Numerical Stability Of Delay Differential Equation

In many realistic models,we need to know some past states of the system,thus it converts to the study of delay differential equation.Delay differential equation is often used in life sciences, control theory,electric system.As we know,few analytic expression of delay differential equation can be obtained.So the numerical treatments of delay differential equation becomes very necessary.Furthermore,numerical stability is an important part in numerical analysis.There are many effective numerical methods for solving delay differential equations.The blockθ-methods have good stability properties without requiting high order detivatives.The method have great potential.Rosenbrock H.H gave the Rosenbrock method in 1963.It is also an effectively numerical method for stiff ordinary differential equations.Among the methods which already gave satisfactory results for stiff equations,Rosenbrock methods are easier to program.In this paper we analyse the numerical stability of blockθ-method and Rosenbrock method.In Chapter 2,we discuss the asymptotic stability of blockθ-method for several kinds of neutral delay system.We show that blockθ-method is NGP_m-stable or NGP_G-stable if and only if it is A-stable.In Chapter 3,we discuss the generalized delay system and equations with many delays. On the premise of asymptotic stability of theory solution,We show the necessary and sufficient condition of numerical stability of Rosenbrock method.