两类非线性演化方程的精确解和守恒律问题研究

发布时间：2018-05-01 12:10

本文选题：Qiao方程 + Gardner-KP方程　；参考：《昆明理工大学》2017年硕士论文

【摘要】：非线性演化方程在数学、物理学、化学、流体力学、振动力学、天体力学、生物学、生态学和财政金融等自然科学和社会科学领域有着广泛的应用.一大批数学和其他领域科学工作者长期以来致力于非线性演化方程的研究,在获得了丰硕成果的基础上不断推进非线性问题的持续发展,为人类社会的生产、生活和科学研究提供了具有重要指导意义和应用价值的结果.本文应用经典的李对称分析方法,研究两类非线性演化方程的解析解和守恒律问题.李对称分析方法是求解非线性偏微分方程和计算方程守恒律的重要理论之一,本文首先介绍了该方法的产生背景和基础理论,给出了论文后续研究过程中用到的一系列关键定理和公式.本文第二部分重点研究Qiao方程,用李对称分析求出了Qiao方程的无穷小生成子,选择无穷小生成子的两种线性组合对原方程进行对称约化,获得了对应的约化方程,并求出了群不变解.在群不变解的基础上,通过采用一种新的构造方式,获得了一批原方程的非群不变解,并通过数值模拟画图对比了群不变解和非群不变解的区别.论文根据特殊的对称,还求了方程的迭代解.最后对应于每个生成子求出了Qiao方程的非局部守恒律公式.本文第三部分研究了Gardner-KP方程.基于李对称分析,首先求出了方程的部分群不变解和迭代解.由于方程有三个独立变量,一次对称约化的方程仍然是偏微分形式,求解较为困难,为此对约化方程再次利用对称分析将方程化为各种形式的常微分方程,并采用幂级数方法,求出常微分方程的幂级数解且证明了解的收敛性,从而获得了一批原方程的幂级数解.对应于第一次约化的每个生成子,本章还求出了相应的非局部守恒律公式.论文最后对研究方法、过程和研究结果进行了必要的总结,并提出了下一步研究工作的目标和方向.
[Abstract]:Nonlinear evolution equations are widely used in the fields of mathematics, physics, chemistry, fluid dynamics, vibration dynamics, astromechanics, biology, ecology, finance and finance. For a long time, a large number of scientists in mathematics and other fields have devoted themselves to the study of nonlinear evolution equations, and on the basis of obtaining fruitful results, they have continuously promoted the sustainable development of nonlinear problems for the production of human society. Life and scientific research provide important guiding significance and application value of the results. In this paper, the classical lie symmetry analysis method is used to study the analytical solutions and conservation laws of two kinds of nonlinear evolution equations. The lie symmetry analysis method is one of the important theories for solving nonlinear partial differential equations and the conservation law of computing equations. In this paper, the background and basic theory of the method are introduced. A series of key theorems and formulas are given. In the second part of this paper, the Qiao equation is studied, and the infinitesimal generator of Qiao equation is obtained by using the lie symmetry analysis. Two linear combinations of the infinitesimal generator are selected to reduce the original equation and the corresponding reductive equation is obtained. The group invariant solution is obtained. On the basis of group invariant solution, a group of nongroup invariant solutions of the original equation are obtained by adopting a new construction method, and the difference between group invariant solution and nongroup invariant solution is compared by numerical simulation drawing. According to the special symmetry, the iterative solution of the equation is also obtained. Finally, the nonlocal conservation law formula of Qiao equation is obtained for each generator. In the third part of this paper, we study the Gardner-KP equation. Based on lie symmetry analysis, the partial group invariant solution and iterative solution of the equation are first obtained. Because the equation has three independent variables, the equation of the first degree symmetry reduction is still a partial differential form, so it is difficult to solve the equation. Therefore, the reduced equation is transformed into ordinary differential equation of various forms by symmetric analysis again, and the power series method is adopted. The power series solutions of ordinary differential equations are obtained and the convergence of the solutions is proved, and the power series solutions of some original equations are obtained. For each generator of the first reduction, the corresponding nonlocal conservation law formula is also obtained in this chapter. Finally, the paper summarizes the research methods, processes and results, and puts forward the goal and direction of the next research.
【学位授予单位】：昆明理工大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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