Fractional-order dependent Radial basis functions meshless methods for the integral fractional Laplacian

TL;DR

This paper introduces a novel meshless numerical method using fractional order-dependent radial basis functions for efficiently computing the integral fractional Laplacian and solving fractional diffusion equations with high accuracy.

Contribution

It develops a pseudo-spectral formula based on fractional order-dependent RBFs, enabling efficient, nearly integration-free solutions for fractional Laplacian problems.

Findings

Demonstrates high accuracy and efficiency in numerical experiments.

Shows improved performance over existing Gaussian RBF-based methods.

Provides convergence analysis via Galerkin formulations.

Abstract

We study the numerical evaluation of the integral fractional Laplacian and its application in solving fractional diffusion equations. We derive a pseudo-spectral formula for the integral fractional Laplacian operator based on fractional order-dependent, generalized multi-quadratic radial basis functions (RBFs) to address efficient computation of the hyper-singular integral. We apply the proposed formula to solving fractional diffusion equations and design a simple, easy-to-implement and nearly integration-free meshless method. We discuss the convergence of the novel meshless method through equivalent Galerkin formulations. We carry out numerical experiments to demonstrate the accuracy and efficiency of the proposed approach compared to the existing method using Gaussian RBFs.