• Ronny Ramlau, Lothar Reichel (Hg.)

ETNA - Electronic Transactions on Numerical Analysis

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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
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ETNA - Electronic Transactions on Numerical Analysis



ISBN 978-3-7001-8258-0
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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: bestellung.verlag@oeaw.ac.at
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A new Legendre polynomial-based approach for non-autonomous linear ODEs

    Stefano Pozza, Niel Van Buggenhout

ETNA - Electronic Transactions on Numerical Analysis, pp. 292-326, 2024/06/03

doi: 10.1553/etna_vol60s292

doi: 10.1553/etna_vol60s292


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doi:10.1553/etna_vol60s292



doi:10.1553/etna_vol60s292

Abstract

We introduce a new method with spectral accuracy to solve linear non-autonomous ordinary differential equations (ODEs) of the kind $ \frac{d}{dt}\tilde{u}(t) = \tilde{f}(t) \tilde{u}(t)$, $\tilde{u}(-1)=1$, with $\tilde{f}(t)$ an analytic function.The method is based on a new analytical expression for the solution $\tilde{u}(t)$ given in terms of a convolution-like operation, the $\star$-product. We prove that, by representing this expression in a finite Legendre polynomial basis, the solution $\tilde{u}(t)$ can be found by solving a matrix problem involving the Fourier coefficients of $\tilde{f}(t)$.An efficient procedure is proposed to approximate the Legendre coefficients of $\tilde{u}(t)$, and the truncation error and convergence are analyzed. We show the effectiveness of the proposed procedure through numerical experiments. Our approach allows for a generalization of the method to solve systems of linear ODEs.

Keywords: Legendre polynomials, spectral accuracy, ordinary differential equations

Published Online: 2024/06/03 11:19:52

Object Identifier: 0xc1aa5572 0x003f234e

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