Computational study for the Caputo sub-diffusive and Riesz super-diffusive processes with a fractional order reaction–diffusion equation

Abstract

A Numerical solution of the Caputo-time and Riesz-space fractional reaction–diffusion model is considered in this paper. Based on finite difference schemes, we formulate both second-order and fourth-order numerical methods for the approximation of the Riesz space fractional reaction–diffusion-like equation of Fisher type. In the experiment, it was observed that the fourth-order scheme has better accuracy than the second-order method when applied to solve the fractional diffusion equation. It should be mentioned that the lower order scheme computes rapidly and save more computational time as displayed in the table of results. Finally, some simulation results are presented to justify the effectiveness and applicability of the numerical methods. The one- two- and three-dimensional results obtained for some instances of fractional order (α,β) depict some amazing complex and spatiotemporal patterns which are applicable in applied sciences and engineering.

Keywords

• Caputo-time derivative
• Riesz-space derivative
• Fourier spectral method
• Fractional reaction–diffusion equation
• Numerical simulations