Solving the random Legendre differential equation: Mean square power series solution and its statistical functions

Abstract

In this paper we construct, by means of random power series, the solution of second order linear differential equations of Legendre-type containing uncertainty through its coefficients and initial conditions. By assuming appropriate hypotheses on the data, we prove that the constructed random power series solution is mean square convergent. In addition, the main statistical functions of the approximate solution stochastic process generated by truncation of the exact power series solution are given. Finally, we apply the proposed method to some illustrative examples to compare the numerical results for the average and the variance with respect to those obtained by the Monte Carlo approach.