The Numerical Stability Analysis Of Delay Integro-differential Equations

Nondelay integro-differential equations (IDEs) arise widely in many fields such as Physics, Biology, Medical Science, Engineering, Economics etc. Up to now, the numerical and analytical solutions have been studied for more than twenty years. Numerous excellent achievements have been presented in all kinds of literature and have been applied in practical problems. Recently, with the development of the computational implementation and the theoretical analysis of delay system, it's shown that it would be more efficient to use DIDEs to simulate practical problems in the science engineering. Thus, many researchers have turned their attention to the numerical computation of DIDEs. As for the theoretical analysis of numerical methods, Lyapunov functions, Halanay inequality are used to study the stability of all kinds of new methods, and classical Lipschitz condition is used to study the convergence of the methods. Comparatively, DIDEs is a new field, and hasn't been advanced developed, so there is much of creative work to be done. To this paper, I discussed the stability of Runge-Kutta methods and Multi-Runge-Kutta methods for DIDEs.We focus on the research work as follows. Firstly, the paper introduces stability of linear DIDEs. Secondly, with regard to BDIDEs, I discuss the stability of the system and the corresponding numerical methods. In addition, as for NDIDEs, new results of stability have been given. Finally, recurring to numerical test, I present the error analysis of Runge-Kutta methods for DIDEs.In theory, the above research has enriched the connotation of stability of DIDEs. In practice, it provides the theoretical ground for constructing efficient practical methods/...