On the linear advection equation subject to random velocity fields

Dorini, Fabio Antonio; Cunha, M. Cristina C.

doi:10.1016/j.matcom.2011.10.008

Cited by 20 publications

(19 citation statements)

References 19 publications

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“…It is worthy highlighting that in the context of RDEs, the RVT technique has been successfully applied to determine the 1-PDF of the solution stochastic process of relevant problems, formulated via RDEs, that appear in different areas [28,29,30,31].…”

Section: Motivation and Preliminariesmentioning

confidence: 99%

Analysis of random non-autonomous logistic-type differential equations via the Karhunen–Loève expansion and the Random Variable Transformation technique

López

Navarro‐Quiles

Romero

et al. 2019

Communications in Nonlinear Science and Numerical Simulation

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This paper deals with the study, from a probabilistic point of view, of logistic-type differential equations with uncertainties. We assume that the initial condition is a random variable and the diffusion coefficient is a stochastic process. The main objective is to obtain the first probability density function, f 1 (p, t), of the solution stochastic process, P(t, ω). To achieve this goal, first the diffusion coefficient is represented via a truncation of order N of the Karhunen-Loève expansion, and second, the Random Variable Transformation technique is applied. In this manner, approximations, say f N 1 (p, t), of f 1 (p, t) are constructed. Afterwards, we rigorously prove that f N 1 (p, t) −→ f 1 (p, t) as N → ∞ under mild conditions assumed on input data (initial condition and diffusion coefficient). Finally, three illustrative examples are shown.

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Section: Motivation and Preliminariesmentioning

confidence: 99%

Analysis of random non-autonomous logistic-type differential equations via the Karhunen–Loève expansion and the Random Variable Transformation technique

López

Navarro‐Quiles

Romero

et al. 2019

Communications in Nonlinear Science and Numerical Simulation

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“…Regarding the application of RVT method to account for the 2-p.d.f., to the best of our knowledge, it is has only been addressed by assuming specific standard distributions for random inputs. Some interesting contributions dealing with random differential equations include [15]. RVT technique has also been applied in combination with other techniques like Taylor series expansion to compute approximations to the p.d.f.…”

Section: Motivationmentioning

confidence: 99%

Solving Random Homogeneous Linear Second-Order Differential Equations: A Full Probabilistic Description

et al. 2016

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Roselló, M. (2016). Solving random homogeneous linear second-order differential equations: a full probabilistic description. Mediterranean Journal of Mathematics. 13(6): 3817-3836. doi:10.1007/s00009-016-0716-6 Solving random homogeneous linear secondorder differential equations: A full probabilistic description Abstract. In this paper a complete probabilistic description for the solution of random homogeneous linear second-order differential equations via the computation of its two first probability density functions is given. As a consequence, all unidimensional and two-dimensional statistical moments can be straightforwardly determined, in particular, mean, variance and covariance functions, as well as the first-order conditional law. With the aim of providing more generality, in a first step, all involved input parameters are assumed to be statistically dependent random variables having an arbitrary joint probability density function. Secondly, the particular case that just initial conditions are random variables is also analysed. Both problems have common and distinctive feature which are highlighted in our analysis. The study is based on Random Variable Transformation method. As a consequence of our study, the well-known deterministic results are nicely generalized. Several illustrative examples are included.Mathematics Subject Classification (2010). 60H35, 60H10, 37H10. Keywords. Random Variable Transformation method, first and second probability density functions, random homogeneous linear second-order differential equations. MotivationThe quantification of uncertainty in dynamic models is currently playing an important role in many applied areas. Classical deterministic differential equations, which have demonstrated to be powerful tools for analysing problems that appear in areas such as Physics, Engineering, Chemistry, Epidemiology, etc., need to consider randomness in their formulation in order to account for measurement errors and inherent complexity of problems underThis work was completed with the support of our T E X-pert. [4][5][6]. However, it is important to point out that in the random context besides computing the solution stochastic process (s.p.), say Z(t), it is also of great interest the determination of its main statistical properties. Most of the contributions focus on the computation of the mean, µ Z (t) = E[Z(t)], and the variance, σ 2 Z (t) = V[Z(t)], functions of the solution s.p. However, a more convenient goal is the determination of its first probability density function (1-p.d.f.), f 1 (z, t), since from it one can easily compute not just these two first statistical moments,but also the one-dimensional statistical moment of any orderThe 1-p.d.f. characterizes, from a probabilistic point of view, the solution s.p. Z(t) at every time instant t. In general, more challenging is the determination of the rest of the so-called fidis (finite dimensional distributions), i.e., the n-dimensional p.d.f.'s of the solution s.p. for n ≥ 2, because it usually involves complex developments...

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“…In particular, a major challenge is to determine the probability density function of the solution, since from it one can obtain a full characterization of all one-dimensional statistical moments of the solution (hence including, just as particular cases, the mean and the variance). We point out that the computation of the probability density function of the solution stochastic process of some random ordinary and partial differential equations describing relevant problems in Physics and Engineering has been achieved, [6,7,8,9,10,11,12,13,14].…”

Section: Introductionmentioning

confidence: 99%

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

Calatayud

López

Jornet

2018

Physica A: Statistical Mechanics and its Applications

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This paper deals with the damped pendulum random differential equation:Ẍ(t) + 2ω 0 ξẊ(t) + ω 2 0 X(t) = Y (t), t ∈ [0, T ], with initial conditions X(0) = X 0 andẊ(0) = X 1 . The forcing term Y (t) is a stochastic process and X 0 and X 1 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 L p senses. To understand the probabilistic behaviour 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 f X(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 {Y N (t)} ∞ N =1 in L 2 ([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 X 0 and X 1 and a specific stochastic process Y (t), and then, we find the probability density function of X(t).

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On the linear advection equation subject to random velocity fields

Cited by 20 publications

References 19 publications

Analysis of random non-autonomous logistic-type differential equations via the Karhunen–Loève expansion and the Random Variable Transformation technique

Analysis of random non-autonomous logistic-type differential equations via the Karhunen–Loève expansion and the Random Variable Transformation technique

Solving Random Homogeneous Linear Second-Order Differential Equations: A Full Probabilistic Description

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

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