{
  "id": "1807.03141",
  "version": "v1",
  "published": "2018-07-03T19:12:06.000Z",
  "updated": "2018-07-03T19:12:06.000Z",
  "title": "Improving the approximation of the first and second order statistics of the response process to the random Legendre differential equation",
  "authors": [
    "J. Calatayud",
    "J. -C. Cortés",
    "M. Jornet"
  ],
  "comment": "13 pages; 6 tables",
  "categories": [
    "math.NA",
    "cs.NA",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "In this paper, we deal with uncertainty quantification for the random Legendre differential equation, with input coefficient $A$ and initial conditions $X_0$ and $X_1$. In a previous study [Calbo G. et al, Comput. Math. Appl., 61(9), 2782--2792 (2011)], a mean square convergent power series solution on $(-1/e,1/e)$ was constructed, under the assumptions of mean fourth integrability of $X_0$ and $X_1$, independence, and at most exponential growth of the absolute moments of $A$. In this paper, we relax these conditions to construct an $\\mathrm{L}^p$ solution ($1\\leq p\\leq\\infty$) to the random Legendre differential equation on the whole domain $(-1,1)$, as in its deterministic counterpart. Our hypotheses assume no independence and less integrability of $X_0$ and $X_1$. Moreover, the growth condition on the moments of $A$ is characterized by the boundedness of $A$, which simplifies the proofs significantly. We also provide approximations of the expectation and variance of the response process. The numerical experiments show the wide applicability of our findings. A comparison with Monte Carlo simulations and gPC expansions is performed.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-03T19:12:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34F05",
      "60H10",
      "60H35",
      "65C05",
      "65C60",
      "93E03"
    ],
    "keywords": [
      "random legendre differential equation",
      "second order statistics",
      "response process",
      "approximation",
      "square convergent power series solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}