A fourth-order accurate numerical scheme for distributed-order Riesz space fractional diffusion equations involving the time-fractional regularized Caputo–Prabhakar derivative

Abstract

This paper introduces a fourth-order accurate finite difference scheme developed for the distributed-order Riesz space fractional diffusion equation involving the time-fractional regularized Caputo-Prabhakar derivative in both one- and two-dimensional settings. The method begins by discretizing the distributed-order integral terms using Simpson’s quadrature rule, transforming the original problem into a system of multi-term Riesz fractional diffusion equations. Subsequently, a fourth-order difference scheme is formulated to accurately approximate these equations. The stability and convergence of the proposed schemes are rigorously established in the L 2 norm for both 1D and 2D cases. Finally, numerical experiments are conducted to demonstrate the effectiveness and accuracy of the approach]