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Entropy 2015, 17(6), 4439-4453; doi:10.3390/e17064439

Analysis of the Keller–Segel Model with a Fractional Derivative without Singular Kernel

1
Institute for Groundwater Studies, Faculty of Natural and Agricultural Sciences, University of the Free State, 9300 Bloemfontein, South Africa
2
Department of Mathematics, Colleges of Sciences, King Saud University, P.O. Box 1142, Riyadh, 11989, Saudi Arabia,
*
Author to whom correspondence should be addressed.
Academic Editor: Carlo Cattani
Received: 15 May 2015 / Revised: 5 June 2015 / Accepted: 9 June 2015 / Published: 23 June 2015
(This article belongs to the Special Issue Wavelets, Fractals and Information Theory)
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Abstract

Using some investigations based on information theory, the model proposed by Keller and Segel was extended to the concept of fractional derivative using the derivative with fractional order without singular kernel recently proposed by Caputo and Fabrizio. We present in detail the existence of the coupled-solutions using the fixed-point theorem. A detailed analysis of the uniqueness of the coupled-solutions is also presented. Using an iterative approach, we derive special coupled-solutions of the modified system and we present some numerical simulations to see the effect of the fractional order. View Full-Text
Keywords: Keller–Segel model; Caputo–Fabrizio fractional derivative; fixed-point theorem; special solution Keller–Segel model; Caputo–Fabrizio fractional derivative; fixed-point theorem; special solution
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Atangana, A.; Alkahtani, B.S.T. Analysis of the Keller–Segel Model with a Fractional Derivative without Singular Kernel. Entropy 2015, 17, 4439-4453.

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