Multiscale Analysis To A Class Of Impulsive Differential Equation

Impulsive differential equations have the characteristics of continuous and discrete. Usually the solution near a pulse point is discrete. We use multiscale method to form a asymptotic solution of a class of impulsive differential equations. Mainly use the boundary layer function to modify solutions near the pulse point, looking forward to get a better solution which maybe continuous or even smooth. To simplify, we discuss impulsive differential systems with only one pulse point.We start our discussion with a class of impulsive differential equations with initial-value prob-lem. Using singularly perturbed theory, we extend the considered impulsive differential equation into a singularly perturbed equation with infinite initial values. Using the boundary layer function method to construct the continuous asymptotic solution, and presenting its remainder estimation, which means that this asymptotic solution is an excellent approximation to the original equation. Next we divide the original equation into two problems. We extend the considered impulsive differ-ential equations into singularly perturbed equations with infinite initial values from both sides, then construct the smooth asymptotic solution. We present a example to demonstrate that this method is valid. In order to demonstrate this method has universal applicability, we continue our discuss with a kind of impulsive differential equations with time-lag. Using the same method we can construct the smooth asymptotic solution in the entire domain. In the last, as an extension of a class of impulsive d-ifferential equations, we study about a certain kind of singularly perturbed equation with impulse. We divide the original problem into two semi-linear two-point boundary problems. By solving the two singularly perturbed equations we construct the continuous asymptotic solution in the entire domain.