Legendre Expansions of Products of Functions with Applications to Nonlinear Partial Differential Equations
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866909148265840640 |
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| author | Djellouli, Rabia Klein, David Levy, Matthew |
| author_facet | Djellouli, Rabia Klein, David Levy, Matthew |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2110_01372 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Legendre Expansions of Products of Functions with Applications to Nonlinear Partial Differential Equations Djellouli, Rabia Klein, David Levy, Matthew Numerical Analysis 42C10, 41A25, 65L06, 65N35, 40-08 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. |
| title | Legendre Expansions of Products of Functions with Applications to Nonlinear Partial Differential Equations |
| topic | Numerical Analysis 42C10, 41A25, 65L06, 65N35, 40-08 |
| url | https://arxiv.org/abs/2110.01372 |