ETNA - Electronic Transactions on Numerical Analysis
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Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |
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DATUM, UNTERSCHRIFT / DATE, SIGNATURE
BANK AUSTRIA CREDITANSTALT, WIEN (IBAN AT04 1100 0006 2280 0100, BIC BKAUATWW), DEUTSCHE BANK MÜNCHEN (IBAN DE16 7007 0024 0238 8270 00, BIC DEUTDEDBMUC)
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ETNA - Electronic Transactions on Numerical Analysis ISBN 978-3-7001-8258-0 Online Edition Research Article
Carlota M. Cuesta,
Francisco de la Hoz,
Ivan Girona
S. 59 - 98 doi:10.1553/etna_vol60s59 Verlag der Österreichischen Akademie der Wissenschaften doi:10.1553/etna_vol60s59
Abstract: In this paper we develop a fast and accurate pseudospectralmethod to numerically approximate the half Laplacian $(-\Delta)^{1/2}$ of afunction on $\mathbb{R}$, which is equivalent to the Hilbert transform of thederivative of the function. The main ideas are as follows. Given a twicecontinuously 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 expansionof $u(x(s))$, 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 weconsidered 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, weexpress $(-\Delta)_s^{1/2}e^{iks}$ for $k$ odd in terms of the Gaussianhypergeometric function ${}_2F_1$ and as a well-conditioned finite sum.Then we use a fast convolution result that enable us to compute veryefficiently $\sum_{l = 0}^Ma_l(-\Delta)_s^{1/2}e^{i(2l+1)s}$ for extremelylarge 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 havingfront solutions whose speed grows exponentially. Keywords: half Laplacian, pseudospectral method, Gaussian hypergeometric functions, fast convolution, fractional Fisher's equation Published Online: 2024/02/21 08:46:30 Object Identifier: 0xc1aa5572 0x003ed577 Rights: . Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613. …
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Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |