Modeling Anomalous Diffusion with the Fractional Brusselator

Abstract

This paper studies the fractional reaction-diffusion Brusselator model, which incorporates fractional-time derivatives to describe memory effects and anomalous diffusion in pattern formation. A fully discrete numerical scheme is developed using an L1 approximation for the fractional derivative and a finite difference method for spatial discretization. Theoretical analysis proves the uniqueness, asymptotic stability, and convergence of the scheme. Numerical simulations demonstrate the emergence of stationary Turing patterns under appropriate conditions, validating the model’s ability to capture complex spatiotemporal dynamics. The work provides a reliable computational framework for exploring fractional reaction-diffusion systems in two dimensions.