Electronic Journal of Differential Equations,
Vol. 2019 (2019), No. 85, pp. 1-40.

Title: Approximate solutions of randomized non-autonomous complete 
       linear differential equations via probability density functions

Authors: Julia Calatayud    (Univ. Politecnica de Valencia, Spain)
         Juan Carlos Cortes (Univ. Politecnica de Valencia, Spain)
         Marc Jornet        (Univ. Politecnica de Valencia, Spain)

Abstract:
 Solving a random differential equation means to obtain an exact or
 approximate expression for the solution stochastic process, and
 to compute its statistical properties, mainly the mean and the variance
 functions. However, a major challenge is the computation of the probability
 density function of the solution. In this article we construct reliable
 approximations of the probability density function to the randomized
 non-autonomous complete linear differential equation by assuming that the
 diffusion coefficient and the source term are stochastic processes and the
 initial condition is a random variable. The key tools to construct these
 approximations are the random variable transformation technique and
 Karhunen-Loeve expansions. The study is divided into a large number of
 cases with a double aim: firstly, to extend the available results in the
 extant literature and, secondly, to embrace as many practical situations
 as possible. Finally, a wide variety of numerical experiments illustrate the
 potentiality of our findings.

Submitted July 2, 2018. Published July 16, 2019.
Math Subject Classifications: 34F05, 60H35, 60H10, 65C30, 93E03.
Key Words: Random non-autonomous complete linear differential equation;
           random variable transformation technique; Karhunen-Loeve expansion;
           probability density function.