{
  "id": "2110.01372",
  "version": "v2",
  "published": "2021-09-18T19:31:21.000Z",
  "updated": "2024-03-24T15:16:06.000Z",
  "title": "Legendre Expansions of Products of Functions with Applications to Nonlinear Partial Differential Equations",
  "authors": [
    "Rabia Djellouli",
    "David Klein",
    "Matthew Levy"
  ],
  "comment": "38 pages, 25 figures",
  "journal": "Applied Numerical Mathematics, Vol. 201, p. 301-321 (2024)",
  "doi": "10.1016/j.apnum.2024.03.014",
  "categories": [
    "math.NA",
    "cs.NA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-03-24T15:16:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C10",
      "41A25",
      "65L06",
      "65N35",
      "40-08"
    ],
    "keywords": [
      "nonlinear partial differential equations",
      "legendre expansions",
      "fourier-legendre expansion",
      "applications",
      "nonlinear pdes"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}