In this paper, we propose a fast algorithm for the variable-order (VO) Caputo fractional derivative based on a shifted binary block partition and uniform polynomial approximations of degree $r$. Compared with the general direct method, the proposed algorithm can reduce the memory requirement from $\mathcal{O}\left(n\right)$ to $\mathcal{O}(rlogn)$ storage and the complexity from $\mathcal{O}\left(n^2\right)$ to $\mathcal{O}(rnlogn)$ operations, where $n$ is the number of time steps. As an application, we develop a fast finite difference method for solving a class of VO time-fractional diffusion equations. The computational workload is of $\mathcal{O}(rmnlogn)$ and the active memory requirement is of $\mathcal{O}(rmlogn)$, where $m$ denotes the size of spatial grids. Theoretically, the unconditional stability and error analysis for the proposed fast finite difference method are given. Numerical results of one and two dimensional problems are presented to demonstrate the well performance of the proposed method.

在本文中，我们提出了一种基于移位二进制块分区和度的均匀多项式近似的变阶Caputo分数阶导数的快速算法。 $\mathrm{[R}$。与一般直接方法相比，该算法可以减少内存需求。$\mathcal{Ø}（ñ）$ 至 $\mathcal{Ø}（\mathrm{[R}日志ñ）$ 存储和复杂性 $\mathcal{Ø}（{ñ}^2）$ 至 $\mathcal{Ø}（\mathrm{[R}ñ日志ñ）$ 操作，在哪里 $ñ$是时间步数。作为一种应用，我们开发了一种快速有限差分方法来求解一类VO时间分数阶扩散方程。计算工作量为$\mathcal{Ø}（\mathrm{[R}米ñ日志ñ）$ 并且活动内存需求为 $\mathcal{Ø}（\mathrm{[R}米日志ñ）$，在哪里 $米$表示空间网格的大小。从理论上讲，给出了所提出的快速有限差分方法的无条件稳定性和误差分析。给出了一维和二维问题的数值结果，以证明该方法的良好性能。