A Fast Adaptive Method with a Sum-of-Exponentials Approximation for Fractional Derivative Diffusion Equation

The high numerical computing cost of time-fractional diffusion equation (tFDE) models over long time periods is a major obstacle to their real-world applications. Therefore, this study presents a rapid adaptive finite difference method, which uses the sum-of-exponentials (SOE) technique to quickly evaluate the kernel function and adopts the trial-and-error (T&E) method to select optimal time steps. For a uniform number of time steps N_T with T >> 1, the cumulative computational cost of the approximate fractional derivative can be reduced from O($N_T^2$) for the T&E method to O(N_T log N_T). To evaluate the accuracy and computational efficiency of the proposed method, a comprehensive comparison is conducted based on three numerical examples. Numerical results show that the SOE-T&E technique provides more accurate results with fewer grid points, compared with uniform mesh method. Moreover, the SOE-T&E technique reduces the computation time by 88.98% compared to the T&E method for the same error level in our numerical examples.