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A simple and fast finite difference method for the integral fractional Laplacian of variable order

Zhaopeng Hao, Siyuan Shi, Zhongqiang Zhang, Rui Du

Abstract

For the fractional Laplacian of variable order, an efficient and accurate numerical evaluation in multi-dimension is a challenge for the nature of a singular integral. We propose a simple and easy-to-implement finite difference scheme for the multi-dimensional variable-order fractional Laplacian defined by a hypersingular integral. We prove that the scheme is of second-order convergence and apply the developed finite difference scheme to solve various equations with the variable-order fractional Laplacian. We present a fast solver with quasi-linear complexity of the scheme for computing variable-order fractional Laplacian and corresponding PDEs. Several numerical examples demonstrate the accuracy and efficiency of our algorithm and verify our theory.

A simple and fast finite difference method for the integral fractional Laplacian of variable order

Abstract

For the fractional Laplacian of variable order, an efficient and accurate numerical evaluation in multi-dimension is a challenge for the nature of a singular integral. We propose a simple and easy-to-implement finite difference scheme for the multi-dimensional variable-order fractional Laplacian defined by a hypersingular integral. We prove that the scheme is of second-order convergence and apply the developed finite difference scheme to solve various equations with the variable-order fractional Laplacian. We present a fast solver with quasi-linear complexity of the scheme for computing variable-order fractional Laplacian and corresponding PDEs. Several numerical examples demonstrate the accuracy and efficiency of our algorithm and verify our theory.
Paper Structure (20 sections, 7 theorems, 67 equations, 7 figures, 8 tables)

This paper contains 20 sections, 7 theorems, 67 equations, 7 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Motivation for the variable-order model
  3. Related definitions of variable-order fractional Laplacian
  4. Literature review and research gap
  5. Contributions and outline of this work
  6. A second-order finite difference approximation
  7. Variable-order fractional Laplacian defined via Fourier transform
  8. A second-order finite difference approximation
  9. Implementation
  10. Fast computation of the discrete variable-order Laplacian
  11. Numerical experiments
  12. Application on fractional PDEs
  13. Finite difference scheme for the elliptic equations
  14. Numerical Experiments
  15. Proofs
  16. ...and 5 more sections

Key Result

Theorem 2.1

Let $u(x)\in C_b^2(\mathbb{R})$, which is the space of bounded and twice continuously differentiable functions. The $\alpha(x)$-th variable-order fractional Laplacian is equivalent to the one defined in the Fourier transform, i.e., when both sides of the equality def-v-frac-lap are well-defined.

Figures (7)

  • Figure 1: The profile of the function $\alpha(x)=0.8+1.2g(x)$ with $g(x) = \max\{|x_1|,|x_2|\}$. (Example \ref{['Ex-known-solution']}).
  • Figure 2: Dynamics of the two kissing bubbles for variable-order fractional Allen-Cahn equation (left: $\alpha(x)=1.5-0.2\tanh(|x|);$ middle: $\alpha(x)=1.8+|x|/8;$ right: $\alpha(x)=0.2\tanh(10(x_1-0.5)) + 0.2\tanh(10(x_2-0.5)) + 1.5$). (Example \ref{['AppliedtoFACequation']}).
  • Figure 3: Evolution of the two “kissing” bubbles for variable fractional Allen-Cahn equation with $\alpha(x)=1.5-0.2\tanh(|x|)$. (Example \ref{['AppliedtoFACequation']}).
  • Figure 4: Profiles of order $\alpha(x) \, {\rm on} \, \Omega = [-1,1]^2$. (left: $\alpha(x)=1.5-0.2\tanh(|x|);$ middle: $\alpha(x)=1.8+|x|/8;$ right: $\alpha(x)=0.2\tanh(10(x_1-0.5)) + 0.2\tanh(10(x_2-0.5)) + 1.5$). (Example \ref{['AppliedtoFACequation']}).
  • Figure 5: The profile of solution at the initial time (Example \ref{['Ex-general-domain']}).
  • ...and 2 more figures

Theorems & Definitions (21)

  • Theorem 2.1: Equivalence of the definitions
  • Remark 2.2
  • Theorem 2.3: Approximation properties
  • Lemma 2.4
  • proof
  • Example 2.6: Approximations of variable-order fractional Laplacian in multi-dimensions
  • Remark 3.1
  • Theorem 3.2: Stability and Convergence
  • Example 3.3: Finite difference scheme for variable-order fractional elliptic equations
  • Example 3.4: Time-dependent problem
  • ...and 11 more