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 $(-\Delta)^{\alpha({\bf x})/2}$ with $0 < \alpha({\bf x}) \le 2$, which will also be referred as the variable-order fractional Laplacian if $\alpha({\bf x})$ 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 $(-\Delta)^{\alpha({\bf x})/2}$. 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.

翻译：变阶分数阶拉普拉斯算子在异质性系统研究中具有重要作用。本文首次提出变阶拉普拉斯算子$(-\Delta)^{\alpha({\bf x})/2}$（其中$0 < \alpha({\bf x}) \le 2$）的数值方法，当$\alpha({\bf x})$严格小于2时，该算子亦称为变阶分数阶拉普拉斯算子。我们给出了一类可解析表达其变阶拉普拉斯算子的超几何函数。基于这些解析结果，我们设计了基于全局支撑径向基函数（RBFs，包括高斯型、广义逆多二次型及贝塞尔型RBF）的无网格方法，用于逼近变阶拉普拉斯算子$(-\Delta)^{\alpha({\bf x})/2}$。所提无网格方法融合了变阶分数阶拉普拉斯算子的伪微分形式与超奇异积分形式的优势，从而避免了超奇异积分的数值近似。此外，该方法具有域几何适应性简单灵活的特点，且对于任意维度$d \ge 1$，其计算机实现方式保持一致。与有限差分方法相比，本方法能以更少的节点实现期望精度，这一特性使得本方法在节点数量成为关键计算成本的变阶分数阶拉普拉斯问题中极具吸引力。我们随后将该方法应用于研究不同领域中变阶分数阶偏微分方程的解行为，包括经典介质与分数阶介质之间的波传播转换，以及扩散方程和Allen-Cahn方程中反常扩散与正常扩散的共存现象。这些结果将为变阶分数阶导数的深入理解和应用提供新的见解。