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

Julia Calatayud,Juan Carlos Cortés,Francisco Rodríguez,Marc Jornet

doi:10.3390/math8091417

Julia Calatayud, Juan Carlos Cortés + Show 2 more

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https://doi.org/10.3390/math8091417

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Abstract

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.

Highlights

Modeling physical systems for which the future state depends on history due to hereditary characteristics, such as aftereffects or time lags, usually requires the use of delay differential models.The delay may be discrete or continuous, depending on whether a specific or complete past information is used
Mickens observed that traditional standard finite difference schemes may be modified, on the basis of exact numerical schemes for basic ordinary differential equations, so that the essential properties of the Mathematics 2020, 8, 1417; doi:10.3390/math8091417
In the context of delay random differential equations, we focus on the linear case (2) studied in this paper assuming, as in Theorem 3, that α and β are bounded random variables, so that kαk∞ and k βk∞ are finite

Summary

Introduction

Modeling physical systems for which the future state depends on history due to hereditary characteristics, such as aftereffects or time lags, usually requires the use of delay differential models. In [28] the authors studied (2), but considered the solution in the sample-path sense and computed its probability density function via the random variable transformation technique, for certain forms of the initial condition process. Standard finite difference methods have already been applied to random ordinary, partial and fractional differential equations, by establishing the m.s. convergence, and even the convergence of densities, of the numerical discretizations towards the stochastic process solution [29,30,31,32,33,34]. M.s. convergence gives rise to m.s. calculus, where continuity, differentiability and Riemann integrability of a stochastic process are naturally defined by taking m.s. limits in the classical definitions. For two random variables y1 ∈ L∞ (Ω) and y2 ∈ L2 (Ω), it holds ky y2 k2 ≤ ky k∞ ky k2 < ∞

NSFD Methods for Linear Deterministic Differential Equations with Delay

NSFD Methods for Linear Random Differential Equations with Delay

Numerical Examples

Findings

Conclusions