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arXiv · 2110.01372

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

Rabia Djellouli·David Klein·Matthew Levy
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Abstract

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 efficiency and performance accuracy of this Fourier-Legendre based solution methodology for solving an important class of nonlinear PDEs.

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Rabia Djellouli, David Klein, Matthew Levy. 2021-09-18. Legendre Expansions of Products of Functions with Applications to Nonlinear Partial Differential Equations. https://doi.org/10.1016/j.apnum.2024.03.014

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