基于两族二阶数值公式的分数阶Cable模型光滑解的有限元法
Finite element methods based on two families of second-order numerical formulas for the fractional Cable model with smooth solutions
摘要：
本文将作者在前期工作中提出的两类新的分数阶$θ$方法FBT- $θ$和FBN- $θ$应用于分数阶Cable模型，其中时间方向用分数阶$θ$方法近似，空间方向用有限元法近似.这两种方法的系数的一些积极的性质，这是至关重要的证明的稳定性估计。我们分析了格式的稳定性，并得到了用$O(τ^2+h^{r+1})$求解光滑解的最优收敛结果，其中$τ$是时间网格尺寸，$h$是空间网格步长.对光滑解和非光滑解进行了数值实验，验证了理论分析的正确性。为了克服初始值的奇异性，增加了起始部分，使算法在时间上恢复到二阶收敛速度。
英文摘要：
We apply two families of novel fractional $θ$-methods, the FBT-$θ$ and FBN-$θ$ methods developed by the authors in previous work, to the fractional Cable model, in which the time direction is approximated by the fractional $θ$-methods, and the space direction is approximated by the finite element method. Some positivity properties of the coefficients for both of these methods are derived, which are crucial for the proof of the stability estimates. We analyse the stability of the scheme and derive an optimal convergence result with $O(τ^2+h^{r+1})$ for smooth solutions, where $τ$ is the time mesh size and $h$ is the spatial mesh size. Some numerical experiments with smooth and nonsmooth solutions are conducted to confirm our theoretical analysis. To overcome the singularity at initial value, the starting part is added to restore the second-order convergence rate in time.
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