Solution of the stochastic radiative transfer equation with Rayleigh scattering using RVT technique

Hussein, A.M.; Selim, M. M.

doi:10.1016/j.amc.2011.12.088

Cited by 57 publications

(74 citation statements)

References 20 publications

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“…method has been recently applied in order to determine the 1-p.d.f. of random ordinary differential equations [6,7], random partial differential equations [8,9,10,11] and random difference equations [12]. In [13], a number of illustrative examples show that R.V.T.…”

Section: Motivationmentioning

confidence: 99%

Probabilistic solution of random SI-type epidemiological models using the Random Variable Transformation technique

Casabán

López

Romero

et al. 2015

Communications in Nonlinear Science and Numerical Simulation

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This paper presents a full probabilistic description of the solution of random SI-type epidemiological models which are based on nonlinear differential equations. This description consists of determining: the first probability density function of the solution in terms of the density functions of the diffusion coefficient and the initial condition, which are assumed to be independent random variables; the expectation and variance functions of the solution as well as confidence intervals and, finally, the distribution of time until a given proportion of susceptibles remains in the population. The obtained formulas are general since they are valid regardless the probability distributions assigned to the random inputs. We also present a pair of illustrative examples including in one of them the application of the theoretical results to model the diffusion of a technology using real data.

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Section: Motivationmentioning

confidence: 99%

Probabilistic solution of random SI-type epidemiological models using the Random Variable Transformation technique

Casabán

López

Romero

et al. 2015

Communications in Nonlinear Science and Numerical Simulation

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“…via RVT method within the context of r.d.e. 's, it is worth pointing out that some significant contributions are [8][9][10][11][12][13][14]. Most of these contributions usually assume that uncertainty enters via a single r.v.…”

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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“…method is a powerful technique. This method has been used to study relevant continuous models [9,8,5,10,4]. In this paper we will determine the 1-p.d.f.…”

Section: Motivationmentioning

confidence: 99%

Probabilistic solution of random homogeneous linear second-order difference equations

Casabán

López

Romero

et al. 2014

Applied Mathematics Letters

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Solution of the stochastic radiative transfer equation with Rayleigh scattering using RVT technique

Cited by 57 publications

References 20 publications

Probabilistic solution of random SI-type epidemiological models using the Random Variable Transformation technique

Probabilistic solution of random SI-type epidemiological models using the Random Variable Transformation technique

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

Probabilistic solution of random homogeneous linear second-order difference equations

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