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

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Highlights

  • 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.

Abstract

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).

Section snippets

Introduction and motivation

The study of the damped pendulum differential equation with uncertainties has been tackled using different approaches, namely, the random and the Itô approaches, [1, pp. 96–97], [2]. In the former case, uncertainty is manifested in coefficients, initial/boundary conditions and/or forcing term via random variables and/or functions whose sample behavior is regular (e.g., continuous). This approach leads to the so-called Random Differential Equations (RDEs). While in the latter case, uncertainty

Probability density function of the solution stochastic process

Our main goal in this paper is to establish conditions under which the solution stochastic process X(t) given by (1.3)–(1.4) is an absolutely continuous random variable for each t[0,T], and then to compute its probability density function, fX(t)(x). Physically, the existence and computation of the probability density function of X(t) means that the probability for the response to lie in a certain set A at time t can be calculated as P(X(t)A)=AfX(t)(x)dx. This allows computing the main

Applications

In this section, we showcase the proposed approach on several examples where we apply our theoretical findings to particular random problems (1.1). The examples will cover a wide variety of situations which are of mathematical and physical interest. The main objective will be to test the methodology reported in this paper.

For the sake of clarity, throughout Example 3.3, Example 3.4, Example 3.5, Example 3.6, Example 3.7 we will fix the constants in (1.1): we choose the upper time T=1, the

Conclusions

In this paper we have provided a comprehensive probabilistic analysis of the damped pendulum differential equation in the case that the initial conditions (position, X0, and velocity, X1) are random variables and the forcing term, Y(t), is a stochastic process. To the best of our knowledge, a major difference of our contribution with respect to the ones available in the extant literature is that we have provided exact or approximate expressions for the probability density function of the

Acknowledgments

This work has been supported by the Spanish Ministerio de Economía y Competitividad grant MTM2017–89664–P. Marc Jornet acknowledges the doctorate scholarship granted by Programa de Ayudas de Investigación y Desarrollo (PAID), Universitat Politècnica de València. The authors are grateful for the valuable comments raised by the reviewers that have improved the final version of the paper.

Conflict of interest statement

The authors declare that there is no conflict of interests regarding the publication of this article.

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