In this paper, an exact expression for the first probability density function of the solution stochastic process to a randomized homogeneous linear second-order complex differential equation is ...determined. To complete the probabilistic analysis, the first probability density functions of the real and complex contributions of the solution stochastic process are also calculated. To compute the densities, the random variable transformation method is applied under general hypothesis, all coefficients and initial conditions are absolutely continuous complex random variables. The capability of the theoretical results established is demonstrated by several numerical examples. Finally, we show the applicability of the method in engineering, by analysing the solution of a randomized simple harmonic oscillator, defined on the complex domain, and comparing our results with those obtained by using Monte Carlo simulations.
This paper provides a comprehensive probabilistic analysis of a full randomization of approximate SIR‐type epidemiological models based on discrete‐time Markov chain formulation. The randomization is ...performed by assuming that all input data (initial conditions, the contagion, and recovering rates involved in the transition matrix) are random variables instead of deterministic constants. In the first part of the paper, we determine explicit expressions for the so called first probability density function of each subpopulation identified as the corresponding states of the Markov chain (susceptible, infected, and recovered) in terms of the probability density function of each input random variable. Afterwards, we obtain the probability density functions of the times until a given proportion of the population remains susceptible, infected, and recovered, respectively. The theoretical analysis is completed by computing explicit expressions of important randomized epidemiological quantities, namely, the basic reproduction number, the effective reproduction number, and the herd immunity threshold. The study is conducted under very general assumptions and taking extensive advantage of the random variable transformation technique. The second part of the paper is devoted to apply our theoretical findings to describe the dynamics of the pandemic influenza in Egypt using simulated data excerpted from the literature. The simulations are complemented with valuable information, which is seldom displayed in epidemiological models. In spite of the nonlinear mathematical nature of SIR epidemiological model, our results show a strong agreement with the approximation via an appropriate randomized Markov chain. A justification in this regard is discussed.
This paper provides a full probabilistic solution of the randomized fractional linear nonhomogeneous differential equation with a random initial condition via the computation of the first probability ...density function of the solution stochastic process. To account for most generality in our analysis, we assume that uncertainty appears in all input parameters (diffusion coefficient, source term, and initial condition) and that a wide range of probabilistic distributions can be assigned to these parameters. Throughout our study, we will consider that the fractional order of Caputo derivative lies in 0,1, that corresponds to the main standard case. To conduct our analysis, we take advantage of the random variable transformation technique to construct approximations of the first probability density function of the solution process from a suitable infinite series representation. We then prove these approximations do converge to the exact density assuming mild conditions on random input parameters. Our theoretical findings are illustrated through 2 numerical examples.
•Random non-autonomous logistic-type differential equations are studied.•Random Variable Transformation method and Karhunen–Love expansion are combined.•First probability density function of the ...solution stochastic process is determined.•Numerical simulations for the mean, variance and PDF of the solution are performed.•A wide range of PDFs for input data are considered in numerical experiments.
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, f1(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 f1N(p,t), of f1(p, t) are constructed. Afterwards, we rigorously prove that f1N(p,t)⟶f1(p,t) as N → ∞ under mild conditions assumed on input data (initial condition and diffusion coefficient). Finally, three illustrative examples are shown.
In this paper we perform a complete probabilistic study of a finite dimensional linear control system with uncertainty. The controllability condition with random initial data and final target is ...analysed. To conduct this investigation we determine the first probability density function of the control and the solution of the random control problem under different scenarios. To achieve this objective, Random Variable Transformation technique is extensively applied. Several examples illustrate the theoretical results.
The classical kinetic equation has been broadly used to describe reaction and deactivation processes in chemistry. The mathematical formulation of this deterministic nonlinear differential equation ...depends on reaction and deactivation rate constants. In practice, these rates must be calculated via laboratory experiments, hence involving measurement errors. Therefore, it is more realistic to treat these rates as random variables rather than deterministic constants. This leads to the randomization of the kinetic equation, and hence its solution becomes a stochastic process. In this paper we address the probabilistic analysis of a randomized kinetic model to describe reaction and deactivation by catalase of hydrogen peroxide decomposition at a given initial concentration. In the first part of the paper, we determine closed-form expressions for the probability density functions of important quantities of the aforementioned chemical process (the fractional conversion of hydrogen peroxide, the time until a fixed quantity of this fractional conversion is reached and the activity of the catalase). These expressions are obtained by taking extensive advantage of the so called Random Variable Transformation technique. In the second part, we apply the theoretical results obtained in the first part together with the principle of maximum entropy to model the hydrogen peroxide decomposition and
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catalase deactivation using real data excerpted from the recent literature. Our results show full agreement with previous reported analysis but having as additional benefit that they provide a more complete description of both model inputs and outputs since we take into account the intrinsic uncertainties involved in modelling process.
This paper is addressed to give a generalization of the classical Markov methodology allowing the treatment of the entries of the transition matrix and initial condition as random variables instead ...of deterministic values lying in the interval
. This permits the computation of the first probability density function (1-PDF) of the solution stochastic process taking advantage of the so-called Random Variable Transformation technique. From the 1-PDF relevant probabilistic information about the evolution of Markov models can be calculated including all one-dimensional statistical moments. We are also interested in determining the computation of distribution of some important quantities related to randomized Markov chains (steady state, hitting times, etc.). All theoretical results are established under general assumptions and they are illustrated by modelling the spread of a technology using real data.
Random initial value problems to non-homogeneous first-order linear differential equations with complex coefficients are probabilistically solved by computing the first probability density of the ...solution. For the sake of generality, coefficients and initial condition are assumed to be absolutely continuous complex random variables with an arbitrary joint probability density function. The probability of stability, as well as the density of the equilibrium point, are explicitly determined. The Random Variable Transformation technique is extensively utilized to conduct the overall analysis. Several examples are included to illustrate all the theoretical findings.
The random variable transformation technique is a powerful method to determine the probabilistic solution for random differential equations represented by the first probability density function of ...the solution stochastic process. In this paper, that technique is applied to construct a closed form expression of the solution for the Bernoulli random differential equation. In order to account for the general scenario, all the input parameters (coefficients and initial condition) are assumed to be absolutely continuous random variables with an arbitrary joint probability density function. The analysis is split into two cases for which an illustrative example is provided. Finally, a fish weight growth model is considered to illustrate the usefulness of the theoretical results previously established using real data.
This paper deals with the determination of the first probability density function of the solution stochastic process to the homogeneous Riccati differential equation taking advantage of both ...linearization and Random Variable Transformation techniques. The study is split in all possible casuistries regarding the deterministic/random character of the involved input parameters. An illustrative example is provided for each one of the considered cases.
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