Fast Algorithms and Error Analysis of Caputo Derivatives with Small Factional Orders

In this paper, we investigate fast algorithms in the small fraction order regime to approximate the Caputo derivative $^C_0D_t^\alpha u(t)$ when $\alpha$ is small. We focus on two fast algorithms, i.e. FIR and FIDR, both relying on the sum-of-exponential approximation to reduce the cost of evaluating the history part. FIR is the numerical scheme originally proposed in [16], and FIDR is an alternative scheme proposed in [26], and we show that the latter is superior when $\alpha$ is small. With quantitative estimates, we prove that given a certain error threshold, the computational cost of evaluating the history part of the Caputo derivative can be decreased as $\alpha$ gets small. Hence, only minimal cost for the fast evaluation is required in the small $\alpha$ regime, which matches prevailing protocols in engineering practice. We also present improved stability and error analysis of FIDR for solving linear fractional diffusion equations, which achieves clear dependence of the error bound on the fraction order $\alpha$. Finally, we carry out systematic numerical studies for the performances of both FIR and FIDR schemes, where we explore the trade-off between accuracy and efficiency when $\alpha$ is small.