Numerical Stability Of Linear Multistep Methods For Linear Neutral Integro-differential Equations With Many Delays

Delay integro-differential equations (DIDEs) arise widely in many fields of the society, such as economics, biology, physics, engineering, aviation and so on. Up to now, there are many results of the stability of numerical methods for DIDEs. DIDEs can describe rules in these fields more accurately. Neutral delay integro-differential equations (NDIDEs) is an important branch of DIDEs. This thesis studies the stability of numerical methods of NDIDEs with multiple delays.This paper is organized as follow.Firstly, we present many applications of delay differential equations and integral equations. We also review the research of both analytic stability and numerical stability of delay differential equation. Moreover, we introduce the background of the problem.Secondly, we are concerned with of linear neutral delay integro-differential equations with many delays. We establish a new result for the distribution of the zeros of its characteristic function; and then obtain a sufficient condition for its delay-dependent asymptotic stability. Based on above, we present a sufficient condition for its delay-independent stability. We also discuss the theory of DIDEs (where Ni=0).Thirdly, we investigate the corresponding numerical stability of linear multistep methods applied to linear neutral delay integro-differential equations with many delays. The linear multistep methods, with the compound quadrature formula, are numerically stable under suitable conditions. In addition, numerical examples and simulation of Matlab are given for checking the stability of the linear multistep methods. And the results confirm the theoretical results in this paper.Finally, The end part concludes studies above, and shows the directions of researching in future.