Volume 60, pp. 59-98, 2024.

Numerical computation of the half Laplacian by means of a fast convolution algorithm

Carlota M. Cuesta, Francisco de la Hoz, and Ivan Girona


In this paper we develop a fast and accurate pseudospectral method to numerically approximate 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 $(-\Delta)_s^{1/2}e^{iks} \equiv (-\Delta)^{1/2}[(x + i)^k/(1+x^2)^{k/2}]$. In a previous work we considered the case with $k$ even for more general powers $\alpha/2$, 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 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.

Full Text (PDF) [4.1 MB], BibTeX

Key words

half Laplacian, pseudospectral method, Gaussian hypergeometric functions, fast convolution, fractional Fisher's equation

AMS subject classifications

26A33, 33C05, 65T50

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