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Corpus ID: 238259817

Legendre Expansions of Products of Functions with Applications to Nonlinear Partial Differential Equations

@article{Djellouli2021LegendreEO,
title={Legendre Expansions of Products of Functions with Applications to Nonlinear Partial Differential Equations},
author={Rabia Djellouli and David Klein and Matthew Levy},
journal={ArXiv},
year={2021},
volume={abs/2110.01372}
}

Given the Fourier-Legendre expansions of f and g, and mild conditions on f and g, we derive the Fourier-Legendre expansion of their product in terms of their corresponding Fourier-Legendre coefficients. In this way, expansions of whole number powers of f may be obtained. We establish upper bounds on rates of convergence. We then employ these expansions to solve semi-analytically a class of nonlinear PDEs with a polynomial nonlinearity of degree 2. The obtained numerical results illustrate the… Expand

1. It is known that any polynomial in μ. can be expanded as a linear function of Legendre polynomials [1]. In particular, we have The earlier coefficients, say A 0 , A 2 , A 4 may easily be found by… Expand

A new and sharper bound for the Legendre coefficients of differentiable functions is provided and a new error bound of the truncated Legendre series in the uniform norm is derived.Expand

The expression for the product of two Legendre’s coefficients which is the subject of the present paper, was found by induction on the 13th of February, 1873, and on the following day I succeeded in… Expand

SUMMARY
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