Neutral Delay Differential Equations Stability

Delay differential equations(DDEs) appear in a wide range of applications such as speciesreproduction, population growth, cybernetics, energy loss in the electric power net, neural net-works and so on. Long time ago , people always believe that there isn't any difference betweenthe numerical treatment to ODEs and DDEs . But that isn't the fact . Actually , the analysis of thestability behavior of a numerical method for DDEs is more complex than for ODEs. As we know,few analytic expression of delay differential equation can be obtained. So the numerical treatmentsof delay differential equation becomes very necessary. Furthermore, numerical stability play animportant role in numerical analysis.Neutral delay differential equation is an important class ofdelay differential system. Because of the neutral item, the numerical treatments of neutral delaydifferential equation is different from delay differential equation. Rosenbrock method is one ofeffective numerical methods for stiff differential equations . Among the methods which alreadygave satisfactory results for stiff equations , Rosenbrock methods are the easiest to program .Inthis paper we analysis the numerical stability of Rosenbrock andθ-method.In Chapter 2 we discuss the NGPLm-stability of Rosenbrock method for NDDEs with manydelays, we show that Rosenbrock method is NGPLm-stable if and only if it is L-stable. InChapter 3 we discuss the NGPG-stability of Rosenbrock method for generalized neutral delaydifferential equations, we show that Rosenbrock method is NGPG-stable if and only if Rosen-brock method is A-stable. In Chapter 4 we discuss the NGPm-stability ofθ- method for NDDEswith many delays, we show thatθ-method is NGPm-stable if and only if it is A-stable.