Fractional Left Local General M-Derivative

  • George A. AnastassiouEmail author
Part of the Studies in Computational Intelligence book series (SCI, volume 886)


Here is introduced and studied the left fractional local general M-derivative of various orders. All basic properties of an ordinary derivative are established here. We also define the corresponding left fractional M-integrals. Important theorems are established such as: the inversion theorem, the fundamental theorem of fractional calculus, the mean value theorem, the extended mean value theorem, the Taylor’s formula with integral remainder, the integration by parts. Our left fractional derivative generalizes the alternative fractional derivative and the local M-fractional derivative. See also [3].


  1. 1.
    T. Abdeljawad, On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    G.A. Anastassiou, Fractional Differentiation Inequalities (Springer, Heidelberg, 2009)CrossRefGoogle Scholar
  3. 3.
    G.A. Anastassiou, On the left fractional local general \(M\)-Derivative, submitted for publication (2019)Google Scholar
  4. 4.
    R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications (Springer, Berlin, 2014)CrossRefGoogle Scholar
  5. 5.
    U.N. Katugampola, A new fractional derivative with classical properties (2014). arXiv:1410.6535v2
  6. 6.
    R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    G.W. Leibniz, Letter from Hanover, Germany, to G.F.A L’Hospital, September 30, 1695, Leibniz Mathematische Schriften (Olms-Verlag, Hildescheim), pp. 301–302 (First published in 1849)Google Scholar
  8. 8.
    G.M. Mittag-Leffler, Sur la nouvelle fonction \( \mathbb{E}_{\alpha }\left( x\right) \). CR Acad. Sci. Paris 137, 554–558 (1903)zbMATHGoogle Scholar
  9. 9.
    J.V.C. Sousa, E.C. de Oliveira, On the local \(M\)-derivative. Progr. Fract. Differ. Appl. 4(4), 479–492 (2018)Google Scholar

Copyright information

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of MemphisMemphisUSA

Personalised recommendations