Numerical analysis and pattern formation process for space-fractional superdiffusive systems

In this paper, we consider the numerical solution of fractional-in-space reaction-diffusion equation, which is obtained from the classical reaction-diffusion equation by replacing the second-order spatial derivative with a fractional derivative of order $ α∈(1, 2] $. We adopt a class of second-order approximations, based on the weighted and shifted Grünwald difference operators in Riemann-Liouville sense to numerically simulate two multicomponent systems with fractional-order in higher dimensions. The efficiency and accuracy of the numerical schemes are justified by reporting the norm infinity and norm relative errors as well as their convergence. The complexity of the dynamics in the equation is theoretically discussed by conducting its local and global stability analysis and Numerical experiments are performed to back-up the theoretical claims.