In this paper, a novel numerical scheme is proposed to numerically solve the fractional activator-inhibitor system, which is a coupled nonlinear model. In the temporal direction, we employ two families of novel fractional θ-methods, the FBT-θ and FBN-θ methods, in spatial direction, the Legendre spectral method is used. Based on some positivity properties of the coefficients of both methods, the stability and convergence of the numerical scheme are proved. We derive an optimal convergence rate in space and second order accuracy in time. Finally, some numerical results are given to confirm the theoretical analysis.

中文翻译：

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基于两类新颖的二阶数值公式的分数活化剂-抑制剂系统的勒让德光谱方法

本文提出了一种新颖的数值方案来数值求解分数活化剂-抑制剂系统，它是一个耦合的非线性模型。在时间方向上，我们采用了两类新颖的分数θ方法：FBT- θ和FBN- θ方法，在空间方向上，使用了勒让德谱方法。基于两种方法的系数的正性，证明了数值格式的稳定性和收敛性。我们得出空间的最佳收敛速度和时间的二阶精度。最后，给出一些数值结果以证实理论分析。