Density function of random differential equations via finite difference schemes: a theoretical analysis of a random diffusion-reaction Poisson-type problem

Abstract

A computational approach to approximate the probability density function of random differential equations is based on transformation of random variables and finite difference schemes. The theoretical analysis of this computational method has not been performed in the extant literature. In this paper, we deal with a particular random differential equation: a random diffusion-reaction Poisson-type problem of the form −u″(x)+αu(x)=φ(x), , with boundary conditions , . Here, α, A and B are random variables and is a stochastic process. The term is a stochastic process that solves the random problem in the sample path sense. Via a finite difference scheme, we approximate with a sequence of stochastic processes in both the almost sure and senses. This allows us to find mild conditions under which the probability density function of can be approximated. Illustrative examples are included.