Random Airy type differential equations: Mean square exact and numerical solutions

Abstract

This paper deals with the construction of power series solutions of random Airy type differential equations containing uncertainty through the coefficients as well as the initial conditions. Under appropriate hypotheses on the data, we establish that the constructed random power series solution is mean square convergent over the whole real line. In addition, the main statistical functions, such as the mean and the variance, of the approximate solution stochastic process generated by truncation of the exact power series solution are given. Finally, we apply the proposed technique to several illustrative examples which show substantial speed-up and improvement in accuracy compared to other approaches such as Monte Carlo simulations.