This paper deals with developing a fast and robust numerical formulation to simulate a system of fractional PDEs. At the first stage, the time variable is approximated by a finite difference method with first-order accuracy. At the second stage, the spectral Galerkin method based upon the fractional Jacobi polynomials is employed to discretize the spatial variables. We apply a reduced-order method based upon the proper orthogonal decomposition technique to decrease the utilized computational time. The unconditional stability property and the order of convergence of the new technique are analyzed in detail. The proposed numerical technique is well known as the reduced-order spectral Galerkin scheme. Furthermore, by employing the Newton–Raphson method and semi-implicit schemes, the proposed method can be used for solving linear and nonlinear ODEs and PDEs. Finally, some examples are provided to confirm the theoretical results.

中文翻译：

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求解带误差估计的空间分数Gray-Scott模型的基于谱Galerkin方法的POD降阶模型

本文涉及开发一种快速且稳健的数值公式来模拟分数阶偏微分方程系统。在第一阶段，时间变量通过一阶精度的有限差分方法逼近。在第二阶段，采用基于分数雅可比多项式的谱伽辽金方法对空间变量进行离散化。我们应用基于适当正交分解技术的降阶方法来减少使用的计算时间。详细分析了新技术的无条件稳定性和收敛性。所提出的数值技术被称为降阶谱伽辽金方案。此外，通过采用 Newton-Raphson 方法和半隐式方案，所提出的方法可用于求解线性和非线性 ODE 和 PDE。最后，给出了一些例子来证实理论结果。