Modeling Anomalous Diffusion With Riesz Fractional Derivatives: Applications to Pattern Formation

This work introduces a high-order numerical framework for solving time-dependent partial differential equations involving space-fractional operators, with a focus on the Riesz fractional derivative. Motivated by the growing need to accurately model anomalous diffusion and nonlocal transport phenomena, we formulate an efficient scheme that approximates the Riesz derivative—which unifies left- and right-sided Riemann–Liouville derivatives—within a spectral or pseudo-spectral context. The proposed method is carefully designed to ensure high accuracy, numerical stability, and computational efficiency. We conduct a comprehensive analysis of its convergence properties and demonstrate its versatility through detailed numerical simulations applied to several benchmark fractional reaction-diffusion models, including the Gray–Scott, BVAM, and FitzHugh–Nagumo systems. These examples exhibit a range of complex spatiotemporal behaviors such as Turing patterns, labyrinths, and spiral structures, particularly under variations in the fractional order. By leveraging the inherent nonlocality of the Riesz operator, our approach captures long-range interactions and memory effects that are otherwise neglected in classical diffusion models. The results highlight the capacity of fractional modeling to reproduce richer and more diverse pattern formation dynamics. Overall, the method offers a powerful and scalable tool for simulating a wide variety of fractional PDEs in mathematical biology, physics, and other applied sciences.