Numerical Methods For Impulsive Delay Differential Equations

From the last decade of last century to now, the qualitative theory of impulsive delaydi?erential equations has developed rapidly. But in fact the solving of the simple linearimpulsive delay di?erential equations is very complex. So when people can't get theexplicit forms of the analytical solutions, numerical simulation breaks a feasible way forpeople.The most di?cult thing in studying the numerical methods for impulsive delay dif-ferential equations is choosing the numerical nodes. Due to the variety of impulse instantsand the dependence on the"history"brought by delay, the comprehensive dispose be-comes more di?cult. Poor means in dealing with the nodes can lower the convergenceorders of numerical methods and may bring a result that these methods can't hold theanalytical properties of original systems.This paper firstly considers a class of linear impulsive delay di?erential equations,which have fixed impulse instants and one delay. Using an existing method to deal withthe impulse instants, we give a class ofθ-methods with fixed stepsizes and the conver-gence orders of theseθ-methods are one. If the stepsizes, coe?cients of these equationsand the parameterθsatisfy a condition, then theseθ-methods hold the exponential sta-bilities of original systems. Secondly, a special class of nonlinear impulsive delay di?er-ential equations is studied, of which the impulse intervals and the delay are equal. Forthis system, this paper o?ers a new method to deal with the impulse instants, and by theRunge-Kutta methods with order p, a class of new Runge-Kutta methods with order pfor these systems are given, the prove of which is achieved by transforming these sys-tems into systems of ordinary equations with increasing dimensions. And a conditionamong the stepsizes, the coe?cients of the equations and the parameters of Runge-Kuttamethods to make the methods maintain the exponential stability of original systems areobtained. Moreover, in the numerical experiments, the data of the global truncation er-ror and the error curves clearly display the convergence orders of these methods, and thefigures directly show the stability of numerical solutions.