Capturing Nonlocal Effects in Fractional Dynamics: Riesz Derivatives as a New Frontier in Modeling Complex Diffusion and Pattern Formation

This work addresses the numerical solution of fractional time-dependent partial differential equations (FTPDEs) that involve the Riesz fractional derivative in space. Motivated by the effectiveness of space-fractional operators in modeling anomalous diffusion and dispersion phenomena in mathematical physics, we extend this framework to describe classical Brownian motion using a fractional-order formulation based on the Riesz derivative. To this end, we develop a high-order, robust, and efficient numerical scheme for approximating the Riesz derivative, which combines both the left- and right-sided Riemann–Liouville derivatives in a symmetric formulation. We perform a comprehensive analysis of the proposed method, particularly examining its stability and convergence. Furthermore, we apply this method to explore the complex dynamics of pattern formation in two important fractional reaction-diffusion equations, which remain of significant interest in the field. Our experimental results, presented for various fractional parameter values, highlight the method's effectiveness and reveal the intricate behaviors of the system. By utilizing the Riesz fractional derivative, our approach captures the nonlocal and memory effects characteristic of fractional dynamics. This allows for more accurate modeling of phenomena where standard integer-order methods fall short, particularly in capturing the subtleties of anomalous diffusion and pattern formation. The high-order approximation scheme not only ensures numerical accuracy but also enhances computational efficiency, making it a valuable tool for researchers dealing with fractional partial differential equations.