Numerical Modeling of Fractional-order Diffusion for Complex Systems in Applied Mathematics

This study presents an enhanced and comprehensive approach to modeling fractional-order diffusion processes in complex systems using a numerical method based on the Grünwald-Letnikov (GL) approximation. The proposed model aims to bridge the theoretical foundations of fractional calculus with efficient simulation techniques applicable to heterogeneous and memory-dependent phenomena. Compared to classical integer-order models, fractional models offer greater flexibility in capturing anomalous diffusion, long-range interactions, and nonlocal behavior observed in real-world systems. The research investigates the influence of the fractional order parameter on diffusion dynamics across various applied scenarios, including heat conduction in porous media, pollutant transport in groundwater, epidemic spread in network structures, drug release through biological tissues, and petroleum flow in stratified reservoirs. Numerical simulations demonstrate that tuning the parameter allows for accurate modeling of both sub-diffusive and super-diffusive behaviors, improving the fidelity of results compared to classical models. The methodology employs an implicit Euler time integration scheme and adaptive mesh refinement to enhance stability, accuracy, and computational efficiency. The results confirm the robustness of the GL-based scheme in preserving mass conservation, achieving second-order spatial accuracy, and maintaining stability over a wide range of values. This approach provides practical tools for engineers, physicists, and biomedical researchers seeking precise numerical modeling of complex transport phenomena.