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

TL;DR

This paper presents a novel meshfree Gaussian RBF collocation method for fractional PDEs, featuring efficient computation, stability analysis, and error estimates, validated through numerical experiments.

Contribution

It introduces a new RBF collocation approach with rigorous stability and error analysis for fractional PDEs, addressing computational efficiency and numerical stability.

Findings

Efficient Toeplitz-structured stiffness matrix enabling fast computations

Rigorous stability analysis and error estimates for the method

Numerical experiments validating theoretical results

Abstract

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.