Computing the probability density function of non-autonomous first-order linear homogeneous differential equations with uncertainty

Abstract This paper is devoted to construct approximations of the probability density function of the non-autonomous first-order homogeneous linear random differential equation, where the initial condition and the diffusion coefficient are assumed to be a random variable and a stochastic process, respectively. We combine Random Variable Transformation technique and Karhunen–Loeve expansion to construct reliable approximations under general conditions. Several numerical examples illustrate our theoretical findings.