Electron. J. Differential Equations,
Vol. 2019 (2019), No. 85, pp. 140.
Approximate solutions of randomized nonautonomous complete
linear differential equations via probability density functions
Julia Calatayud, Juan Carlos Cortes, Marc Jornet
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
nonautonomous 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
KarhunenLoeve 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 nonautonomous complete linear differential equation;
random variable transformation technique; KarhunenLoeve expansion;
probability density function.
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Julia Calatayud
Instituto Universitario de Matemática Multidisciplinar
Universitat Politècnica de València
Camino de Vera s/n, 46022, Valencia, Spain
email: jucagre@doctor.upv.es


Juan Carlos Cortés
Instituto Universitario de Matemática Multidisciplinar
Universitat Politècnica de València
Camino de Vera s/n, 46022, Valencia, Spain
email: jccortes@imm.upv.es


Marc Jornet
Instituto Universitario de Matemática Multidisciplinar
Universitat Politècnica de València
Camino de Vera s/n, 46022, Valencia, Spain
email: marjorsa@doctor.upv.es

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