Operational Calculus for the nth-Level Prabhakar Type Fractional Derivative with Applications
Imtiaz Waheed, Erkinjon Karimov, Mujeeb ur Rehman
TL;DR
This work introduces the $n$-th level Prabhakar fractional derivative as a unified operator that subsumes RL, Caputo, and Hilfer derivatives and connects to the Prabhakar integral. It builds a Mikusinski-style operational calculus in the convolution-quotient field $ ext{M}_{-1}$, with a semigroup-structured family of operators $ ext{P}_{eta,eta}$ that enables algebraic manipulation and inversion of the derivative. The authors derive explicit solution formulas for generalized fractional integro-differential equations and a time-fractional heat equation in terms of convolution powers and Mittag-Leffler kernels, recovering the Hilfer-Prabhakar special case when $n=1$. The framework provides analytic, closed-form solutions and a rigorous algebraic approach to complex nonlocal fractional models with potential applications in physics and engineering.
Abstract
This study investigates the nth-level Prabhakar fractional derivative, a generalization encompassing some well-known fractional derivatives. We establish its fundamental properties, particularly its relationship with the corresponding Prabhakar fractional integral. Furthermore, we develop Mikusinski-type operational calculus for this derivative, providing a framework for solving differential equations involving this operator. To illustrate its application, we present analytical solutions of two problems: a fractional order ordinary differential equation and the time fractional heat equation, both of which include the nth-level Prabhakar derivative.