Eur. Phys. J. Plus (2019) 134: 324
https://doi.org/10.1140/epjp/i2019-12703-2

Regular Article

Numerical simulation of Turing patterns in a fractional hyperbolic reaction-diffusion model with Grünwald differences

¹ Departamento de Matemáticas y Física, Universidad Autónoma de Aguascalientes, Aguascalientes, Mexico
² Department of Computational Mathematics and Computer Science, Institute of Natural Sciences and Mathematics, Ural Federal University, Yekaterinburg, Russia
³ Department of Mathematics, Faculty of Science, Benha University, Benha City, Egypt

^* e-mail: jemacias@correo.uaa.mx

Received: 13 March 2019
Accepted: 24 April 2019
Published online: 10 July 2019

Abstract

Departing from a two-dimensional hyperbolic system that describes the interaction between some activator and inhibitor substances in chemical reactions, we investigate a general form of that model using a finite-difference approach. The model under investigation is a nonlinear system consisting of two coupled partial differential equations with generalized reaction terms. The presence of two-dimensional diffusive terms consisting of fractional operators of the Riesz type is considered here, using spatial differentiation orders in the set (1, 2] . We impose initial conditions on a closed and bounded rectangle, and a finite-difference methodology based on the use of weighted-shifted Grünwald differences is proposed. Among the most important results of this work, we establish analytically the second-order consistency of our scheme. Moreover, the discrete energy method is employed to prove the stability and the quadratic convergence of the technique. Some numerical simulations obtained through our method show the appearance of Turing patterns and wave instabilities, in agreement with some reports found in the literature on superdiffusive hyperbolic activator-inhibitor systems.