The Diagonal Implicit Runge-kutta Methods For Singular Perturbation Functional Differential Equations

The functional differential equations are widely applied to various fields, such aspopulation ecology, genetic problems and epidemiology. Due to the universal applicationsof the equations in actual, the related theoretical analysis and computational issues arepaid great attention by many researchers. As a type of special case, delay singularperturbation equations were also studied by some researchers. For example, Hairer,Wanner, Xiao, Gan et al. constructed various effective numerical methods, and derived theconvergence theory. So far, but, their methods and theory are restricted to the case offix-stepsize. However, in practice, the methods with variable stepsize are more effectivefor solving delay singular perturbations. Especially, it has a broad prospect how toimplement the diagonal implicit Runge-Kutta methods with variable stepsize to solve theequations.This article is concerned with that how to effectively use variable stepsizeRunge-Kutta methods to solve the singular perturbation functional differential equationand the singular perturbation functional integro-differential equation. Firstly, the firstchapter introduces the related research background and the development of singularperturbation problems and variable stepsize Runge-Kutta methods. Secondly, the generalvariable stepsize Runge-Kutta methods are summarized in this chapter. In second chapter,the starting algorithm, the selection of initial stepsize, the rule of automatically adjustingthe stepsize and the procedure of calculation are derived. In third and fourth chapters, thevariable Runge-Kutta methods are future extended to solve the singular perturbation delaydifferential equations and delay integro-differential equations, and the properties of thealgorithm are analyzed by numerical experiments.