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

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], 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 L p solution (1 ≤ p ≤ ∞) 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.

On the Legendre differential equation with uncertainties at the regular‐singular point 1: L p ( Ω ) random power series solution and approximation of its statistical moments

“…This result has been extended in the recent contribution, 2 where the solution has been constructed on the whole domain (−1, 1) with weaker assumptions on the random input coefficients. A common hypotheses in both works 1,2 was that the random variable A has statistical absolute moments that increase at most exponentially, equivalently, that A is a bounded random variable (see Lemma 2.2 in the work of Calatayud et al). 2 This assumption of boundedness for A will be essential in our subsequent development.…”

“…In the work of Calbo et al, a mean square power series solution to was constructed on (−1/e,1/e), being e the Euler constant. This result has been extended in the recent contribution, where the solution has been constructed on the whole domain (−1,1) with weaker assumptions on the random input coefficients. A common hypotheses in both works was that the random variable A has statistical absolute moments that increase at most exponentially, equivalently, that A is a bounded random variable (see Lemma 2.2 in the work of Calatayud et al) .…”

“…As it has been previously indicated, our goal in this section is to provide an analogous analysis to the works of Calbo et al and Calatayud et al for the regular‐singular point t =1.…”

“…A common hypotheses in both works 1,2 was that the random variable A has statistical absolute moments that increase at most exponentially, equivalently, that A is a bounded random variable (see Lemma 2.2 in the work of Calatayud et al). 2 This assumption of boundedness for A will be essential in our subsequent development.…”

Some Notes to Extend the Study on Random Non-Autonomous Second Order Linear Differential Equations Appearing in Mathematical Modeling

“…In this sense, some numerical experiments illustrating and demonstrating the potentiality of our main findings are also included. The study of random non-autonomous second order linear differential equations has been carried out for particular cases, such as Airy, Hermite, Legendre, Laguerre, and Bessel equations (see [8][9][10][11][12][13], respectively), and the general case [14][15][16][17]. Alternative approaches to study this class of random/stochastic differential equations include the so-called probabilistic transformation method [18] and stochastic numerical schemes [19,20], for example.…”