Numerical computation of the half Laplacian by means of a fast convolution algorithm
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2023
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| _version_ | 1866917566011670528 |
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| author | Cuesta, Carlota M. de la Hoz, Francisco Girona, Ivan |
| author_facet | Cuesta, Carlota M. de la Hoz, Francisco Girona, Ivan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_05009 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Numerical computation of the half Laplacian by means of a fast convolution algorithm Cuesta, Carlota M. de la Hoz, Francisco Girona, Ivan Numerical Analysis 26A33, 33C05, 65T50 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. |
| title | Numerical computation of the half Laplacian by means of a fast convolution algorithm |
| topic | Numerical Analysis 26A33, 33C05, 65T50 |
| url | https://arxiv.org/abs/2306.05009 |