A fast convolution method for the fractional Laplacian in ℝ
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In this article, we develop a new method to approximate numerically the fractional Laplacian of functions defined on ℝ, as well as some more general singular integrals. After mapping 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.
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