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.

翻译：对于变阶分数阶拉普拉斯算子，由于其奇异积分的本质，在多维空间中进行高效精确的数值计算是一项挑战。本文针对由超奇异积分定义的多维变阶分数阶拉普拉斯算子，提出了一种简单且易于实现的有限差分格式。我们证明了该格式具有二阶收敛性，并将所发展的有限差分格式应用于求解各类含有变阶分数阶拉普拉斯算子的方程。我们提出了一种具有拟线性计算复杂度的快速求解器，用于计算变阶分数阶拉普拉斯算子及相应的偏微分方程。多个数值算例验证了我们算法的精度与效率，并证实了相关理论。