Efficient Numerical Methods for Time-Fractional Diffusion Equations with Caputo-Type Erd\’{e}lyi-Kober Operator

This study proposes an L1 discretization scheme to solve time-fractional diffusion equations involving Caputo-type Erd\'{e}lyi-Kober operator, which are fundamental in modeling anomalous diffusion phenomena. To facilitate analysis, we first reformulate the original problem as an equivalent fractional integral equation. Within this transformed setting, the temporal Caputo-type Erd\'{e}lyi-Kober operator is discretized via the L1 formula, with rigorous demonstration of its second-order temporal accuracy and detailed analysis of the coefficient properties in the discrete system. The spatial derivative is approximated using the classical second-order centered difference, culminating in a fully discrete scheme that achieves second-order accuracy in both temporal and spatial dimensions. To address computational challenges arising from the nonlocal nature of fractional operators, we further develop a fast implementation algorithm based on sum-of-exponential approximation for the fractional kernel function, significantly reducing memory requirements and computational costs. The numerical experiment validates the theoretical convergence rates and substantiates the computational superiority of the proposed fast algorithm.