Foundations of generalized Prabhakar-Hilfer fractional calculus with applications
- George A. Anastassiou ganastss@memphis.edu
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DOI:
https://doi.org/10.4067/S0719-06462021000300423Abstract
Here we introduce the generalized Prabhakar fractional calculus and we also combine it with the generalized Hilfer calculus. We prove that the generalized left and right side Prabhakar fractional integrals preserve continuity and we find tight upper bounds for them. We present several left and right side generalized Prabhakar fractional inequalities of Hardy, Opial and Hilbert-Pachpatte types. We apply these in the setting of generalized Hilfer calculus.
Keywords
G. A. Anastassiou, Fractional differentiation inequalities, New York: Springer-Verlag, 2009.
G. A. Anastassiou, Intelligent Computations: abstract fractional calculus inequalities, approximations, Cham: Springer, 2018.
A. Giusti, I. Colombaro, R. Garra, R. Garrappa, F. Polito, M. Popolizio and F. Mainardi, “A practical guide to Prabhakar fractional calculus”, Fract. Calc. Appl. Anal., vol. 23, no. 1, pp. 9–54, 2020.
R. Gorenflo, A. Kilbas, F. Mainardi and S. Rogosin, Mittag-Leffler functions, related topics and applications, Heidelberg: Springer, 2014.
E. Hewith and K. Stromberg, Real and abstract analysis. A modern treatment of the theory of functions of a real variable, New York: Springer, 1965.
F. Polito and Ž. Tomovski, “Some properties of Prabhakar-type fractional calculus operators”, Fract. Differ. Calc., vol. 6, no. 1, pp. 73–94, 2016.
T. R. Prabhakar, “A singular integral equation with a generalized Mittag-Leffler function in the kernel”, Yokohama Math. J., vol. 19, pp. 7–15, 1971.
J. Vanterler da C. Sousa, E. Capelas de Oliveira, “On the ψ-Hilfer fractional derivative”, Commun. Nonlinear Sci. Numer. Simul., vol. 60, pp. 72–91, 2018.
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