Analytical and numerical stability of nonlinear neutral delay integro-differential equations

https://doi.org/10.1016/j.jfranklin.2011.04.007Get rights and content

Abstract

In this paper, we are concerned with the analytical and numerical stability of nonlinear neutral delay integro-differential equations (NDIDEs). First, sufficient conditions for the analytical stability of nonlinear NDIDEs with a variable delay are derived. Then, we show that any A-stable linear multistep method can preserve the asymptotic stability of the analytical solution for nonlinear NDIDEs with a constant delay. At last, we validate our conclusions by numerical experiments.

Introduction

In this paper, we study the analytical and numerical stability of nonlinear neutral delay integro-differential equations (NDIDEs) of the form:y(t)=f(t,y(t),y(tτ(t)),y(tτ(t)),tτ(t)tg(t,s,y(s))ds),tt0y(t)=φ(t),ttt0,where t=inftt0(tτ(t)). Functions f:[t0,+)×CN×CN×CN×CNCN, g:[t0,+)×[t,+)×CNCN are smooth enough, and initial function φ:[t,t0]CN is continuously differentiable. The delay function τ(t) satisfies the following hypotheses (see [33]):

    (H1)

    τ(t) is a continuous function and there exists a constant τ0>0 such that τ(t)τ0, tt0.

    (H2)

    α(t)tτ(t) is strictly increasing for all tt0 and divergent as t.

This class of equations play an important role in modeling phenomena of the real world. They arise widely in scientific and engineering fields such as biology, physics, control theory, ecology and so on (see [4], [5], [40]). So it is valuable to investigate the properties of the solution of these equations. Since most of these equations cannot be solved exactly, it is necessary to study efficient numerical methods to solve these equations. An important problem in this context is the investigation of the stability of numerical methods.

In recent years, many papers discussed analytical stability and numerical stability of delay differential equations (DDEs) (see [3], [11], [12], [13], [14], [17], [18], [20], [32], [33] and the references therein), neutral delay differential equations (NDDEs) (see [1], [2], [9], [19], [22], [23], [24], [25], [26], [38] and the references therein) and delay integro-differential equations (DIDEs) (see [6], [10], [15], [16], [34], [35] and the references therein). A significant number of important results have already been found. But for neutral delay integro-differential equations (NDIDEs), only a few of results have been presented in the literature. Zhao et al. [39] considered the linear neutral Volterra delay integro-differential system:Au(t)+Bu(t)+Cu(tτ)+Du(tτ)+Gtτtu(x)dx=0,t>0,u(t)=ϕ(t),τt0,where A,B,C,D,GRd×d, τ>0 and the matrix A may be singular. They gave a sufficient condition such that the system is asymptotically stable, and proved that every linear θmethod with 1/2θ1 and A-stable BDF methods preserve the delay-independent stability of the analytical solution. Later, Xu et al. [29] further considered Runge–Kutta methods for system (1.2). It is proved that every A-stable natural Runge–Kutta method preserves the delay-independent stability of the analytical solution. In 2008, Zhang et al. [36] investigated the asymptotic stability of exact and discrete solutions of neutral multidelay-integro-differential equations. A sufficient condition for the asymptotic stability of the analytical solution has been derived. And the asymptotic stability criteria of Runge–Kutta methods and linear multistep methods were established. Wu et al. [28] considered a test equation for one-dimension NDIDEs and got some delay-dependent stability results.

In the nonlinear case, this topic has also attracted the attention of some authors. Yu et al. [31] and Zhang et al. [37] investigated the stability of Runge–Kutta methods for the equation:ddt[y(t)Ny(tτ)]=f(t,y(t),y(tτ),tτtg(t,ξ,y(ξ))dξ),t0,y(t)=φ(t),τt0.For the numerical stability analysis of general nonlinear NDIDEs with a constant delay, we refer the reader to [30], where many methods, including one-leg methods, Runge–Kutta methods, general linear methods and linear θmethods, are considered. For the analytical stability analysis of nonlinear NDIDEs, we mention the paper by Wen et al. [27] and the doctoral thesis by Wang [21]. In [27], by applying a generalized Halanay inequality to Eq. (1.3), some dissipativity and asymptotic stability results were obtained. In [21], the analytical stability of more general neutral functional differential equations (NFDEs) was studied and the results obtained there are also applicable to NDIDEs.

In this paper, we further investigate the analytical and numerical stability of NDIDEs. We first study the analytical stability of problem (1.1). The approach we use is based on the theory of stability analysis which was first introduced by Torelli [20] and developed by Zennaro [33], Bellen et al. [1] and Wang et al. [22]. We derive some new sufficient conditions for boundedness stability and asymptotic stability of the exact solution. Then, we focus our attention on the nonlinear stability of the numerical solutions produced by linear multistep methods. By using the equivalence between A-stability and G-stability, which was established by Dahlquist [8], it is shown that every A-stable linear multistep method can preserve the asymptotic stability of a class of NDIDEs with a constant delay. At last, we give some numerical experiments to confirm our theoretical results.

Section snippets

Analytical stability of nonlinear NDIDEs

Let ·,· be an inner product on CN and · be the corresponding norm. We consider the following initial value problem:z(t)=f(t,z(t),z(tτ(t)),z(tτ(t)),tτ(t)tg(t,s,z(s))ds),tt0,z(t)=ϕ(t),ttt0,which is a corresponding perturbed problem of Eq. (1.1) with a different continuously differentiable initial function. In this paper we assume that the functions f and g satisfy the following one-sided Lipschitz or global Lipschitz conditions: Rey1y2,f(t,y1,u,v,w)f(t,y2,u,v,w)R(t)y1y22,tt

Linear multistep methods for nonlinear NDIDEs

In this section, we consider linear multistep methods for NDIDEs. Let stepsize h satisfy h=τ/m, with m a positive integer, and let tn=nh. A linear multistep method for problem (2.43) is given byρ(E)yn=hσ(E)f(tn,yn,ynm,y¯nm,y˜n),where E is the translation operator defined by Eyn=yn+1 and the argument yn is an approximation to the exact solution y(tn). And ρ(ξ)=j=0kajξj,σ(ξ)=j=0kbjξjare generating polynomials of the method, which are assumed to have real coefficients and no common divisor.

Numerical stability analysis

In this section, we investigate the asymptotic stability of A-stable linear multistep methods for nonlinear NDIDEs. We will make use of the conclusion that A-stability is equivalent to G-stability which was established by Dahlquist [8] in 1978.

For any real symmetric positive definite k×k matrix G=[gij], the norm ·G on (CN)k is defined byUG=i,j=1kgijui,uj1/2,U=(u1T,u2T,,ukT)T,uiCN.

The following result shows the asymptotic stability of linear multistep methods when applied to problem

Numerical experiment

In this section, we test some numerical methods to validate Theorem 4.1. We consider the following nonlinear NDIDE:y(t)=ay(t)+bsin2(y(tτ))cos(y(tτ))+ctτty(s)ds+sin2(t),t0,y(t)=cos(t),τt0,and its corresponding perturbed problemz(t)=az(t)+bsin2(z(tτ))cos(z(tτ))+ctτtz(s)ds+sin2(t),t0,z(t)=sin(t),τt0.Here we choose the parameters a=−8, b=0.1, c=0.5 and τ=1, and it is easy to get R=8,β1=β2=0.2,β3=0.5,γ=1 and ϱ=1 so that R=8<0,β2=0.2<1,R+ϱ+τβ3γ1β2=8+1+1×0.5×110.2<0,i.e.,

Acknowledgments

The authors are indebted to the referees and the editors for their carefully reading of this paper and their valuable comments.

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    This work was supported by the NSF of China (No. 10971077).

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