The damped pendulum random differential equation: A comprehensive stochastic analysis via the computation of the probability density function
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https://doi.org/10.1016/j.physa.2018.08.024 |
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Title
The damped pendulum random differential equation: A comprehensive stochastic analysis via the computation of the probability density functionDate
2018-08-13Publisher
Elsevier B.V.ISSN
0378-4371Bibliographic citation
Calatayud, J., Cortés, J. C., & Jornet, M. (2018). The damped pendulum random differential equation: A comprehensive stochastic analysis via the computation of the probability density function. Physica A: Statistical Mechanics and its Applications, 512, 261-279.Type
info:eu-repo/semantics/articlePublisher version
https://www.sciencedirect.com/science/article/pii/S0378437118309762?via%3Dihub#d1e711Version
info:eu-repo/semantics/publishedVersionSubject
Abstract
This paper deals with the damped pendulum random differential equation: X¨(t)+2ω0ξX˙(t)
+ ω
2
0
X(t) = Y(t), t ∈ [0, T ], with initial conditions X(0) = X0 and X˙(0) = X1. The
forcing term Y(t) is a stochastic ... [+]
This paper deals with the damped pendulum random differential equation: X¨(t)+2ω0ξX˙(t)
+ ω
2
0
X(t) = Y(t), t ∈ [0, T ], with initial conditions X(0) = X0 and X˙(0) = X1. The
forcing term Y(t) is a stochastic process and X0 and X1 are random variables in a common
underlying complete probability space (Ω, F, P). The term X(t) is a stochastic process that
solves the random differential equation in both the sample path and in the Lp
senses.
To understand the probabilistic behavior of X(t), we need its joint finite-dimensional
distributions. We establish mild conditions under which X(t) is an absolutely continuous
random variable, for each t, and we find its probability density function fX(t)(x). Thus,
we obtain the first finite-dimensional distributions. In practice, we deal with two types
of forcing term: Y(t) is a Gaussian process, which occurs with the damped pendulum
stochastic differential equation of Itô type; and Y(t) can be approximated by a sequence
{YN (t)}
∞
N=1
in L2
([0, T ] × Ω), which occurs with Karhunen–Loève expansions and some
random power series. Finally, we provide numerical examples in which we choose specific
random variables X0 and X1 and a specific stochastic process Y(t), and then, we find the
probability density function of X(t). [-]
Is part of
Physica A: Statistical Mechanics and its Applications, Vol. 512 (15 December 2018)Funder Name
Ministerio de Ciencia e Innovación | Universitat Politècnica de València
Project code
MTM2017–89664–P
Project title or grant
Programa de Ayudas de Investigación y Desarrollo (PAID)
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© 2018 Elsevier B.V. All rights reserved.
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