Abstract
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].
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
T. Abdeljawad, On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)
G.A. Anastassiou, Fractional Differentiation Inequalities (Springer, Heidelberg, 2009)
G.A. Anastassiou, On the left fractional local general \(M\)-Derivative, submitted for publication (2019)
R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications (Springer, Berlin, 2014)
U.N. Katugampola, A new fractional derivative with classical properties (2014). arXiv:1410.6535v2
R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)
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)
G.M. Mittag-Leffler, Sur la nouvelle fonction \( \mathbb{E}_{\alpha }\left( x\right) \). CR Acad. Sci. Paris 137, 554–558 (1903)
J.V.C. Sousa, E.C. de Oliveira, On the local \(M\)-derivative. Progr. Fract. Differ. Appl. 4(4), 479–492 (2018)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2020 The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Anastassiou, G.A. (2020). Fractional Left Local General M-Derivative. In: Intelligent Analysis: Fractional Inequalities and Approximations Expanded. Studies in Computational Intelligence, vol 886. Springer, Cham. https://doi.org/10.1007/978-3-030-38636-8_27
Download citation
DOI: https://doi.org/10.1007/978-3-030-38636-8_27
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-38635-1
Online ISBN: 978-3-030-38636-8
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)