Electron. J. Differential Equations,
Vol. 2019 (2019), No. 85, pp. 1-40.
Approximate solutions of randomized non-autonomous 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
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.
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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
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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
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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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