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

Abstract

In this paper we develop a fast and accurate pseudospectral method to numerically approximate the half Laplacian $(-\mathrm{\Delta }{)}^{1/2}$ 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 $(-\mathrm{\Delta }{)}_s^{1/2}u(x(s))\equiv (-\mathrm{\Delta }{)}^{1/2}u(x)$. Therefore, by performing a Fourier series expansion of $u(x(s))$, the problem is reduced to computing $(-\mathrm{\Delta }{)}_s^{1/2}{e}^{iks}\equiv (-\mathrm{\Delta }{)}^{1/2}[(x+i{)}^k/(1+x^2{)}^{k/2}]$. 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 $(-\mathrm{\Delta }{)}_s^{1/2}{e}^{iks}$ for $k$ odd in terms of the Gaussian hypergeometric function ${}_2{F}_1$ and as a well-conditioned finite sum. Then we use a fast convolution result that enable us to compute very efficiently $\sum _{l=0}^M{a}_l(-\mathrm{\Delta }{)}_s^{1/2}{e}^{i(2l+1)s}$ for extremely large values of $M$. This enables us to approximate $(-\mathrm{\Delta }{)}_s^{1/2}u(x(s))$ 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 , DOI: 10.1553/etna_vol60s59

Key words

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

AMS subject classifications

26A33, 33C05, 65T50