Abstract
In this paper, the authors introduce the Prabhakar derivative associated with the generalised Mittag-Leffler function. Some properties of the Prabhakar integrals, Prabhakar derivatives and some of their extensions, like fractional Fourier transform of Prabhakar integrals and fractional Fourier transform of Prabhakar derivatives are introduced. This note aims to study the Mittag-Leffler-Hyers-Ulam stability of the linear and nonlinear fractional differential equations with the Prabhakar derivative. Furthermore, we give a brief definition of the Mittag-Leffler-Hyers-Ulam problem and a method for solving fractional differential equations using the fractional Fourier transform. We show that the fractional differential equations are Mittag-Leffler-Hyers-Ulam stable in the sense of Prabhakar derivatives.