A Generalized Definition of Fractional Derivative with Applications

TL;DR

This paper introduces a generalized fractional derivative (GFD) that unifies various definitions, simplifies solving fractional differential equations, and offers improved accuracy over existing derivatives like conformable derivatives.

Contribution

The paper proposes a new GFD that generalizes existing fractional derivatives, enabling straightforward solutions for fractional differential equations and demonstrating superior accuracy.

Findings

GFD coincides with Caputo and Riemann-Liouville derivatives for certain functions.

Solutions to Riccati fractional differential equations are simplified using GFD.

GFD provides better accuracy than conformable derivatives.

Abstract

A generalized fractional derivative (GFD) definition is proposed in this work. For a differentiable function that can be expanded by Taylor series, we show that D^Elafa*D^Beta f(t)=D^(Elafa+Beta)f(t). GFD is applied for some functions in which we investigate that GFD coincides with Caputo and Riemann-Liouville fractional derivatives' results. The solutions of Riccati fractional differential equation are simply obtained via GFD. A comparison with other definitions is also discussed. The results show that the proposed definition in this work gives better accuracy than the commonly known conformable derivative definition. Therefore, GFD has some advantages in comparison with other definitions in which a new path is provided for simple analytical solutions of many problems in the context of fractional calculus.