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.
翻译:在本文中,我们调查小片段顺序制度中的快速算法,以在美元小时接近Caputo衍生物 $C_0D_t ⁇ alpha u(t)美元。我们侧重于两个快速算法,即FIR和FIDR,两者都依靠耗资总和近似来降低历史部分的评估成本。FIR是最初在[16]中提议的数值方案,FIDR是[26]中提议的替代方案,我们表明后者在美元小时优于后者。根据定量估计,我们证明如果存在一定的错误阈值,那么评估Caputo衍生物历史部分的计算成本可以随着美元小一些而降低。因此,在小部分的alpha$制度下,只需要最低限度的快速评估成本来降低历史部分的评估成本,这符合工程实践中通行的规程。我们还对FIDR的稳定性和错误分析进行了改进,以解决线性分数扩散方方程的公式,在分数顺序上明显依赖$Rpha$的错误。最后,我们进行系统化的FIRA和美元贸易效率研究时,我们在这方面进行系统的数字研究。