In this paper, we study the variable-order (VO) time-fractional diffusion equations. For a VO function $\alpha(t)\in(0,1)$, we develop an exponential-sum-approximation (ESA) technique to approach the VO Caputo fractional derivative. The ESA technique keeps both the quadrature exponents and the number of exponentials in the summation unchanged at the different time levels. Approximating parameters are properly selected to achieve efficient accuracy. Compared with the general direct method, the proposed method reduces the storage requirement from $\mathcal{O}(n)$ to $\mathcal{O}(\log^2 n)$ and the computational cost from $\mathcal{O}(n^2)$ to $\mathcal{O}(n\log^2 n)$, respectively, with $n$ being the number of the time levels. When this fast algorithm is exploited to construct a fast ESA scheme for the VO time-fractional diffusion equations, the computational complexity of the proposed scheme is only of $\mathcal{O}(mn\log^2 n)$ with $\mathcal{O}(m\log^2n)$ storage requirement, where $m$ denotes the number of spatial grids. Theoretically, the unconditional stability and error analysis of the fast ESA scheme are given. The effectiveness of the proposed algorithm is verified by numerical examples.

中文翻译：

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时变分数阶扩散方程的指数和逼近技术

在本文中，我们研究了可变阶（VO）时间分数扩散方程。对于VO函数$ \ alpha（t）\ in（0,1）$，我们开发了一种指数和近似（ESA）技术来逼近VO Caputo分数导数。ESA技术使正交指数和求和中的指数数量在不同的时间级别保持不变。逼近参数正确选择，以实现高效的准确性。与一般直接方法相比，该方法将存储需求从$ \ mathcal {O}（n）$减少到$ \ mathcal {O}（\ log ^ 2 n）$，并将计算成本从$ \ mathcal {O }（n ^ 2）$到$ \ mathcal {O}（n \ log ^ 2 n）$，其中$ n $是时间级别数。当利用此快速算法为VO时间分数扩散方程构建快速ESA方案时，所提出方案的计算复杂度仅为$ \ mathcal {O}（mn \ log ^ 2 n）$和$ \ mathcal {O}（m \ log ^ 2n）$的存储要求，其中$ m $表示数字空间网格。从理论上讲，给出了快速ESA方案的无条件稳定性和误差分析。数值算例验证了所提算法的有效性。