Eur. Phys. J. Plus (2018) 133: 94
https://doi.org/10.1140/epjp/i2018-11947-6

Regular Article

Comparing the Caputo, Caputo-Fabrizio and Atangana-Baleanu derivative with fractional order: Fractional cubic isothermal auto-catalytic chemical system

¹ Department of Mathematics, Faculty of Arts and Sciences, Najran University, Najran, Saudi Arabia
² Department of Mathematics, Faculty of Applied Science, Taiz University, Taiz, Yemen

^* e-mail: khaledma_sd@hotmail.com

Received: 1 November 2017
Accepted: 24 January 2018
Published online: 7 March 2018

Abstract

In this work we extend the standard model for a cubic isothermal auto-catalytic chemical system (CIACS) to a new model of a fractional cubic isothermal auto-catalytic chemical system (FCIACS) based on Caputo (C), Caputo-Fabrizio (CF) and Atangana-Baleanu in the Liouville-Caputo sense (ABC) fractional time derivatives, respectively. We present approximate solutions for these extended models using the q -homotopy analysis transform method (q -HATM). We solve the FCIACS with the C derivative and compare our results with those obtained using the CF and ABC derivatives. The ranges of convergence of the solutions are found and the optimal values of h , the auxiliary parameter, are derived. Finally, these solutions are compared with numerical solutions of the various models obtained using finite differences and excellent agreement is found.