An Efficient High-Accuracy RBF-HFD Scheme for Caputo Time-Fractional Sub-Diffusion Problems with Integral Boundaries

This study presents an efficient high-order radius function Hermite finite difference (RBF-HFD) scheme for the numerical solution of Caputo time-fractional sub-diffusion equations with integral boundary conditions. The spatial derivatives are approximated using a fourth-order RBF-HFD scheme, while the Caputo fractional derivative in time is discretized via the L2−1σ formula. To ensure global fourth-order spatial accuracy, the integral boundary conditions are discretized with the composite Simpson rule. As a result, we obtain an unconditionally stable numerical scheme that achieves fourth-order convergence in space and second-order convergence in time. The solvability, stability, and convergence of the scheme are rigorously established using the discrete energy method. The proposed method is validated through three numerical examples and is compared with existing approaches. The numerical results demonstrate that the proposed scheme achieves higher accuracy than the methods available in the literature.