In this paper, we develop a fast and accurate pseudospectral method to approximate numerically the half Laplacian $(-Δ)^{1/2}$ 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 $s\in[0,π]$, which maps $\mathbb{R}$ into $[0,π]$, and denote $(-Δ)_s^{1/2}u(x(s)) \equiv (-Δ)^{1/2}u(x)$. Therefore, by performing a Fourier series expansion of $u(x(s))$, the problem is reduced to computing $(-Δ)_s^{1/2}e^{iks} \equiv (-Δ)^{1/2}[(x + i)^k/(1+x^2)^{k/2}]$. On a previous work, we considered the case with $k$ even for the more general power $α/2$, with $α\in(0,2)$, so here we focus on the case with $k$ odd. More precisely, we express $(-Δ)_s^{1/2}e^{iks}$ 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 $\sum_{l = 0}^Ma_l(-Δ)_s^{1/2}e^{i(2l+1)s}$, for extremely large values of $M$. This enables us to approximate $(-Δ)_s^{1/2}u(x(s))$ in a fast and accurate way, especially when $u(x(s))$ is not periodic of period $π$. As an application, we simulate a fractional Fisher's equation having front solutions whose speed grows exponentially.

34 pages, 13 figures, 3 Matlab listings