Analysis of the solutions of coupled nonlinear fractional reaction–diffusion equations

• Published in: Chaos, solitons and fractals solitons and fractals, 2009-08, Vol.41 (3), p.1095-1104
• Main Authors: Gafiychuk, V., Datsko, B., Meleshko, V., Blackmore, D.
• Format: Article
• Language: English
• ISSN: 0960-0779; 1873-2887
• DOI: 10.1016/j.chaos.2008.04.039

This paper is concerned with analysis of coupled fractional reaction–diffusion equations. As an example, the reaction–diffusion model with cubic nonlinearity and Brusselator model are considered. It is shown that by combining the fractional derivatives index with the ratio of characteristic times, it is possible to find the marginal value of the index where the oscillatory instability arises. Computer simulation and analytical methods are used to analyze possible solutions for a linearized system. A computer simulation of the corresponding nonlinear fractional ordinary differential equations is presented. It is shown that an increase of the fractional derivative index leads to periodic solutions which become stochastic as the index approaches the value of 2. It is established by computer simulation that there exists a set of stable spatio-temporal structures of the one-dimensional system under the Neumann and periodic boundary condition. The characteristic features of these solutions consist in the transformation of the steady state dissipative structures to homogeneous oscillations or spatio-temporal structures at certain values of the fractional index.