The Numerical Stability Analysis For Two Classes Of Delay Differential Equations

Since delay differential equations (DDEs) are widely applied in many fields,such as physics,economics,biology,neural networks,dynamic systems and so on,the stability research of their numerical solutions plays an important role in both theory and practice.In this paper,the research summary of delay differential equations (DDEs) is given first.In addition,some results about the stability of different numerical methods for the delay system are also itemized.Then the structure of this paper is given.This paper mainly discusses the numerical stability of DDEs..As for the linear DDEs,we discuss the asymptotic stability of DDEs with variable coefficients,and gain a sufficient condition of asymptotic stability for this class. Furthermore,it is proved that,under this condition,the implicit Euler method is asymptotic stable.In regard to nonlinear delay differential equations,the concepts GR(p,q)- and GAR(p,q)- stability,which are introduced in nonlinear DDEs with constant delays,called Kα,β,γ,are applied to DDEs with variable delay.Then we introduce the new concepts of GR(p,q,v)- and GAR(p,q,v)- stability,and research the numerical stability of the more general G(c,p,q)- algebraically stable one-leg method.And it is proved that,when c≤1,the one-leg method is GR(p/2,q/2,2) stable, and when c<1,the method is GAR(p/2,q/2,2) stable.In the end,through some numerical experiments,we prove that the above theories are correct.