Asymptotic Solution To A Class Of Impulsive Differential Equation

As one kind of instantaneous mutation phenomena, the impulsive phenomenon exists widespread in practical problems of areas of the modern technology, and its mathematical model is always due to the impulsive differential system. Therefore, the impulsive differ-ential equation, developed in recent years, becomes an important branch of the differential equations. And many properties, such as the existence and uniqueness, stability of the so-lution to the impulsive differential equation have been extensively studied by most scholars. However, the property in the impulsive point is little-noticed. In view of this, the author con-struct a smooth and continuous solution to approximate to the impulsive solution, and provide a new way for the study of impulsive differential equation.In this paper, a class of impulsive differential equation with only one impulsive point is discussed. By the method of step-by-step, the original problem is divided into two prob-lems:left problem and right problem. We expand the right problem to a singularly perturbed problem with infinite initial value by means of singular perturbation theory. By virtue of boundary function method, the continuous formal asymptotic solution is constructed, the ex-istence of the solution is proved and the remainder estimation is presented. Meanwhile, we also prove that the continuous solution to the left problem and the expanded problem are well approximated to the solution to the original impulsive problem.Then, we have improved the property of the solution from continuously approximate to to smoothly approximate to the solution to the original impulsive differential problem. In this part, the impulsive differential equation with only one impulsive point is expanded to a second order singularly perturbed problem. This expanded problem can be discussed as left problem and right problem with infinite initial value each. With the help of the section function and the method of boundary layer function, the formal asymptotic solutions to the left problem and the right problem are constructed respectively and the uniform validity of the formal asymptotic solutions are presented. Since the left problem and the right problem compose an inner layer solution similar to the solution in the space contrast structure, sewing connection method is used to prove the existence of the solution. Meanwhile, we prove that the smooth solution to the expanded problem defined in the whole interval is well approximated to the solution to the original impulsive problem. After theoretical analysis, some specific examples were given to illustrate our main results.