Applications of generalized formable transform with $$\Psi $$ -Hilfer–Prabhakar derivatives

Abstract

This paper introduces the \(\Psi \)-formable integral transform, discusses the several essential properties and results—Convolution, \(\Psi \)-formable transform of tth derivative, \(\Psi \)-Riemann Liouville fractional integration and differentiation, \(\Psi \)-Caputo fractional differentiation, \(\Psi \)-Hilfer fractional differentiation, \(\Psi \)-Prabhakar fractional integration and differentiation, and \(\Psi \)-Hilfer–Prabhakar fractional derivatives. Next, we use the Fourier integral and \(\Psi \)-Modifiable conversions to solve some Cauchy-type fractional differential equations using the generalized three-parameter Mittag–Leffler function and \(\Psi \)-Hilfer–Prabhakar fractional derivatives.