Spatiotemporal chaos in diffusive systems with the Riesz fractional order operator

Abstract

A reliable and efficient numerical technique for the approximation of Riesz fractional partial differential equations with chaotic and spatiotemporal chaos properties is developed in this work. In such a system, the standard anomalous diffusion terms are modeled using the Riesz fractional derivative which can be extended from one to high dimensional cases. The methodology adopts finite difference schemes, as well as the novel Fourier spectral methods for the approximation of the Riesz fractional derivatives in space. The resulting system of ordinary differential equations is advanced in time with the exponential time-differencing Runge–Kutta method. The performance and applicability of these methods were tested with some practical and real-life problems which are of current and recurring interests arising from computational physics, engineering and other areas of applied sciences. Simulation results are given for different instances of fractional parameter α in the intervals (0,1) and (1,2) which correspond to subdiffusion and superdiffusion scenarios, respectively.