Stability analysis of numerical methods for linear neutral Volterra delay-integro-differential system

Abstract

This paper is concerned with the stability of numerical methods for linear neutral Volterra delay-integro-differential system. A sufficient condition such that the system is asymptotically stable is derived. Furthermore, it is proved that every linear θ-method with θ ∈ (1/2, 1] and A-stable BDF method preserve the delay-independent stability of its exact solutions. Finally, the numerical experiments are given to demonstrate the conclusions.

Introduction

In this paper, we consider the delay-integro-differential equations

$${\mathit{My}}^{\prime }(t)=f\left(t\text{,}y(t)\text{,}{y}^{\prime }(t-\tau )\text{,}G\left(t\text{,}y(t-\tau )\text{,}{y}^{\prime }(t-\tau )\text{,}\underset{t-\tau }{\overset{t}{\int }}g(t\text{,}s\text{,}y(s))\mathrm{d}s\right)\right)\text{,}$$

where f:R⁺ × R^d × R^d × R^d → R^d is continuous, M ∈ R^d×d may be singular.

This system can be found in a wide variety of scientific and engineering fields such as biology, physics, ecology and so on (cf. [6], [9]). Particularly, it is observed that the delay-integro-differential-algebraic system (when M is singular) plays an important role in modelling many different phenomena of circuit analysis and chemical process simulation, which have a comprehensive list in [15]. Recently, there is a growing interest in developing numerical methods for solving the system (1.1) (cf. [2], [3], [4]).

As for the linear delay-integro-differential equations

$$\frac{\mathrm{d}u}{\mathrm{d}t}=\lambda u(t)+\mu u(t-\tau )+\kappa \underset{t-\tau }{\overset{t}{\int }}u(\sigma )\mathrm{d}\sigma \text{,}$$

Koto dealt with the stability of θ-methods and Runge–Kutta methods with a given quadrature formula (cf. [16], [17]).

Moreover, Zhang studied the nonlinear Volterra delay-integro equations

$$y(t)=g(t)+\underset{0}{\overset{t}{\int }}f(\xi \text{,}y(\xi )\text{,}y(\xi -\tau ))\mathrm{d}\xi \text{,}$$

the criteria on stability of BDF method is given (cf. [21]). In Ref. [22], [23], Zhang extended the research to Volterra delay-integro-differential equations

$$y^{\prime }(t)=f\left(t\text{,}y(t)\text{,}G\left(t\text{,}y(t-\tau )\text{,}\underset{t-\tau }{\overset{t}{\int }}g(t\text{,}s\text{,}y(s))\mathrm{d}s\right)\right)\text{,}$$

by investigating the BDF methods and Runge–Kutta methods.

Up to now, few papers discuss the stability analysis of neutral delay-integro-differential system (cf. [9], [10]).

For the system (1.1) without integro term, Ref. [24] considered the asymptotic stability of some numerical methods for the linear delay-differential-algebraic equation

$${\mathit{Ax}}^{\prime }(t)+\mathit{Bx}(t)+{\mathit{Cx}}^{\prime }(t-\tau )+\mathit{Dx}(t-\tau )=0\text{,}$$

where the matrix A is singular.

In this paper, we focus our attention on the stability of numerical methods for the linear neutral Volterra delay-integro-differential system. The stability analysis of exact solutions to the equation is considered. A sufficient condition is given for the neutral Volterra delay-integro-differential system. Especially, this condition is also necessary for the system with algebraic constraints. Furthermore, the linear θ-method and BDF method are applied to the system, respectively. It is proved that the numerical method preserve the delay-independent stability of system if the linear θ-method satisfies θ ∈ (1/2,1] and the BDF method is A-stable. Finally, the numerical experiments are presented to illustrate the conclusions.