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

• Published in: Communications in nonlinear science & numerical simulation Vol. 82; p. 104903
• Main Authors: Cresson, Jacky, Szafrańska, Anna
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.03.2020; Elsevier Science Ltd; Elsevier
• Subjects: Chain rule; Chains; Continuity (mathematics); Derivatives; Dynamical Systems; Fractional derivations; Fractions; Jumarie fractional derivative; Leibniz property; Linear equations; Linearity; Local fractional derivative; Mathematical functions; Mathematical models; Mathematics; Obstruction Lemma; Operators (mathematics); Leibniz property; Jumarie fractional derivative; Obstruction Lemma; 26A33; Chain rule; 49M25; Fractional derivations; Local fractional derivative; 65Q30; fractional derivations; obstruction Lemma; local fractional derivative AMS subject classification: 26A33; Ju- marie fractional derivative; chain rule
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2019.104903

•The Leibniz and chain rule properties are discussed in the light of various extensions of the Riemann–Liouvile fractional derivative.•We derive the obstruction Lemma and use it to prove the triviality of some fractional operators defined on continuous functions as long as the linearity and the Leibniz property are preserved.•We explain that the operator which satisfies the chain rule property and is zero on a constant function needs to be trivial.•The Jumarie’s fractional derivative and the local fractional derivative proposed by Kolwankar and Gangal are discussed as examples. 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.