一类脉冲微分方程的多尺度研究

发布时间：2018-03-31 14:53

本文选题：奇异摄动　切入点：脉冲微分方程　出处：《华东师范大学》2015年硕士论文

【摘要】：脉冲微分方程兼具连续和离散的特征,其在脉冲点附近的解通常是不连续的.本文用多尺度方法构造了一类脉冲微分方程的近似解.主要方法是利用奇摄动理论中的边界层函数来修正脉冲点附近的解,以期得到性质较好的连续甚至光滑渐近解.为了简明起见,本文所讨论的方程均只含一个脉冲点.我们从一类脉冲微分方程的初值问题出发.运用奇异摄动理论,从单侧将原脉冲微分方程扩充成具有无穷大初值的奇掇动问题,利用边界层函数法,构造了原问题的连续渐近解,并给出了相应的余项估计,表明了得到的渐近解是原问题很好的近似；进一步,我们将原问题分成两个子问题,从双侧扩充成具有无穷大初值的奇摄动问题,构造出了原问题的光滑渐近解,并通过实例验证了结果的正确性；为了说明上述方法具有普遍适用性,接下来讨论一类含时滞的脉冲微分方程,运用前面的方法可构造出在整个定义域上的光滑渐近解；最后作为脉冲微分方程问题讨论的深入,研究了一类奇摄动脉冲微分方程,通过对原问题适当分解化成了两个奇摄动两点边值问题,通过解两个奇摄动问题得到脉冲微分方程在整个定义域上的连续渐近解.
[Abstract]:Impulsive differential equations have both continuous and discrete characteristics, and their solutions near impulsive points are usually discontinuous.In this paper, the approximate solutions of a class of impulsive differential equations are constructed by using multi-scale method.The main method is to use the boundary layer function of singularly perturbed theory to correct the solution near the pulse point in order to obtain a better continuous and even smooth asymptotic solution.For the sake of brevity, the equations discussed in this paper contain only one pulse point.We proceed from the initial value problem of a class of impulsive differential equations.By using singular perturbation theory, the original impulsive differential equation is extended from one side to the singularly ejected problem with infinite initial value. By using the boundary layer function method, the continuous asymptotic solution of the original problem is constructed, and the corresponding remainder estimates are given.It is shown that the obtained asymptotic solution is a good approximation of the original problem. Furthermore, we divide the original problem into two sub-problems, extend from two sides to a singularly perturbed problem with infinite initial value, and construct the smooth asymptotic solution of the original problem.In order to prove that the above method is universal, a class of impulsive differential equations with time delay is discussed, and the smooth asymptotic solution on the whole domain can be constructed by using the previous method.Finally, as an in-depth discussion of impulsive differential equations, a class of singularly perturbed impulsive differential equations is studied. By properly decomposing the original problem into two singularly perturbed two-point boundary value problems,By solving two singularly perturbed problems, the continuous asymptotic solutions of impulsive differential equations in the whole domain are obtained.
【学位授予单位】：华东师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175

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