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Nonlinear Dynamics

, Volume 93, Issue 3, pp 1757–1763 | Cite as

A comment on some new definitions of fractional derivative

  • Andrea GiustiEmail author
Original Paper

Abstract

After reviewing the definition of two differential operators which have been recently introduced by Caputo and Fabrizio and, separately, by Atangana and Baleanu, we present an argument for which these two integro-differential operators can be understood as simple realizations of a much broader class of fractional operators, i.e. the theory of Prabhakar fractional integrals. Furthermore, we also provide a series expansion of the Prabhakar integral in terms of Riemann–Liouville integrals of variable order. Then, by using this last result we finally argue that the operator introduced by Caputo and Fabrizio cannot be regarded as fractional. Besides, we also observe that the one suggested by Atangana and Baleanu is indeed fractional, but it is ultimately related to the ordinary Riemann–Liouville and Caputo fractional operators. All these statements are then further supported by a precise analysis of differential equations involving the aforementioned operators. To further strengthen our narrative, we also show that these new operators do not add any new insight to the linear theory of viscoelasticity when employed in the constitutive equation of the Scott–Blair model.

Keywords

Prabhakar function Mittag–Leffler function Caputo–Fabrizio derivative Atangana–Baleanu derivative 

Notes

Acknowledgements

The work of the authors has been carried out in the framework of the activities of the National Group of Mathematical Physics (GNFM, INdAM). Moreover, the work of A.G. has been partially supported by GNFM/INdAM Young Researchers Project 2017 “Analysis of Complex Biological Systems”.

Compliance with ethical standards

Conflict of interest

The author declares that he has no conflict of interest concerning the publication of this manuscript.

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Physics and AstronomyUniversity of Bologna and INFNBolognaItaly
  2. 2.Arnold Sommerfeld CenterLudwig-Maximilians-UniversitätMunichGermany

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