A New Approach to Generalized Fractional Derivatives

TL;DR

This paper introduces a new generalized fractional derivative that unifies Riemann-Liouville and Hadamard derivatives, providing new representations and an illustrative example.

Contribution

It presents a novel fractional derivative that encompasses Riemann-Liouville and Hadamard derivatives within a single unified framework.

Findings

Introduces a generalized fractional derivative unifying existing types.

Provides two different representations of the new derivative.

Includes an example demonstrating the application of the derivative.

Abstract

The author \mbox{(Appl. Math. Comput. 218(3):860-865, 2011)} introduced a new fractional integral operator given by, \[ \big({}^\rho \mathcal{I}^\alpha_{a+}f\big)(x) = \frac{\rho^{1- \alpha }}{\Gamma({\alpha})} \int^x_a \frac{\tau^{\rho-1} f(\tau) }{(x^\rho - \tau^\rho)^{1-\alpha}}\, d\tau, \] which generalizes the well-known Riemann-Liouville and the Hadamard fractional integrals. In this paper we present a new fractional derivative which generalizes the familiar Riemann-Liouville and the Hadamard fractional derivatives to a single form. We also obtain two representations of the generalized derivative in question. An example is given to illustrate the results.