【作者】 汪娜；

【导师】 倪明康；

【作者基本信息】 华东师范大学 ， 运筹学与控制论， 2012， 博士

【摘要】 本文利用边界层函数法、多元缝接法、隐函数定理以及其他方法,构造了四类时滞奇摄动问题解的渐近表达式,得到原问题解存在的充分条件。探讨了内部层出现的区域及原因,证明了形式渐近解的对原问题精确解的一致有效性.并给出其在系统控制理论、化学反应动力学、半导体理论等领域的应用模型.对已有的一些结论做了一定程度的推广全文共分五章.第一章叙述了奇摄动问题的发展过程,引入了与本文研究内容相关的一些概念与定理.着重介绍了本文工作的特色和创新之处.第二章,第五章分别讨论了一类拟线性时滞奇摄动初边值问题和一类具有时滞的捕食-食饵种群问题.利用边界层函数构造了问题的形式渐近解,并利用逐次逼近和压缩映照不动点定理证明了解的存在性,最后结合数值模拟验证了所得结果.第三章着重研究高维时滞奇摄动初边值问题.本章共分两节.我们首先对一类快系统的内部层问题进行研究.再对具有快慢变量的Tikhonov系统的内部层问题进行探讨.利用边界层函数法分别得到了两个系统的近似解,再运用多元缝接法对轨道进行了光滑缝接.不同于第二、五章,本章我们运用隐函数定理证明了在退化解附近,对于充分小的参数μ,原问题解的存在性和渐近解的一致有效性.第四章致力于探论了临界情况下的时滞奇摄动问题.我们分别对弱非线性初值问题的内部层和高维方程组初边值问题的内部层进行研究.构造了原文的形式渐近解.并利用逐次逼近和压缩映照不动点定理得到解的存在性及其近似解的一致有效性.最后举例说明所得结果.更多还原

【Abstract】 By applying the boundary layer function method, patching connections, implicit function theorem and other methods, this dissertation mainly aims to construct the asymptotic expansion of the solution for four kinds of singu-larly perturbed problems. The sufficient conditions to obtain the existence of the solution are also found. We consider the region and reason for the inner layer, and give the proof of the uniformly valid asymptotic expansion. Meanwhile, we give the corresponding application models in the system of control theory, chemical kinetics, semiconductor theory and the other fields. The result obtained in the thesis generalizes and extends the corresponding known results.The whole thesis contains fives chapters. The first chapter introduces concisely the background of the subjects relevant to this dissertation and some definitions and theorems, emphatically introducing the main work to be done in this thesis.Chapter2and Chapter5mainly consider a kind of quasi-linear singu-larly perturbed differential difference equation with initial values and bound-ary values and a class of predator-prey models. By using the boundary layer function method, the asymptotic solutions of these problems are constructed. The existence of the solution to the original problem has also been proved by successive approximation method and the contraction mapping theorem.Chapter3is devoted to the problems for vector singularly perturbed delay-differential equations. This Chapter contains two sections. The first section narrate the internal layer problem of a kind of fast system. In the sec-ond section, we study the singularly perturbed delayed systems of Tikhonov’s type with fast and slow variables. By means of the boundary layer function method, patching connections and implicit function theorem we prove the existence of solution of our problems near the degenerate solution for suffi-ciently small μ and obtain the uniformly valid solution in the whole interval.Chapter4is focused on singularly perturbed delayed boundary value problems in the critical case. We consider the internal layers of a weakly nonlinear initial value problem and high dimensional differential equations respectively. We not only construct the asymptotic expansion of the solu-tion for the original equation but also give the proof of the uniformly valid asymptotic expansion by using successive approximations and the contrac-tion mapping theorem. Meanwhile, we give examples to demonstrate our results.更多还原