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

Zhaopeng Hao, Zhiqiang Cai, Zhongqiang Zhang

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.

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

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.

Paper Structure

This paper contains 17 sections, 5 theorems, 72 equations, 5 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Fractional Laplacian of some generalized multi-quadratic functions
  3. Derivations
  4. Collocation methods for fractional Poisson equations
  5. Implementation
  6. Stability and error estimates
  7. Stability in Sobolev spaces
  8. Well-posedness
  9. Convergence of an equivalent method
  10. A conforming Galerkin method for $\alpha\in (0,1)$
  11. A non-conforming Galerkin method for $\alpha \in (1,2)$
  12. An equivalent formulation and implementation
  13. Numerical experiments
  14. One-dimensional examples
  15. Two-dimensional steady problems
  16. ...and 2 more sections

Key Result

Lemma 2.1

For function $v$, assume that $\mathcal{V}(x):=(-\Delta)^{\alpha/2} v(x)$ is well defined for $x\in \mathbb{R}^d$. Then it satisfies the following properties: for any fixed point $x_0\in \mathbb{R}^d$, and for any scalar $c\in \mathbb{R}$.

Figures (5)

  • Figure 1: Numerical solution of the fractional Poisson problem cross reference on a unit disk domain for $f=1$ by GMQ with the shape parameter $\varepsilon=1$.
  • Figure 2: Pointwise errors of GMQ with the shape parameter $\varepsilon=1$ solving the fractional Poisson problem cross reference on a unit disk.
  • Figure 3: Numerical solution of the fractional Poisson problem cross reference on a square domain for $f=1$ by GMQ with the shape parameter $\varepsilon=0.05$ and the uniform grid points step-size $h=1/32$.
  • Figure 4: Comparison of the dynamics between the standard diffusion ($\chi=0$, Left), the mixed diffusion ($\chi=0.5$, Middle) and fractional diffusion ($\chi=1$, Right) for the model problem \ref{['mixed-diffusion']} with $\alpha=1$.
  • Figure 5: The time evolution of the field $\theta$ for the model problem \ref{['eq-quasi-geotrophic']} with $\kappa=0.001$.

Theorems & Definitions (15)

  • Lemma 2.1: Translation and Scaling Properties
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Remark 3.1
  • Lemma 3.2
  • proof
  • ...and 5 more