arXiv subjects

J. -C. Cortés

Publications and source records attributed to J. -C. Cortés.

6 recordsLinked to original sources

arXiv2019

A probabilistic analysis of a Beverton-Holt type discrete model: Theoretical and computing analysis

In this paper a randomized version of the Beverton-Holt type discrete model is proposed. Its solution stochastic process and the random steady state are determined. Its first probability density function and second probability density function are obtained by means of the random variable transformation method, providing a full probabilistic description of the solution. Finally, several numerical examples are shown.

math.GM↗

arXiv2019

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

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 $f_1^N(p,t)$, of $f_1(p,t)$ are constructed. Afterwards, we rigorously prove that $f_1^N(p,t) \longrightarrow f_1(p,t)$ as $N\to \infty$ under mild conditions assumed on input data (initial condition and diffusion coefficient). Finally, three illustrative examples are shown.

math.PR↗

arXiv2019

Solving the Random Pielou Logistic Equation with the Random Variable Transformation Technique: Theory and Applications

The study of the dynamics of the size of a population via mathematical modelling is a problem of interest and widely studied. Traditionally, continuous deterministic methods based on differential equations have been used to deal with this problem. However discrete versions of some models are also available and sometimes more adequate. In this paper, we randomize the Pielou logistic equation in order to include the inherent uncertainty in modelling. Taking advantage of the method of transformation of random variables, we provide a full probabilistic description to the randomized Pielou logistic model via the computation of the probability density functions of the solution stochastic process, the steady state and the time until a certain level of population is reached. The theoretical results are illustrated by means of two examples, the first one consists of a numerical experiment and the second one shows an application to study the diffusion of a technology using real data.

math.PR↗

arXiv2018

Some notes to extend the study on random non-autonomous second order linear differential equations appearing in Mathematical Modeling

The objective of this paper is to complete certain issues from our recent contribution [J. Calatayud, J.-C. Cortés, M. Jornet, L. Villafuerte, Random non-autonomous second order linear differential equations: mean square analytic solutions and their statistical properties, Advances in Difference Equations, 2018:392, 1--29 (2018)]. We restate the main theorem therein that deals with the homogeneous case, so that the hypotheses are clearer and also easier to check in applications. Another novelty is that we tackle the non-homogeneous equation with a theorem of existence of mean square analytic solution and a numerical example. We also prove the uniqueness of mean square solution via an habitual Lipschitz condition that extends the classical Picard Theorem to mean square calculus. In this manner, the study on general random non-autonomous second order linear differential equations with analytic data processes is completely resolved. Finally, we relate our exposition based on random power series with polynomial chaos expansions and the random differential transform method, being the latter a reformulation of our random Fröbenius method.

math.NA↗

arXiv2018

Improving the approximation of the first and second order statistics of the response process to the random Legendre differential equation

In this paper, we deal with uncertainty quantification for the random Legendre differential equation, with input coefficient $A$ and initial conditions $X_0$ and $X_1$. In a previous study [Calbo G. et al, Comput. Math. Appl., 61(9), 2782--2792 (2011)], a mean square convergent power series solution on $(-1/e,1/e)$ was constructed, under the assumptions of mean fourth integrability of $X_0$ and $X_1$, independence, and at most exponential growth of the absolute moments of $A$. In this paper, we relax these conditions to construct an $\mathrm{L}^p$ solution ($1\leq p\leq\infty$) to the random Legendre differential equation on the whole domain $(-1,1)$, as in its deterministic counterpart. Our hypotheses assume no independence and less integrability of $X_0$ and $X_1$. Moreover, the growth condition on the moments of $A$ is characterized by the boundedness of $A$, which simplifies the proofs significantly. We also provide approximations of the expectation and variance of the response process. The numerical experiments show the wide applicability of our findings. A comparison with Monte Carlo simulations and gPC expansions is performed.

math.NA↗

arXiv2016

A mean square chain rule and its applications in solving the random Chebyschev differential equation

In this paper a new version of the chain rule for calculating the mean square derivative of a second-order stochastic process is proven. This random operational calculus rule is applied to construct a rigorous mean square solution of the random Chebyshev differential equation (r.C.d.e.) assuming mild moment hypotheses on the random variables that appear as coefficients and initial conditions of the corresponding initial value problem. Such solution is represented through a mean square random power series. Moreover, reliable approximations for the mean and standard deviation functions to the solution stochastic process of the r.C.d.e. are given. Several examples, that illustrate the theoretical results, are included.

math.PR↗