Numerical simulation of distributed-order Caputo–Prabhakar time and Riesz space fractional models using cubic spline and meshless approximations

In this work, we propose a novel distributed-order fractional partial differential equation model with Caputo–Prabhakar time derivatives and variable-order Riesz spatial derivatives, capable of capturing complex memory and nonlocal effects. A fully discrete numerical method is developed, employing cubic spline approximation in time and meshless Moving Least Squares for spatial discretization. Convergence and stability analyses confirm second-order accuracy in space and third-order accuracy in time. Two illustrative examples are presented: one with an exact solution to validate accuracy, and one without, demonstrating the method’s robustness. Numerical results show that the proposed approach is efficient, accurate, and suitable for solving complex fractional PDEs in scientific and engineering applications.