Linear Random Differential Equations Research Articles

A note on the mean-square solution of the hypergeometric differential equation with uncertainties

The Fröbenius method of power series has been applied to several linear random differential equations. The interest relies on the derivation of a closed-form mean-square solution and on the possibility of approximating statistical measures at exponential convergence rate. In this paper, we deal with the hypergeometric differential equation with random coefficients and initial conditions. On the interval (0, 1), random power series centered at the regular singular point 0 are employed, which are given in terms of the hypergeometric function. We find the stochastic basis of mean-square solutions and solve random initial-value problems. The approximation of the expectation and the variance is studied and illustrated computationally.

Probabilistic analysis of a class of impulsive linear random differential equations forced by stochastic processes admitting Karhunen-Loève expansions

<p style='text-indent:20px;'>We study a full randomization of the complete linear differential equation subject to an infinite train of Dirac's delta functions applied at different time instants. The initial condition and coefficients of the differential equation are assumed to be absolutely continuous random variables, while the external or forcing term is a stochastic process. We first approximate the forcing term using the Karhunen-Loève expansion, and then we take advantage of the Random Variable Transformation method to construct a formal approximation of the first probability density function (1-p.d.f.) of the solution. By imposing mild conditions on the model parameters, we prove the convergence of the aforementioned approximation to the exact 1-p.d.f. of the solution. All the theoretical findings are illustrated by means of two examples, where different types of probability distributions are assumed to model parameters.</p>

Probabilistic analysis of a class of impulsive linear random differential equations via density functions

An important class of non-homogeneous first-order linear random differential equations subject to an infinite sequence of square impulses with random intensity is studied. In applications, these equations are useful to model the dynamics of a population with periodic harvesting and migration under uncertainties. The solution is explicitly obtained via the first probability density function assuming an arbitrary joint density for all model parameters. Probabilistic stability analysis is carried out through the densities of the random sequences of minima and maxima. All the theoretical results are fully illustrated through two numerical examples.

Mean Square Convergent Non-Standard Numerical Schemes for Linear Random Differential Equations with Delay

In this paper, we are concerned with the construction of numerical schemes for linear random differential equations with discrete delay. For the linear deterministic differential equation with discrete delay, a recent contribution proposed a family of non-standard finite difference (NSFD) methods from an exact numerical scheme on the whole domain. The family of NSFD schemes had increasing order of accuracy, was dynamically consistent, and possessed simple computational properties compared to the exact scheme. In the random setting, when the two equation coefficients are bounded random variables and the initial condition is a regular stochastic process, we prove that the randomized NSFD schemes converge in the mean square (m.s.) sense. M.s. convergence allows for approximating the expectation and the variance of the solution stochastic process. In practice, the NSFD scheme is applied with symbolic inputs, and afterward the statistics are explicitly computed by using the linearity of the expectation. This procedure permits retaining the increasing order of accuracy of the deterministic counterpart. Some numerical examples illustrate the approach. The theoretical m.s. convergence rate is supported numerically, even when the two equation coefficients are unbounded random variables. M.s. dynamic consistency is assessed numerically. A comparison with Euler’s method is performed. Finally, an example dealing with the time evolution of a photosynthetic bacterial population is presented.

On integral separation of bounded linear random differential equations

Our aim in this paper is to investigate the openness and denseness for the set of integrally separated systems in the space of bounded linear random differential equations equipped with the $L^{\infty}$-metric. We show that in the general case, the set of integrally separated systems is open and dense. An exception is the case when the base space is isomorphic to the ergodic rotation flow of the unit circle, in which the setof integrally separated systems is open but not dense.

Solving initial and two-point boundary value linear random differential equations: A mean square approach

This paper deals with the construction of mean square real-valued solutions to both initial and boundary value problems of linear differential equations whose coefficients are assumed to be stochastic processes and, initial and boundary conditions are random variables. A key result to conduct our study is the extension of the Leibniz integral rule to the random framework taking advantage of the so-called random Fourth Calculus. Exact expressions for the main statistical functions (average and variance) associated to the solutions to both problems are also provided. Illustrative examples computing the average and standard deviation are included.

Inhomogeneous linear random differential equations with mutual correlations between multiplicative, additive and initial-value terms

The cumulant expansion for linear stochastic differential equations is extended to the general case in which the coefficient matrix, the inhomogeneous part and the initial condition are all random and, moreover, statistically interdependent. The expansion now involves not only the autocorrelation functions of the coefficient matrix (as in the homogeneous case) but also crosscorrelation functions of the coefficient matrix with the inhomogeneous part and with the initial value term. As a first illustration we consider an exactly solvable stochastic differential equation with initial correlations and compare the exact solution with that of the cumulant expansion. Secondly we show in general how the method can be used for the calculation of second moments, and treat the harmonic oscillator with random frequency and driving term as an example.

Solutions of a Class of Random Differential Equations

The analysis of a wide class of physical problems involves linear random differential equations—linear differential equations with stochastic coefficients. Here we consider the determination of probability densities for the solution processes of a class of linear random differential equations.This class is characterized by the property that the stochastic coefficients can be modeled by random processes with finite degrees of randomness. The solutions are examined by means of a Liouville-type equation satisfied by their joint probability densities. This technique is then applied to the study of electromagnetic fields in an inhomogeneous medium with special profiles.