A Comparative Stability and Advancements in Numerical Solutions of Fractional Order Partial Differential Equations

Abstract

Time-space fractional partial differential equations (TSFPDEs) provide a robust framework for modeling anomalous diffusion processes characterized by memory effects and long-range spatial interactions. Due to the intrinsic nonlocal properties of these equations, analytical solutions are often unattainable, making advanced numerical methods essential. This study presents a comprehensive comparison of finite difference schemes for one-dimensional TSFPDEs involving Caputo time-fractional and Riesz space-fractional derivatives. The Caputo derivative is discretized using the L1 finite difference method, while the Riesz derivative is approximated via a shifted Grunwald-Letnikov scheme. Explicit, implicit, and semi-implicit formulations are developed and systematically analyzed. Stability is assessed using discrete energy techniques, and rigorous error analysis establishes convergence. Theoretical analysis confirms first-order accuracy in time and second-order accuracy in space, which is further validated through numerical experiments on benchmark problems. Results indicate that implicit and semi-implicit methods exhibit superior stability and accuracy compared to explicit schemes, especially in long-term simulations. These findings offer valuable guidance for selecting effective finite difference approaches to solve time-space fractional diffusion problems.