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Open Physics

formerly Central European Journal of Physics

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Volume 11, Issue 6


Volume 13 (2015)

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

Guy Jumarie
  • Department of Mathematics, University of Quebec at Montreal, P.O. Box 8888, Downtown Station, Montreal Qc, H3C 3P8, Canada
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Published Online: 2013-10-09 | DOI: https://doi.org/10.2478/s11534-013-0256-7


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.

Keywords: fractional calculus; fractional Taylor’s series; fractional derivative; systems modelling; fractional derivative chain rule

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About the article

Published Online: 2013-10-09

Published in Print: 2013-06-01

Citation Information: Open Physics, Volume 11, Issue 6, Pages 617–633, ISSN (Online) 2391-5471, DOI: https://doi.org/10.2478/s11534-013-0256-7.

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© 2013 Versita Warsaw. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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