【作者】 许琳；

【导师】 许志国；

【作者基本信息】 吉林大学 ， 基础数学， 2023， 博士

【摘要】 近年来,摄动时滞微分方程在生物学、流体力学、工程学、经济学等领域中都有广泛而重要的应用,因此受到众多学者的广泛关注.但时滞的出现为方程的精确求解带来了巨大的困难.为得到一致有效的近似解,人们先后发展了摄动时滞微分方程的平均法、匹配法、多尺度方法等摄动方法,并得到一系列重要研究成果.在本篇论文中,我们将应用奇摄动重整化群方法研究摄动时滞微分方程.全文共分六章,第一章为绪论,对时滞微分方程和奇摄动理论的研究背景做简要概述,并介绍本文主要工作.第二章为预备知识,简要介绍时滞微分方程的谱分解理论和中心流形定理.第三章,考虑单时滞摄动微分方程初值问题,首先应用奇摄动重整化群方法构造摄动问题的近似解,然后证明近似解的一致有效性,最后,通过两个具体例子对本文结果与平均法的结果做数值对比.第四章,进一步考虑具有两个时滞的摄动微分方程的初值问题.首先,对两个时滞可公度情形,通过对时间区间进行更精细更恰当的划分,利用奇摄动重整化群方法构造问题的近似解,并证明其一致有效性.其次,对两个时滞不可公度情形,通过引入近似辅助系统,得到原问题的近似解,并给出了误差估计.最后,通过两个例子说明本文方法的有效性,并与平均法和多尺度方法所得结果做数值对比.第五章,首先对一类非线性项含时滞的弱非线性摄动微分方程初值问题,利用奇摄动重整化群方法得到其近似解,并证明了近似解的一致有效性.其次,对一类线性项和非线性项均含时滞的特殊弱非线性方程,我们结合谱分解理论、中心流形理论和奇摄动重整化群方法,构造了方程的近似解,并得到方程限制在中心流形上约化方程的法形.最后,我们在第六章中简要总结本文主要工作及未来的工作展望.更多还原

【Abstract】 In recent years,perturbed delay differential equations have been widely used in various fields,such as biology,fluid mechanics,engineering,economics,etc.,so they have been widely concerned by many scholars.Due to the emergence of time delay,which brings great difficulties to the exact solution of the equation.In order to obtain uniformly valid approximate solutions,perturbation methods such as the averaging method,matched method and multi-scale method have been developed successively,and a series of important research results have been obtained.In this paper,we apply the singularly perturbed renormalization group method to investigate the perturbed delay differential equations.The full text is divided into six chapters.Chapter 1 is the introduction,which gives a brief overview of the research background of delay differential equations and singular perturbation theory,and introduces the main work of this paper.In Chapter 2,the spectral decomposition theory and center manifold theorem of delay differential equations are introduced.In Chapter 3,we consider the initial value problems of single delay perturbed differential equations.Firstly,by means of singularly perturbed renormalization group method,an approximate solution of the perturbation problem is constructed.Secondly,we proved the uniform validness of the approximate solution.Finally,two examples are given to compare the results of this paper with the results of the averaging method.In Chapter 4,the initial value problem of perturbed differential equations with two delays is further considered.Firstly,for the case of two commensurate delays,we utilized the singularly perturbed renormalization group method to construct the approximate solution and proved the uniform validness of the approximate solution.Secondly,for the case of two incommensurate delays,the approximate solution of the original equation is constructed by introducing an approximate auxiliary system,and the error estimate is given.Finally,two examples are given to illustrate the validness of the proposed method,and the results are numerically compared with those obtained by the averaging method and the multi-scale method.In Chapter 5,firstly,for a class of weakly nonlinear perturbed delay differential equations,the approximate solution is obtained by singularly perturbed renormalization group method,and the uniform validness of the approximate solution is proved.Secondly,for a class of special weak nonlinear equations with delay in both linear and nonlinear terms,we combine spectral decomposition theory,center manifold theory and singularly perturbed renormalization group method to construct the approximate solution,and obtain the normal form of the reduced equation limited to the center manifold.Finally,we present a brief conclusion of the dissertation and the prospect of future work in Chapter 6.更多还原