Numerical Oscillation Analysis For Two Kinds Of Impulsive Differential Equations

Systems described by differential equations with piecewise continuous arguments exist in signal processing,control theory and biomedical infectious disease models.The research of this kind of equations has a good application prospect.The occurrence of many life phenomena and the management and optimal control of some life phenomena are not a continuous process,which cannot be simply described by differential or difference equations.The most outstanding characteristic of impulsive differential equations is that they can fully consider the influence of instantaneous burst phenomenon on the system state and reflect the changing law of things more deeply and accurately.With the rapid development of science and technology,people are more and more aware of the importance of impulsive differential equations and their value in practical application.In real life,oscillation is a common phenomenon,for example,the shaking of the train,the vibration of the steam turbine generator set,the waves of the sea and the earthquake under the action of the crustal structure vibration.etc.Since many models in life can be converted into delay differential equations models,it is of great significance to study the oscillation of delay differential equations.Due to the addition of pulse and piecewise constant arguments,the delay differential equations become more complicated and difficult.At present,there are relatively few literatures to study the oscillation of numerical solutions of impulsive differential equations with piecewise constant arguments.For the mixed type impulsive differential equations with piecewise continuous arguments,using the ?-methods,difference equation is obtained.By discussing the roots of the characteristic equation,we obtain the analytical solutions and numerical solutions of this equation,The sufficient conditions for oscillation and non-oscillation of numerical solutions and analytical solutions are obtained.Moreover,we obtain conditions under which ?-methods can preserve the oscillation and non-oscillation for mixed type impulsive differential equations with piecewise continuous arguments.To confirm the theoretical results,the numerical examples are given.For the nonlinear impulsive differential equations with piecewise continuous arguments,difference equations are obtained by using the ?-method,the theoretical results for oscillation of numerical solutions is obtained.Moreover,we obtain conditions under which ?-methods can preserve the oscillation for nonlinear impulsive differential equations with piecewise continuous arguments.To confirm the theoretical results,the numerical examples are given.