论文标题
$ \ mathbb {r} $中分数拉普拉斯的快速卷积方法
A fast convolution method for the fractional Laplacian in $\mathbb{R}$
论文作者
论文摘要
在本文中，我们开发了一种新方法，以数字上近似于$ \ Mathbb r $定义的函数的分数laplacian，以及一些更通用的单数积分。将$ \ Mathbb r $映射到有限间隔后，我们使用修改后的中点规则将积分操作员离散。该过程的结果可以作为离散的卷积施放，可以使用快速转换（FFT）进行有效评估。该方法提供了有效的，二阶准确的，近似于分数laplacian，而无需截断域。 我们首先证明该方法给出了分数laplacian和其他相关奇异积分的二阶近似；然后，我们使用快速卷积详细介绍了该方法的实现，并提供了支持其功效和效率的数值示例；最后，作为其适用于进化问题的一个例子，我们在聚焦案例中采用了一种方法来离散一维立方分数Schrödinger方程的方法。
In this article, we develop a new method to approximate numerically the fractional Laplacian of functions defined on $\mathbb R$, as well as some more general singular integrals. After mapping $\mathbb R$ into a finite interval, we discretize the integral operator using a modified midpoint rule. The result of this procedure can be cast as a discrete convolution, which can be evaluated efficiently using the Fast-Fourier Transform (FFT). The method provides an efficient, second order accurate, approximation to the fractional Laplacian, without the need to truncate the domain. We first prove that the method gives a second-order approximation for the fractional Laplacian and other related singular integrals; then, we detail the implementation of the method using the fast convolution, and give numerical examples that support its efficacy and efficiency; finally, as an example of its applicability to an evolution problem, we employ the method for the discretization of the nonlocal part of the one-dimensional cubic fractional Schrödinger equation in the focusing case.
