Comments on various extensions of the Riemann-Liouville fractional derivatives : about the Leibniz and chain rule properties

TL;DR

This paper critically examines various extensions of the Riemann-Liouville fractional derivatives, focusing on their Leibniz and chain rule properties, and establishes fundamental obstructions to their existence under certain conditions.

Contribution

It provides a general obstruction lemma showing the limitations of fractional derivatives maintaining linearity and Leibniz property, and analyzes specific derivatives like Jumarie's and local fractional derivatives.

Findings

Obstruction lemma for fractional derivatives preserving linearity and Leibniz rule.

Demonstration of triviality of certain fractional operators under these conditions.

Discussion of the chain rule limitations in fractional calculus.

Abstract

Starting from the Riemann-Liouville derivative, many authors have built their own notion of fractional derivative in order to avoid some classical difficulties like a non zero derivative for a constant function or a rather complicated analogue of the Leibniz relation. Discussing in full generality the existence of such operator over continuous functions, we derive some obstruction Lemma which can be used to prove the triviality of some operators as long as the linearity and the Leibniz property are preserved. As an application, we discuss some properties of the Jumarie's fractional derivative as well as the local fractional derivative. We also discuss the chain rule property in the same perspective.