, Ψ-Hilfer-Prabhakar, and its regularized form are also described in terms of Ψ-Mittage-Leffler type functions. These are then used to solve a number of Cauchy type equations involving Ψ-Hilfer-Prabhakar fractional derivatives and their regularized form, including the generalized fractional free electron laser equation.