Stability Of Analytic And Numerical Solutions For Impulsive Differential Equations

As the science and technology progress, the application of impulsivedifferential equations seem more and more important. The analytic solutions oflinear impulsive differential systems are easier to get than nonlinear ones. So it isnecessary to construct a new technique to find the solutions of nonlinear impulsivedifferential equations. In this paper we mainly study the analytic solutions,numerical solutions and numerical solutions preserve the stability for the equations.We divided four parts in this paper as follows:Firstly, we give the study backgrounds of impulsive differential equations,meanwhile, analyze the established works researchers got, and also give theequations that we investigate in this thesis.Secondly, we give an accurate definition about impulsive differential systems,and some different properties between impulsive differential equations （IDEs） andordinary differential equations （ODEs）.Thirdly, we construct a new technique to study the analytic and numericalsolutions’ stability for the linear IDEs. Lead to conclusions about analytic andnumerical solutions’ stability, after then some numerical experiments are also givento prove the conclusion.Lastly, we study the stability use the same method as the linear ones. Asufficient condition is given under which the analytic solution is exponentialasymptotic stable. It is proved that algebraically stable Runge-Kutta methodssatisfying|1-d^T A^-1e|<1can preserve the stability of the equation, and then givesome numerical experiments to illustrate the technique we use is right.