An Efficient and Stable Numerical Scheme for Three-Dimensional Riemann–Liouville Time-Fractional Integro-Differential Equations

Three-dimensional Riemann–Liouville time-fractional integro-differential equations provide useful prototype models for diffusion and transport processes with temporal memory and weakly singular hereditary effects. Their numerical solution is challenging because of the nonlocal fractional derivative, history-dependent fractional integral term, and large-scale discrete systems arising from three-dimensional spatial discretization. In this work, an efficient high-order compact finite difference scheme is developed for solving such problems. The Riemann–Liouville fractional derivative is approximated by the weighted and shifted Grünwald difference formula, the fractional integral term is discretized by the product trapezoidal formula, and the Laplace operator is approximated by compact difference operators. The proposed scheme achieves second-order accuracy in time and fourth-order accuracy in space. Moreover, the solvability, stability, and convergence of the fully discrete three-dimensional scheme are analyzed under suitable regularity assumptions. Numerical experiments, including examples with smooth and non-smooth solutions, verify the theoretical convergence orders and demonstrate the effectiveness of the proposed method for different fractional parameters.