标题： 分数阶拉普拉斯的有限差分方法 显示英文标题

标题： Finite difference methods for fractional Laplacians

摘要： 分数阶拉普拉斯算子$(-\Delta)^{\alpha/2}$是典型的非局部椭圆算子。 虽然解析理论早已得到发展和理解，但在该算子的数值分析中仍存在许多未解问题。 在本文中，我们研究了一维均匀网格上分数阶拉普拉斯算子的几种不同的有限差分离散化方法，这些方法具有相同的形式。 在这个相对简单的设置中，可以比较和总结许多性质，以解决在实际实现中常见的非局部性、奇异性和平尾等问题。 还研究了这些方法的精度和渐近行为，以及远场边界条件的处理，为该格式在高维中的进一步发展提供了统一的观点。 显示英文摘要

摘要： The fractional Laplacian $(-\Delta)^{\alpha/2}$ is the prototypical non-local elliptic operator. While analytical theory has been advanced and understood for some time, there remain many open problems in the numerical analysis of the operator. In this article, we study several different finite difference discretisations of the fractional Laplacian on uniform grids in one dimension that takes the same form. Many properties can be compared and summarised in this relatively simple setting, to tackle more important questions like the nonlocality, singularity and flat tails common in practical implementations. The accuracy and the asymptotic behaviours of the methods are also studied, together with treatment of the far field boundary conditions, providing a unified perspective on the further development of the scheme in higher dimensions.