The paper introduces a new meshfree pseudospectral method based on Gaussian radial basis functions (RBFs) collocation to solve fractional Poisson equations. Hypergeometric functions are used to represent the fractional Laplacian of Gaussian RBFs, enabling an efficient computation of stiffness matrix entries. Unlike existing RBF-based methods, our approach ensures a Toeplitz structure in the stiffness matrix with equally spaced RBF centers, enabling efficient matrix-vector multiplications using fast Fourier transforms. We conduct a comprehensive study on the shape parameter selection, addressing challenges related to ill-conditioning and numerical stability. The main contribution of our work includes rigorous stability analysis and error estimates of the Gaussian RBF collocation method, representing a first attempt at the rigorous analysis of RBF-based methods for fractional PDEs to the best of our knowledge. We conduct numerical experiments to validate our analysis and provide practical insights for implementation.

翻译：本文提出了一种基于高斯径向基函数（RBF）配点的新型无网格伪谱方法，用于求解分数阶泊松方程。利用超几何函数表示高斯径向基函数的分数阶拉普拉斯算子，从而实现了刚度矩阵元素的高效计算。与现有基于RBF的方法不同，本方法在均匀分布的RBF中心条件下确保刚度矩阵的Toeplitz结构，使得可通过快速傅里叶变换实现高效的矩阵-向量乘法。我们对形状参数选择开展了系统研究，解决了病态条件和数值稳定性相关的挑战。本研究的主要贡献包括对高斯RBF配点法进行了严格的稳定性分析与误差估计，据我们所知，这是首次针对分数阶偏微分方程基于RBF的方法开展严格分析的尝试。我们通过数值实验验证了理论分析结果，并提供了实现中的实用见解。