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高阶常微分方程的仿射周期解
Affine-periodic Solutions for Higher Order Ordinary Differential Equations
【作者】 徐飞；
【导师】 李勇；
【作者基本信息】 吉林大学 ， 应用数学， 2021， 硕士
【摘要】 本文主要研究高阶常微分方程仿射周期解的存在性和唯一性问题.仿射周期问题是近些年来人们关注的动力系统中的一类新的课题,在数学方面具有重要的理论价值,在刚体动力学、台风动力学、螺旋波、Bloch波等方面也具有重要的实际应用价值.全文总计五章.第一章主要介绍关于仿射周期解的历史背景和研究现状,以及关于解方程的一些问题.第二章主要介绍并证明一类用于求解非线性方程的单调迭代技术.第三章是本文的主要创新点,证明了一类适用于在仿射周期边值条件约束下的高阶常微分算子的最大值原理.第四章运用第三章建立的最大值原理和第二章介绍的单调迭代技术建立了一类高阶常微分方程仿射周期解的存在性.第五章对本文作了简要的总结,以及未来的展望.文末主要包括本文的参考文献、作者简介、硕士在读期间发表的学术论文以及致谢.
【Abstract】 This dissertation mainly studies the existence of affine-periodic solutions to higher order ordinary differential equations.Affine-periodic solutions is a new mathematical topic in dynamical system,and it is of great significance from the perspective of pure mathematics.In addition,from the point of practical applications,it plays an important role in the dynamics of rigid bodies,dynamics of typhoon,spiral wave,Bloch wave and so on and so forth.This dissertation is divided into five chapters.The first chapter includes some history and background of affine-periodic solutions,and some problems of solving equations.The second chapter introduces and proves a kind of monotone iterative technique for solving nonlinear operator equation.The third chapter obtains a type of maximum principle for higher order ordinary differential operator with respect to affine-periodic boundary conditions.This is the main innovation of this dissertation.The fourth chapter applies the maximum principle in the third chapter and the monotone iterative technique in the second chapter to the existence problem of affine-periodic solutions for higher order ordinary differential equations.The fifth chapter gives a summary and outlook for this dissertation.The end of this dissertation includes some references,a brief introduction of the author,some papers of the author published,and thanks.