We apply two families of novel fractional \(\theta \)-methods, the FBT-\(\theta \) and FBN-\(\theta \) methods developed by the authors in previous work, to the fractional Cable model, in which the time direction is approximated by the fractional \(\theta \)-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 \(O(\tau ^2+h^{r+1})\) for smooth solutions, where \(\tau \) 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.

我们将作者在先前的工作中开发的两个新颖的分数\（\ theta \）方法家族，FBT- \（\ theta \）和FBN- \（\ theta \）方法应用于分数Cable模型中。时间方向由分数\（\ theta \）方法近似，而空间方向由有限元方法近似。推导了这两种方法的系数的一些正性，这对于稳定性估计的证明至关重要。我们分析该方案的稳定性，并使用\（O（\ tau ^ 2 + h ^ {r + 1}）\）得出最优解的收敛解，其中\（\ tau \）是时间网格大小，而H是空间网格大小。进行了一些具有光滑和非光滑解的数值实验，以证实我们的理论分析。为了克服初始值的奇异性，增加了起始部分以及时恢复二阶收敛速度。