Global Behavior Of Numerical Solutions For Several Classes Of Differential Equations

This paper deals with the global behavior of numerical solutions for differen-tial equations piecewise continuous arguments (EPCA), pantograph equations, a singlespecies population model and a class of nonlinear delay differential equations. Theseclasses of equations have widely applications in many fields, and the analysis of the nu-merical global properties is of important theoretical value and practical significance.The paper presents the background of some applications and research history ofEPCA, pantograph, a single species population model and nonlinear delay differentialequations in detail, and surveys the developments of some global properties of analyticand numerical solutions of these classes of differential equations.Runge-Kutta methods for EPCA are presented. The stability regions for the Runge-Kutta methods are determined. The conditions under which the analytic stability regionis contained in the numerical stability region are obtained.The asymptotical stability of the analytic solution and the numerical methods withthe constant stepsize for the pantograph equations is investigated by using Razumikhintechnique. Especially, the linear pantograph equations with constant coefficients and vari-able coefficients are considered. The stability conditions of the analytic solutions of thoseequations and the numerical solutions of theθ-methods with the constant stepsize areobtained and the sufficiency of Z. Jackiewicz's conjecture is proved.The modified Runge-Kutta method is constructed and it is proved that the modifiedmethod preserves the order of accuracy of the original one. The necessary and sufficientconditions under which the modified Runge-Kutta methods with the variable mesh for thetest pantograph equation are asymptotically stable are given. Especially, the asymptoticalstability of theθ-methods, the Gauss-Legendre methods, the Lobatto IIIA methods andthe Lobatto IIIB methods are improved.The global stability of Runge-Kutta methods with the exponential form for a sin-gle species population is considered. It is shown that the method preserves the order oforiginal methods. Some conditions under which the analytic invariant sets are numeri-cally positively invariant are discussed. It is shown that the implicit Euler method and the two-stage Af(0)-stable methods of order 2 are globally asymptotically stable and thenumerical solutions of these methods monotonically tend to the stable fixed point. Theglobal asymptotical stability of some Runge-Kutta methods for a single species popula-tion with delay term is investigated. At last, the implicit Runge-Kutta methods are solvedby using the Newton method.The oscillation of Lawson numerical methods applied to a class of delay differentialequations is studied. The Lawson numerical methods for nonlinear delay differentialequations are presented. It is shown that these methods preserve their original order. Theconditions under which the Lawson numerical methods preserve or deduce the oscillatoryof the linear delay differential equations and a class of special nonlinear delay differentialequations are given, respectively. The conditions under which the Lawsonθ-methodspreserve the oscillation of nonlinear delay differential equations are discussed.Moreover, at the end of each section, the feasibility of the algorithm and the validityof theoretical derivation are verified through some practical examples.