The damped pendulum random differential equation: A comprehensive stochastic analysis via the computation of the probability density function

This paper deals with the damped pendulum random differential equation: Ẍ(t)+2ω0ξẊ(t)+ω02X(t)=Y(t), t∈[0,T], with initial conditions X(0)=X0 and Ẋ(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...

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Bibliographic Details
Published in:Physica A Vol. 512; pp. 261 - 279
Main Authors: Calatayud, J., Cortés, J.-C., Jornet, M.
Format: Journal Article
Language:English
Published: Elsevier B.V 15.12.2018
Subjects:
Damped pendulum random differential equation
Numerical analysis
Probability density function
Stochastic methods in physics
Damped pendulum random differential equation
65Z05
60H10
Numerical analysis
93E03
Probability density function
Stochastic methods in physics
34F05
60H35
ISSN:0378-4371, 1873-2119
Online Access:Get full text
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Summary:This paper deals with the damped pendulum random differential equation: Ẍ(t)+2ω0ξẊ(t)+ω02X(t)=Y(t), t∈[0,T], with initial conditions X(0)=X0 and Ẋ(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). •Damped pendulum random differential equation is studied.•The probability density function of the solution stochastic process is computed.•Mild hypotheses on the random inputs (forcing term and initial conditions) are assumed.•The analysis considers a wide variety of situations often usual in practice.•A wide range of examples shows that the results are computationally feasible.
ISSN:0378-4371
1873-2119
DOI:10.1016/j.physa.2018.08.024