Operational Calculus for the nth-Level Prabhakar Type Fractional Derivative with Applications

TL;DR

This paper introduces the nth-level Prabhakar fractional derivative, explores its properties, develops an operational calculus for it, and demonstrates its application in solving fractional differential equations and heat equations.

Contribution

It generalizes existing fractional derivatives, establishes their fundamental properties, and provides a new operational calculus framework for solving related differential equations.

Findings

Derived fundamental properties of the nth-level Prabhakar derivative

Developed Mikusinski-type operational calculus for the derivative

Solved fractional differential equations using the new calculus

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.