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 X0 and X1. 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/e, 1/e) was constructed, under the assumptions of mean fourth integrability of X0 and X1, independence, and at most exponential growth of the absolute moments of A. In this paper, we relax these conditions to construct an Lp 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 X0 and X1. 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.