一类发展方程Cauchy问题解曲线演化
发布时间：2018-08-02 18:44
【摘要】：偏微分方程解曲线的演化一直是偏微分方程研究的重要领域和方向。本文首先对一类线性和半线性偏微分方程Cauchy问题解曲线的演化进行研究,得到在给定的光滑条件下,方程初始解曲线上的几何性质可以继承到任意时刻的解曲线上。其次研究KdV方程Cauchy问题解曲线的一些性质的演化,得到了关于单调性,凹凸性和孤立极值点的演化结果。即在给定的光滑条件下,KdV方程初始解曲线上的孤立极值点,单调性和凹凸性可以继承到任意t(29)0时刻的解曲线上。全文共有五个章节:第一章节介绍了KdV方程的研究背景、现状及主要结果,也介绍了一些对于方程演化研究的现状和主要结果。第二章节介绍了相关基本概念和定理。第三章节研究了一类线性和半线性偏微分方程柯西问题解曲线的演化。第四章节研究了KdV方程柯西问题解曲线的演化。第五章节对全文进行总结并对未来研究方向作出展望。
[Abstract]:The evolution of solution curves of partial differential equations has been an important field and direction in the research of partial differential equations. In this paper, we first study the evolution of the solution curves of a class of linear and semilinear partial differential equations for Cauchy problems. Under given smooth conditions, the geometric properties of the initial solution curves of the equations can be inherited to the solution curves at any time. Secondly, the evolution of some properties of the solution curve of the Cauchy problem for KdV equation is studied, and the evolution results of monotonicity, concave convexity and solitary extremum are obtained. The monotonicity and concave convexity can be inherited to the solution curve at any t (29) 0 moment under the given smooth conditions on the isolated extremum of the initial solution curve of KDV equation. There are five chapters in this paper: the first chapter introduces the research background, current situation and main results of KdV equation, as well as the present situation and main results of the evolution of the equation. The second chapter introduces the basic concepts and theorems. In the third chapter, we study the evolution of the solution curve of Cauchy problem for a class of linear and semilinear partial differential equations. In the fourth chapter, the evolution of the solution curve of Cauchy problem for KdV equation is studied. The fifth chapter summarizes the full text and prospects the future research direction.
【学位授予单位】：江苏大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.2
本文编号：2160359
[Abstract]:The evolution of solution curves of partial differential equations has been an important field and direction in the research of partial differential equations. In this paper, we first study the evolution of the solution curves of a class of linear and semilinear partial differential equations for Cauchy problems. Under given smooth conditions, the geometric properties of the initial solution curves of the equations can be inherited to the solution curves at any time. Secondly, the evolution of some properties of the solution curve of the Cauchy problem for KdV equation is studied, and the evolution results of monotonicity, concave convexity and solitary extremum are obtained. The monotonicity and concave convexity can be inherited to the solution curve at any t (29) 0 moment under the given smooth conditions on the isolated extremum of the initial solution curve of KDV equation. There are five chapters in this paper: the first chapter introduces the research background, current situation and main results of KdV equation, as well as the present situation and main results of the evolution of the equation. The second chapter introduces the basic concepts and theorems. In the third chapter, we study the evolution of the solution curve of Cauchy problem for a class of linear and semilinear partial differential equations. In the fourth chapter, the evolution of the solution curve of Cauchy problem for KdV equation is studied. The fifth chapter summarizes the full text and prospects the future research direction.
【学位授予单位】：江苏大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.2
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