Convergenge Of Waveform Relaxation Methods For Delay Integro-Differential Equations

Due to the importance of delay integro-differential equations (DIDEs) in many fields, so in recent years much of research to it appeared. For instance, some methods are applied to delay integro-differential equations (DIDEs), to study its convergence and stability, etc. And waveform relaxation methods (WR), a dynamic iterative methods , are proposed in 1980s, because WR methods can decouple complex system, and make the decoupling systems maintain the original system's some properties, it can make parallel computation etc, so more and more scholars begin to pay close attention to this method. In this paper ,WR methods are used to delay integro-differential equations (DIDEs), to discuss its convergence.First of all, delay differential equations'numerical methods are reviewed, meanwhile WR methods are introduced, and the progress at home and abroad of related theory of WR methods are introduced.Secondly, delay integro-differential equations (DIDEs) model is given, and WR methods are used to this equation, thus getting continuous time WR methods, and prove the continuous time WR methods'convergence. Meanwhile perturbed continuous time WR iteration's convergence and the existence and uniqueness of the solution is discussed.Thirdly, in order to get the discrete time WR iteration, we put Runge - Kutta methods to continuous time WR iteration, and prove the discrete time WR methods'convergence. Given that, through the numerical examples and the simulation of Matlab, the validity of the theoretical analysis is further demonstrated.Finally, summarized to the full text, and WR methods are applied to other delay differential equations'achievements is prospected.