An Analysis of the Modified L1 Scheme for Time-Fractional Partial Differential Equations with Nonsmooth Data
An Analysis of the Modified L1 Scheme for Time-Fractional Partial Differential Equations with Nonsmooth Data
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DOI：
10.1137/16m1094257
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发表时间：
2018-01
期刊：
影响因子：
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通讯作者：
Yubin Yan;Monzorul Khan;N. Ford
中科院分区：
文献类型：
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作者：
Yubin Yan;Monzorul Khan;N. Ford
我们引入了一种改进的 L1 方案来求解时间分数阶偏微分方程，并在均匀和非均匀情况下获得平滑和非平滑初始数据的误差估计。 Jin、Lazarov 和 Zhou [IMA J. Numer。 Anal., 36 (2016), pp. 197--221) 针对齐次问题的平滑和非平滑初始数据建立了 L1 方案的 $O(k)$ 收敛率，其中 $k$ 表示时间步长。我们表明，对于均匀和非均匀情况下的平滑和非平滑初始数据，修改后的 L1 方案的收敛率为 $O(k^{2-\alpha})，0< \alpha <1$。数值算例表明数值结果与理论结果一致。
We introduce a modified L1 scheme for solving time fractional partial differential equations and obtain error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Jin, Lazarov, and Zhou [IMA J. Numer. Anal., 36 (2016), pp. 197--221) established an $O(k)$ convergence rate for the L1 scheme for smooth and nonsmooth initial data for the homogeneous problem, where $k$ denotes the time step size. We show that the modified L1 scheme has a convergence rate of $O(k^{2-\alpha}), 0< \alpha <1$ for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the numerical results are consistent with the theoretical results.