Improving the approximation of the first and second order statistics of the response process to the random Legendre differential equation

TL;DR

This paper develops a more general approach for uncertainty quantification in the random Legendre differential equation, relaxing previous assumptions and providing improved approximations of the response process's mean and variance.

Contribution

It introduces an $ ext{L}^p$ solution framework on the entire domain, relaxing independence and integrability conditions, and offers practical methods for approximating response statistics.

Findings

L^p solutions constructed on (-1,1) under weaker conditions

Approximate expectation and variance of the response process provided

Numerical experiments demonstrate wide applicability and compare favorably with Monte Carlo and gPC methods

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.