Improving the Approximation of the First- and Second-Order Statistics of the Response Stochastic Process to the Random Legendre Differential Equation

  • J. Calatayud
  • J.-C. CortésEmail author
  • M. Jornet


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 et al. in Comput Math Appl 61(9):2782–2792, 2011), a mean square convergent power series solution on \((-1/\mathrm {e},1/\mathrm {e})\) 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 (\(1\le p\le \infty \)) 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.


random Legendre differential equation random power series mean square calculus uncertainty quantification 

Mathematics Subject Classification

34F05 60H10 60H35 65C05 65C60 93E03 



This work has been supported by the Spanish Ministerio de Economía y Competitividad Grant MTM2017-89664-P. Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigación y Desarrollo (PAID), Universitat Politècnica de València.

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this article.


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Authors and Affiliations

  1. 1.Instituto Universitario de Matemática Multidisciplinar, Building 8G, access C, 2nd floorUniversitat Politècnica de ValènciaValenciaSpain

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