Fourier spectral method for higher order space fractional reaction–diffusion equations

• Fractional Fourier Transform method is proposed to solve fractional reaction – diffusion problems.
• We investigate the linear stability analysis of the model, in our analysis results.
• The dynamic richness of spatial pattern formations is explored in the sub-diffusive, diffusive and super-diffusive cases.
• Results show that the proposed method is reliable, efficient and achieves higher-order of accuracy.

Evolution equations containing fractional derivatives can provide suitable mathematical models for describing important physical phenomena. In this paper, we propose a fast and accurate method for numerical solutions of space fractional reaction–diffusion equations. The proposed method is based on an exponential integrator scheme in time and the Fourier spectral method in space. The main advantages of this method are that it yields a fully diagonal representation of the fractional operator, with increased accuracy and efficiency, and a completely straightforward extension to high spatial dimensions. Although, in general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives, we introduce them to describe fractional hyper-diffusions in reaction diffusion. The scheme justified by a number of computational experiments, this includes two and three dimensional partial differential equations. Numerical experiments are provided to validate the effectiveness of the proposed approach.