Analytical And Numerical Stability For Several Classes Of Delay Differential Equations

There are many time-delay systems in biology, economics, physics, chemistry, engineering and autocontrol. All delay systems are characterized by the common feature of being influenced, in their present evolution, by information on their past history. This fact is translated into mathematical models by Delay Differential Equations (DDEs), i.e. Differential Equations containing one or more delayed terms that can be of several types. In general, there are very few DDEs that we can have their analysis solutions, so it is very necessary to get their numerical solutions. So far, there have been made great progress in the research of stability of DDEs systems, numerical solutions of DDEs and the estimates of the delay value. In this paper we study several problems in the filed of stability analysis of differential equations systems and their numerical methods. In the second chapter, we investigate the rational approximations to the function exp(z ), which is correlated with the stability analysis of numerical methods for differential equations. A class of arbitrarily high order rational approximations with two parameters to the function exp(z ) is given, and the necessary and sufficient conditions of A -Acceptability and L-Acceptability are obtained. The third chapter is concerned with neutral DDEs. On the one hand, the nonlinear stability of general linear methods for a class of neutral DDEs is investigated. In particular, it is proven that all algebraically stable general linear methods can preserve the stability of the underlying systems. On the other hand, the system stability of a class of implicit nonlinear neutral multi-delay differential equations is investigated. In the fourth chapter we discuss the stability examination of linear DDEs system. We approximate the roots of the characteristic equations of linear DDEs by discretizing the infinitesimal generator of the solutions operator semigroup of linear DDEs by θ-method, and then we get the result of the asymptotic stability of DDEs by the location of the rightmost characteristic root of DDEs. Moreover, we discretize the abstract Cauchy problem by θ-method, and obtain the numerical solution of linear DDEs system..