Random fractional generalized Airy differential equations: A probabilistic analysis using mean square calculus

摘要

The aim of this paper is to study a generalization of fractional Airy differential equations whose input data (coefficient and initial conditions) are random variables. Under appropriate hypotheses assumed upon the input data, we construct a random generalized power series solution of the problem and then we prove its convergence in the mean square stochastic sense. Afterwards, we provide reliable explicit approximations for the main statistical information of the solution process (mean, variance and covariance). Further, we show a set of numerical examples where our obtained theory is illustrated. More precisely, we show that our results for the random fractional Airy equation are in full agreement with the corresponding to classical random Airy differential equation available in the extant literature. Finally, we illustrate how to construct reliable approximations of the probability density function of the solution stochastic process to the random fractional Airy differential equation by combining the knowledge of the mean and the variance and the Principle of Maximum Entropy.