  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.
