Stability Analysis Of Numerical Methods For Nonlinear Functional Differential And Functional Equations

Functional differential equations(FDEs) arise widely in the fields of biology,control theory, physics, chemistry, economics and so on. It is meaningful to in-vestigate the theory and application of numerical methods for FDEs. In recent 30years, the theory of computational methods for FDEs has been widely discussed bymany authors and a great deal of results have been found. Functional di?erentialand functional equations(FDFEs) are a class of hybrid systems that are more com-plex than that of FDEs. In particular, functional di?erential equations of neutraltype can be transformed into FDFEs. The numerical stability analysis of neutralfunctional di?erential equations has been studied extensively in recent years. Asfor the systems of FDFEs, the asymptotic stability of numerical methods for linearFDFEs has been discussed by several authors. However, little attention has beenfocused on the nonlinear systems of FDFEs. For these reasons, the present paperis devoted to the stability analysis of numerical methods for nonlinear FDFEs.Runge?Kutta methods and general linear methods are fully discussed. The mainwork of this paper are:(1) Applying Runge?Kutta methods to solve the functional di?erential and func-tional equations of the class D(α,β1,β2,γ1,γ2,δ), the results show that Runge-Kutta methods of (k, l)?algebraic stability are stable and asymptotic stableunder suitable conditions. Numerical tests are given to confirm the theoreticalresults.(2) The more extensive methods of general linear methods are adapted for solvinga class of D(α,β1,β2,γ1,γ2,δ), a series of stable and asymptotic stable conditionsare derived. Numerical tests are given to demonstrate the theoretical results .