Mathematical modeling of pattern formation in sub- and supperdiffusive reaction-diffusion systems

Abstract

This paper is concerned with analysis of coupled fractional reaction-diffusion equations. It provides analytical comparison for the fractional and regular reaction-diffusion systems. As an example, the reaction-diffusion model with cubic nonlinearity and Brusselator model are considered. The detailed linear stability analysis of the system with Cubic nonlinearity is provided. 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 the increase of the fractional derivative index leads to the periodic solutions which become stochastic at the index approaching 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 space temporary structures at a certain value of fractional index.