一类脉冲微分方程的光滑多尺度解

Smooth multi-scale solutions of a class of impulsive differential equations

利用奇异摄动方法,提供了一种以光滑的多尺度解的观点来研究一类脉冲微分方程.通过引入适当的奇异摄动项,定义了相应的奇异摄动边值问题,其对应的退化方程即为原脉冲微分方程.利用边界层函数法和缝接法,构造了该奇异摄动边值问题的光滑多尺度解,并有效地刻画原脉冲微分方程的不连续解,同时也证明了多尺度解的存在性及余项估计.最后,通过实例,验证了文中的主要结果.

This paper studies the impulsive differential equation from the viewpoint of the smooth multi-scale solutions by using a singular perturbation method.By introducing suitable singular per-turbation terms and de fining the singularly perturbed boundary value problems,whose reduced equations are exactly the impulsive differential equation,the smooth multi-scale solutions to the singularly perturbed problems are constructed by applying the boundary layer function method and sewing connection method.By doing so,the discontinuous solutions of the impulsive differential equation are described effectively through the smooth solutions of the singularly perturbed problems.Moreover,the existence and remainder estimation of the multi-scale solutions are demonstrated.Finally,an example is carried out to illustrate the main results.