Lie对称在若干非线性偏微分方程组边值问题中的应用

发布时间：2018-06-26 04:45

本文选题：对称 + 微分特征列集算法　；参考：《内蒙古工业大学》2017年硕士论文

【摘要】：自然科学和工程技术中的很多问题本质上就是微分方程,而偏微分方程(组)(简称为PDEs)是微分方程研究的主体,特别是非线性PDEs(简称为NLPDEs),所以求解NLPDEs的研究具有重要的意义.由于非线性方程本身的比较复杂,所以求解具有一定的难度.为了求解PDEs人们提出了众多求解方法,但还没有统一而系统的方法包揽各种解的求解,并且这些方法具有各自的适用范围.从而研究求解方法仍是数学、物理、力学学科中的基础性问题,特别是现有方法的改进、总结归纳、加深认识、接纳优点、摒弃缺陷,尤为必要,是发现新方法的前提.众多方法中Lie对称是通用性最好的方法,它以众多传统方法为其特例.目前PDEs对称理论和方法在数学、物理和力学等学科中得到了广泛的应用.本文将基于微分特征列集算法,对Lie对称方法和对称分类在NLPDEs边值问题中的应用进行研究.具体研究内容有:第一章,着重综述了对称方法的发展现状和在PDEs的研究中的重要性,并介绍了微分特征列集算法、龙格-库塔法和同伦摄动法.第二章,通过有效结合对称方法和数值计算方法(即龙格-库塔法),求解了一个流体力学中的NLPDEs边值问题的数值解.第三章,研究对称分类在NLPDEs边值问题中的应用,具体计算了2个流体力学中的NLPDEs边值问题的对称分类,并对其进行了求解.步骤如下:(1)基于微分特征列集算法,分析确定了含参数的NLPDEs边值问题的对称分类,并根据方程参数的不同取值,分类确定方程的主对称和扩充对称.(2)利用确定的扩充对称将所研究的NLPDEs边值问题约化为ODEs初值问题.(3)借助Mathmatica符号系统,求解了ODEs初值问题的数值解.第四章,通过将对称方法和近似解析解方法(即同伦摄动法)有效的结合,求解了2个NLPDEs边值问题.先利用对称方法把NLPDEs边值问题约化为ODEs初值问题,再利用同伦摄动法对其进行求解,得到了近似解.最后利用数值方法得到了数值解,并与近似解进行比较,验证了近似解收敛于数值解.最后总结文章所研究的内容,并对下一步的相关研究进行了展望.
[Abstract]:Many problems in natural science and engineering technology are essentially differential equations, and partial differential equations (PDEs) (referred to as PDEs) are the main research subjects of differential equations, especially nonlinear PDEs (NLPDEs), so the study of solving NLPDEs is of great significance. Because of the complexity of nonlinear equations, it is difficult to solve them. In order to solve PDEs, many methods have been proposed, but there are no unified and systematic methods to solve all kinds of solutions, and these methods have their own scope of application. Therefore, it is necessary to study the solution method in mathematics, physics and mechanics, especially the improvement of the existing methods, to sum up, deepen the understanding, accept the advantages and abandon the defects, which is the premise of finding the new method. Lie symmetry is the most versatile method among many methods, and it takes many traditional methods as its special case. At present, PDEs symmetry theory and method have been widely used in mathematics, physics and mechanics. In this paper, the application of lie symmetry method and symmetric classification to NLPDEs boundary value problem is studied based on differential characteristic sequence set algorithm. The main contents are as follows: in the first chapter, the development of symmetric methods and their importance in the study of PDEs are reviewed, and the differential characteristic set algorithm, Runge-Kutta method and homotopy perturbation method are introduced. In chapter 2, the numerical solution of a NLPDEs boundary value problem in hydrodynamics is solved by combining the symmetric method with the numerical method (Runge-Kutta method). In chapter 3, the application of symmetric classification in NLPDEs boundary value problem is studied. The symmetric classification of two NLPDEs boundary value problems in hydrodynamics is calculated and solved. The steps are as follows: (1) based on the differential characteristic sequence set algorithm, the symmetric classification of NLPDEs boundary value problems with parameters is analyzed and determined according to the different values of the equation parameters. Classification determines the principal symmetry and extended symmetry of the equation. (2) the NLPDEs boundary value problem is reduced to the ODEs initial value problem by using the deterministic extended symmetry. (3) the numerical solution of the ODEs initial value problem is solved by means of the Mathmatica symbolic system. In chapter 4, two NLPDEs boundary value problems are solved by combining the symmetric method with the approximate analytical solution method (that is, the homotopy perturbation method). The NLPDEs boundary value problem is reduced to an ODEs initial value problem by using the symmetric method, and the approximate solution is obtained by using the homotopy perturbation method. Finally, the numerical solution is obtained by numerical method, and compared with the approximate solution, it is verified that the approximate solution converges to the numerical solution. Finally, the paper summarizes the content of the study, and prospects for the next related research.
【学位授予单位】：内蒙古工业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175.29

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