A Three‐Variable Jacobi Polynomial Approach to Two‐Dimensional Systems of Space‐ and Time‐Space Fractional Reaction‐Diffusion Equations

Space‐fractional and time‐space fractional reaction‐diffusion systems naturally arise in modeling complex physical, biological, and engineering processes where classical diffusion equations fail to capture anomalous transport behavior. These fractional formulations are particularly essential for describing nonlocal interactions, memory effects, and spatiotemporally heterogeneous dynamics. To address these challenges, a pseudo‐operational matrix method is improved for efficiently approximating solutions to a broad class of such systems. The proposed approach constructs trivariate basis functions by combining shifted Jacobi polynomials over the spatial and temporal domains and , respectively. Corresponding pseudo‐operational matrices for fractional integration in both time and space are systematically derived. The adjustable parameters of the Jacobi polynomials provide additional flexibility in capturing solution singularities and boundary behaviors. Furthermore, a Jacobi‐weighted Sobolev space is introduced to establish rigorous error estimates for the proposed scheme. Comprehensive numerical experiments and graphical analyses demonstrate that the method achieves superior accuracy and robustness compared to existing techniques, highlighting its effectiveness in solving fractional reaction‐diffusion systems with complex transport phenomena.