Variable-order fractional Laplacian and its accurate and efficient computations with meshfree methods

TL;DR

This work tackles the challenge of numerically solving variable-order fractional Laplacians $(-\Delta)^{\alpha({\bf x})/2}$ with spatially varying $0<\alpha({\bf x})\le 2$ by deriving analytic VO-Laplacian expressions for a class of hypergeometric functions and introducing meshfree RBF methods. By combining the pseudo-differential and hypersingular integral viewpoints, the authors develop three RBF-based schemes (Gaussian, generalized inverse multiquadric, and Bessel-type) that avoid hypersingular quadrature and are usable in any dimension, achieving spectral accuracy with relatively few points. The methods are validated through Poisson, wave, diffusion, and Allen–Cahn problems in heterogeneous media, revealing how variable order governs transitions between normal and anomalous diffusion and wave propagation across interfaces. Overall, the paper provides foundational analytic tools and a scalable, accurate computational framework for VO-Laplacian problems, with potential impact on modeling heterogeneous materials and processes where diffusion and wave phenomena vary spatially.

Abstract

The variable-order fractional Laplacian plays an important role in the study of heterogeneous systems. In this paper, we propose the first numerical methods for the variable-order Laplacian $(-Δ)^{α({\bf x})/2}$ with $0 < α({\bf x}) \le 2$, which will also be referred as the variable-order fractional Laplacian if $α({\bf x})$ is strictly less than 2. We present a class of hypergeometric functions whose variable-order Laplacian can be analytically expressed. Building on these analytical results, we design the meshfree methods based on globally supported radial basis functions (RBFs), including Gaussian, generalized inverse multiquadric, and Bessel-type RBFs, to approximate the variable-order Laplacian $(-Δ)^{α({\bf x})/2}$. Our meshfree methods integrate the advantages of both pseudo-differential and hypersingular integral forms of the variable-order fractional Laplacian, and thus avoid numerically approximating the hypersingular integral. Moreover, our methods are simple and flexible of domain geometry, and their computer implementation remains the same for any dimension $d \ge 1$. Compared to finite difference methods, our methods can achieve a desired accuracy with much fewer points. This fact makes our method much attractive for problems involving variable-order fractional Laplacian where the number of points required is a critical cost. We then apply our method to study solution behaviors of variable-order fractional PDEs arising in different fields, including transition of waves between classical and fractional media, and coexistence of anomalous and normal diffusion in both diffusion equation and the Allen-Cahn equation. These results would provide insights for further understanding and applications of variable-order fractional derivatives.

Figures (8)

• Figure 1: Illustration of variable-order Laplacian of the one-dimensional Gaussian type function (left column) and generalized inverse multiquadric type function with $\beta = 1$ (right column) for $x \in (-2, 2)$, where $\alpha_1(x) = 1+x/2$ and $\alpha_2(x) = 1+{\rm tanh}(2x+1)$.
• Figure 2: Illustration of variable-order Laplacian of the one-dimensional compactly supported functions in Lemma \ref{['lemma4']}, where we choose $\alpha_1(x) = 1+x/2$ and $\alpha_2(x) = 1+{\rm tanh}(2x+1)$. The symbol '$\circ$' indicates that points $x = \pm 1$ are not included in the above plots.
• Figure 3: Numerical solution of the one-dimensional Poisson problem in (\ref{['diffusion']}) with different $\alpha(x)$, where $f(x) = 2\sin^2(\pi x)$ and $g(x) \equiv 0$.
• Figure 4: Time evolution of wave solutions for different $\alpha(x)$, where (a) $\alpha(x) \equiv 2$; (b) $\alpha(x) \equiv 1.2$; (c) $\alpha(x) = 1.6-0.4{\rm tanh}(5x)$; (d) $\alpha(x) = 1.6+0.4{\rm tanh}(5x)$. For better illustration, we only display the solution on interval $[-8, 8]$, much smaller than the actual computational domain.
• Figure 5: Time evolution of solution $u({\bf x} , t)$ of (\ref{['diff']})--(\ref{['diff-ic']}), where $\alpha({\bf x} ) \equiv 2$ (left column), $\alpha({\bf x} ) \equiv 1.4$ (middle column), and $\alpha({\bf x} ) = 2{\chi_{\{x \le -0.5\}}} + 1.4{\chi_{\{x \ge 0.5\}}} + (1.7-0.6x){\chi_{\{|x| < 0.5\}}}$ (right column).

Theorems & Definitions (5)

• Lemma 2.1
• Lemma 2.2: Laplacian of generalized hypergeometric functions
• Lemma 2.3: Laplacian of globally supported functions
• Lemma 2.4: Laplacian of compactly supported functions on $B_1({\bf 0})$
• Remark 2.1