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Open AccessArticle

Fractional Derivatives with the Power-Law and the Mittag–Leffler Kernel Applied to the Nonlinear Baggs–Freedman Model

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CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490 Cuernavaca, Morelos, Mexico
2
Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, Bloemfontein 9300, South Africa
*
Author to whom correspondence should be addressed.
Fractal Fract 2018, 2(1), 10; https://doi.org/10.3390/fractalfract2010010
Received: 20 November 2017 / Revised: 11 January 2018 / Accepted: 7 February 2018 / Published: 9 February 2018
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering)
This paper considers the Freedman model using the Liouville–Caputo fractional-order derivative and the fractional-order derivative with Mittag–Leffler kernel in the Liouville–Caputo sense. Alternative solutions via Laplace transform, Sumudu–Picard and Adams–Moulton rules were obtained. We prove the uniqueness and existence of the solutions for the alternative model. Numerical simulations for the prediction and interaction between a unilingual and a bilingual population were obtained for different values of the fractional order. View Full-Text
Keywords: Freedman model; Liouville–Caputo derivative; Atangana–Baleanu derivative; fixed point theorem Freedman model; Liouville–Caputo derivative; Atangana–Baleanu derivative; fixed point theorem
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Gómez-Aguilar, J.F.; Atangana, A. Fractional Derivatives with the Power-Law and the Mittag–Leffler Kernel Applied to the Nonlinear Baggs–Freedman Model. Fractal Fract 2018, 2, 10.

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