Neutral Delay Stability Of The Method Of Integral Differential Equations

Delay integro-differential equations (DIDEs) arise widely in the fields of Physics, Engineering, Biology, Medical Science, Aviation, Economics and so on. The theory of computational methods is very important in DIDEs. Recently, many scholars have pay careful attention to it. Neutral delay integro-differential equations (NDIDEs) is special subclass of DIDEs. In this paper, we research the asymptotic stability of NDIDEs and numerical methods for NDIDEs.Recently, there are many results of the stability of numerical methods for DIDEs. But, there are few results of the stability of the Runge-Kutta methods and the multistep Runge-Kutta methods of NDIDE s and the linear multistep methods of Pouzet type for DIDEs. Therefore, it is significant to study the stability of these three classes of numerical methods. In this article, we systematically study the stability of these three classes of numerical methods. The main results in the thesis are as follows:(1) We investigate the asymptotic stability of the Runge-Kutta methods of NDIDEs. It is proven that A-stable Runge-Kutta methods can preserve the asymptotic stability of the underlying linear systems.(2) We study the asymptotic stability of the multistep Runge-Kutta methods of NDIDEs. It is shown that A-stable multistep Runge-Kutta methods can preserve the asymptotic stability of the underlying linear systems.(3) We analyze the asymptotic stability of the linear multistep methods of Pouzet type for DIDEs. It is proven that A-stable and strongly zero-stable linear multistep methods of Pouzet type can preserve the asymptotic stability of the underlying linear systems.(4) Numerical examples and numerical experiments are given for checking the stability of the linear multistep methods and the Runge-Kutt a methods, which confirm the theoretical results obtained in this paper.