Eur. Phys. J. Plus (2018) 133: 200
https://doi.org/10.1140/epjp/i2018-12038-6

Regular Article

Analytical solutions of the Keller-Segel chemotaxis model involving fractional operators without singular kernel

¹ Facultad de Matemáticas, Universidad Autónoma de Guerrero, Av. Lázaro Cárdenas S/N, Cd. Universitaria, Chilpancingo, Guerrero, Mexico
² CONACyT-Tecnológico Nacional de México/CENIDET, Interior Internado Palmira S/N, Col. Palmira, C.P. 62490, Cuernavaca, Morelos, Mexico
³ Department of Mathematics, National Institute of Technology, Jamshedpur, 831014, Jharkhand, India

^* e-mail: jgomez@cenidet.edu.mx

Received: 1 February 2018
Accepted: 25 April 2018
Published online: 29 May 2018

Abstract

This paper discusses the application of analytical techniques, namely the Laplace homotopy perturbation method and the modified homotopy analysis transform method, for solving a coupled one-dimensional time-fractional Keller-Segel chemotaxis model. The first method is based on a combination of the Laplace transform and homotopy methods, while the second method is an analytical technique based on the homotopy polynomial. Fractional derivatives with exponential and Mittag-Leffler laws in Liouville-Caputo sense are considered. The effectiveness of both methods is demonstrated by finding the exact solutions of the Keller-Segel chemotaxis model. Some examples have been presented in order to compare the results obtained with both fractional-order derivatives.