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The paper introduces a new meshfree pseudospectral method based on Gaussianradial basis functions (RBFs) collocation to solve fractional Poissonequations. Hypergeometric functions are used to represent the fractionalLaplacian of Gaussian RBFs, enabling an efficient computation of stiffnessmatrix entries. Unlike existing RBF-based methods, our approach ensures aToeplitz 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, addressingchallenges related to ill-conditioning and numerical stability. The maincontribution of our work includes rigorous stability analysis and errorestimates of the Gaussian RBF collocation method, representing a first attemptat the rigorous analysis of RBF-based methods for fractional PDEs to the bestof our knowledge. We conduct numerical experiments to validate our analysis andprovide practical insights for implementation.

Gaussian radial basis functions collocation for fractional PDEs: methodology and error analysis