In this paper, we develop a fast and accurate pseudospectral method to approximate numerically the half Laplacian $(-\Delta)^{1/2}$ of a function on $\mathbb{R}$, which is equivalent to the Hilbert transform of the derivative of the function. The main ideas are as follows. Given a twice continuously differentiable bounded function $u\in\mathcal C_b^2(\mathbb{R})$, we apply the change of variable $x=L\cot(s)$, with $L>0$ and $s\in[0,\pi]$, which maps $\mathbb{R}$ into $[0,\pi]$, and denote $(-\Delta)_s^{1/2}u(x(s)) \equiv (-\Delta)^{1/2}u(x)$. Therefore, by performing a Fourier series expansion of $u(x(s))$, the problem is reduced to computing $(-\Delta)_s^{1/2}e^{iks} \equiv (-\Delta)^{1/2}[(x + i)^k/(1+x^2)^{k/2}]$. On a previous work, we considered the case with $k$ even for the more general power $\alpha/2$, with $\alpha\in(0,2)$, so here we focus on the case with $k$ odd. More precisely, we express $(-\Delta)_s^{1/2}e^{iks}$ for $k$ odd in terms of the Gaussian hypergeometric function ${}_2F_1$, and also as a well-conditioned finite sum. Then, we use a fast convolution result, that enable us to compute very efficiently $\sum_{l = 0}^Ma_l(-\Delta)_s^{1/2}e^{i(2l+1)s}$, for extremely large values of $M$. This enables us to approximate $(-\Delta)_s^{1/2}u(x(s))$ in a fast and accurate way, especially when $u(x(s))$ is not periodic of period $\pi$. As an application, we simulate a fractional Fisher's equation having front solutions whose speed grows exponentially.

翻译：本文提出一种快速精确的伪谱方法，用于数值逼近定义在$\mathbb{R}$上函数的半拉普拉斯算子$(-\Delta)^{1/2}$，该算子等价于函数导数的希尔伯特变换。核心思路如下：给定二次连续可微有界函数$u\in\mathcal C_b^2(\mathbb{R})$，采用变量替换$x=L\cot(s)$（其中$L>0$，$s\in[0,\pi]$），将$\mathbb{R}$映射至$[0,\pi]$，并记$(-\Delta)_s^{1/2}u(x(s)) \equiv (-\Delta)^{1/2}u(x)$。通过对$u(x(s))$进行傅里叶级数展开，问题转化为计算$(-\Delta)_s^{1/2}e^{iks} \equiv (-\Delta)^{1/2}[(x + i)^k/(1+x^2)^{k/2}]$。在先前工作中，我们已针对更一般的幂次$\alpha/2$（$\alpha\in(0,2)$）处理了$k$为偶数的情况，故本文聚焦于$k$为奇数的情形。具体而言，我们利用高斯超几何函数${}_2F_1$表达$k$为奇数时的$(-\Delta)_s^{1/2}e^{iks}$，同时给出良态有限和表达式。进而采用快速卷积算法，在极大规模参数$M$下高效计算$\sum_{l = 0}^Ma_l(-\Delta)_s^{1/2}e^{i(2l+1)s}$。这使得我们能够快速精确地逼近$(-\Delta)_s^{1/2}u(x(s))$，尤其适用于$u(x(s))$非$\pi$周期函数的情况。作为应用实例，我们模拟了具有指数增长波速前沿解的分数阶费希尔方程。