Volume 60, pp. 292-326, 2024.

A new Legendre polynomial-based approach for non-autonomous linear ODEs

Stefano Pozza and Niel Van Buggenhout


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.

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Key words

Legendre polynomials, spectral accuracy, ordinary differential equations

AMS subject classifications

65F60, 65L05, 35Q41

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