$$\mathrm {L}^p$$-calculus Approach to the Random Autonomous Linear Differential Equation with Discrete Delay

Abstract

In this paper, we provide a full probabilistic study of the random autonomous linear differential equation with discrete delay τ > 0: x (t) = ax(t) + bx(t − τ), t ≥ 0, with initial condition x(t) = g(t), −τ ≤ t ≤ 0. The coefficients a and b are assumed to be random variables, while the initial condition g(t) is taken as a stochastic process. By using L p-calculus, we prove that, under certain conditions, the deterministic solution constructed with the method of steps that involves the delayed exponential function is an L p-solution too. An analysis of L p-convergence when the delay τ tends to 0 is also performed in detail.

Nonstandard finite difference schemes for general linear delay differential systems

“…In particular, the NSFD methods of this work can be applied to high‐order linear delay differential equations, as they can be converted in the usual way into a vector problem with non‐commuting matrix coefficients. Additionally, it could be possible to extend the type of results for deterministic systems obtained in this work to the random setting, considering problems as those in literature 38,39 for delay scalar equations, with coefficients being random variables and initial functions being stochastic processes.…”

Analytic solution to the generalized delay diffusion equation with uncertain inputs in the random Lebesgue sense

“…In this section, we show some results from Calatayud et al 28 The autonomous and homogeneous linear random differential equation with delay is…”

“…The integral from (3) is considered as an L p -Riemann integral. In Calatayud et al, 28 two results for existence and uniqueness of L p solution to (2) were stated and proved. In this paper, we will only need the second result.…”

“…In Section 2, we show the main definitions and results on random Lebesgue calculus that will be required to solve (1). In Section 3, we summarize the main findings obtained in Calatayud et al 28 for the linear random differential equation with delay. After these preliminaries, Section 4 is devoted to solving (1) in the random L p sense, by employing the method of separation of variables and addressing the L p convergence of the series solution.…”

“…In the separate equation for the time variable, a particular linear random differential equation with delay arises, which is explicitly solved by utilizing. 28 Once the L p convergence of the series solution is established, statistical moments up to order p can be approximated by truncating the series. In particular, when p ≥ 2, the expectation and the variance of the solution are estimated.…”