On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling

By Guy Jumarie. 2013

98 citations recorded by the source corpus.

It has been pointed out that the derivative chains rules in fractional differential calculus via fractional calculus are not quite satisfactory as far as they can yield different results which depend upon how the formula is applied, that is to say depending upon where is the considered function and where is the function of function. The purpose of the present short note is to display some comments (which might be clarifying to some readers) on the matter. This feature is basically related to the non-commutativity of fractional derivative on the one hand, and furthermore, it is very close to the physical significance of the systems under consideration on the other hand, in such a manner that everything is right so. As an example, it is shown that the trivial first order system may have several fractional modelling depending upon the way by which it is observed. This suggests some rules to construct the fractional models of standard dynamical systems, in as meaningful a model as possible. It might happen that this pitfall comes from the feature that a function which is continuous everywhere, but is nowhere differentiable, exhibits random-like features.

Subfields: Modeling and Simulation, Numerical Analysis, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/d6eed77d1f1a47ef6c2b37348b0528748ca9c7b1

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Citations in the corpus

A Review of Definitions for Fractional Derivatives and Integral (2014, background): “2 Mathematical Problems in Engineering but more appropriate to discuss problems involving a fractional differential equation with initial conditions [13–21].”
Leibniz rule and fractional derivatives of power functions (2016, unknown): “[20–40].”
On chain rule for fractional derivatives (2016, background): “(68) in Lemma 14 of [18] has the form D x f (g(x)) = (2 − α) gα−1(x) (D g f (g))g=g(x) D x g(x).”