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.

翻译：本文研究积分分数阶拉普拉斯算子的数值计算及其在求解分数阶扩散方程中的应用。针对超奇异积分的高效计算问题，我们基于分数阶依赖的广义多二次径向基函数（RBFs），推导了积分分数阶拉普拉斯算子的伪谱公式。将该公式应用于求解分数阶扩散方程，我们设计了一种简单易实现、几乎无需积分的无网格方法。通过等效的伽辽金变分形式，我们讨论了该新型无网格方法的收敛性。数值实验表明，与现有采用高斯径向基函数的方法相比，本文所提方法在精度与效率上均具有优越性。