The Asymptotical Stability Of Numerical Methods For Impulsive Differential Equation

The investigations of impulsive differential system are dated from the work of V.D. Mil'man and A. D. Myshkisa in 1960. Fairly rich results have been made for thetheories of impulsive differential equations for more than forty years. However, thesetheories regarded to impulsive differential equations solution was searched in formof analytical expression. Significantly results are presented by V. Lakshmikantham,D. Bainov, P. Simeonov, S. Kostadinov and N. van Minh. However, many impulsivedifferential equations can not be solved in this way or their solving is more compli-cated. From the other side, huge number of practical problems do not need solutionof impulsive differential equations in analytical form, but only need numerical valuesof solution. This is reason that impulsive differential equations can be solved nu-merically. The main results obtained in this dissertation may be summarized as thefollowing:In Chapter 1, we give an introduction to the development of impulsive differentialequations and establish the impulsive differential equation that we investigate in thispaper. The stability condition of the analytic solution of this equation is given.In Chapter 2, the description of impulsive differential system is given, some theo-rems for existence and uniqueness of the solutions of impulsive differential equationsare introduced.In Chapter 3, we establishθ-method for impulsive differential equation and in-vestigate the asymptotical stability of this numerical method. The sufficient and nec-essary conditions of asymptotical stability ofθ-methods are obtained, some numericalexperiments are given, which give a hard evidence for main theorems of this chapter.Especially, an interesting example is presented, we can obtain that the difference ofnumerical methods between impulsive differential equations and the common differ-ential equations.In Chapter 4, we investigate the higher order numerical methods, constructingRunge-Kutta methods with the constant stepsize for impulsive differential equation.The stability conditions of the numerical methods are obtained and the A-stable higherorder Runge-Kutta methods are discussed, we give a relation table of the order and...