In this paper, we present a new reduced order model based on radial basis functions (RBFs) and proper orthogonal decomposition (POD) methods for fractional advection-diffusion equations with a Caputo fractional derivative in time. In the proposed scheme, the number of basis functions in the usual RBFs method reduces by the POD technique. Therefore, the computational cost of the RBF-POD method decreases in comparison with usual RBFs method, while the accuracy completely maintains. In the sequel, we provide a complete error analysis in the \(L_{2}\) norm between the exact solution and the RBFs solution, as well as between the exact solution and the proposed RBF-POD model by using the properties of the native space and projection operators. Also, the obtained error estimation is used to choose the number of POD bases for constructing the RBF-POD model with the required accuracy. Numerical examples are given to confirm the accuracy and efficiency of the proposed scheme.

中文翻译：

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基于POD方法的时分形偏微分方程简化RBF模型的误差分析

在本文中，我们针对基于Caputo分数阶导数的分数对流扩散方程，提出了一种基于径向基函数（RBF）和适当正交分解（POD）方法的新降阶模型。在提出的方案中，通过POD技术减少了常规RBFs方法中基函数的数量。因此，与通常的RBFs方法相比，RBF-POD方法的计算成本降低了，而精度却完全得以保持。在续篇中，我们在\（L_ {2} \）中提供了完整的错误分析通过使用本机空间和投影算符的属性，在精确解和RBFs解之间以及在精确解和建议的RBF-POD模型之间建立范数。而且，所获得的误差估计用于选择具有所需精度的用于构造RBF-POD模型的POD碱基的数目。数值算例证实了所提方案的准确性和有效性。