In Several Types Of Delay Differential Equations Numerical Methods Of Stability And Convergence Analysis

Delay differential equations (DDEs) arise widely in Automatic Control, Biology, Medical Science, Aviation, Economics and so on. Neutral delay integro-differential equations (NDIDEs) and singularly perturbed Volterra integro-differential equations (SPVIDEs) are special subclass of DDEs. As far as we know, there is no work on delay-dependent stability of NDIDEs and numerical methods for NDIDEs, and convergence of numerical solutions for SPVIDEs. Therefore, it is meaningful to investigate these problems. In this paper, we research the delay-dependent stability of NDIDEs and numerical methods for NDIDEs, the error behaviour of numerical methods for SPVIDEs and delay-independent stability of numerical methods for NDIDEs. Our main results in the thesis are as follows:(1) We analysis the delay-dependent stability for the real coefficient linear test equations of NDIDEs. We obtain the delay-dependent stability region of the real coefficient linear test equations of NDIDEs.(2) We analysis the trapezium rule for the real coefficient linear test equations of NDIDEs. We prove that the trapezium rule can preserve the delay-dependent stability of the considered test equations. We also get the delay-dependent stability of the continuous problems, the semi-discrete problems and the fully discrete problems of linear NDPDEs.(3) We analyze the error behaviour of linear multistep methods for SPVIDEs. We obtain the global error estimates of A(α) -stable and strictly-stable linear multistep methods.(4) We analysis the Runge-Kutta methods for NDIDEs. We prove that A-stable Runge-Kutta can preserve the delay-independent stability of NDIDEs.Numerical experiments also confirm our theoretical analysis.