论文标题:非线性发展方程精确解的广义分离变量方法
Generalized Separation of Variables for Solving Exact Solutions of Nonlinear Evolution Equations
论文作者
论文导师 特木尔朝鲁,论文学位 硕士,论文专业 计算数学
论文单位 内蒙古工业大学,点击次数 109,论文页数 50页File Size1553K
2007-05-01论文网 http://www.lw23.com/lunwen_458718927/
Generalized separation of variables;; Wu"s method;; Lie-B(a|¨)cklund symmetry;; Finite-dimensional invariant space
本文根据Titov和Galaktionov提出的分离变量法的思想和著名数学家吴文俊提出的吴方法，以符号计算系统(Mathematica)为工作平台，研究了具有物理、力学背景的非线性发展方程的精确解和线性常微分方程的Lie-B(?)cklund对称。 第一章中作为预备知识简单介绍了对称理论和吴方法，并给出了一些基本概念和理论，如变换群、无穷小算子、算子延拓、首次积分、微分方程在对称下不变的判别准则和微分特征列集算法。 第二章中作为本文的主要结果具体给出了广义分离变量算法的步骤，在此基础上推广了方程左侧的形式且给出了证明，从而扩大了算法的应用范围。并且将算法应用于二阶和三阶的非线性发展方程，得到了方程的一些新解，同时找到了常微分方程的三类常见的线性不变空间，其中用到了吴方法和我们编制的软件包。 第三章中讨论了基于算法产生的两个互逆问题，即(1)构造在已知的微分算子下不变的有限维线性函数空间；(2)讨论拥有相同线性不变空间的微分算子的一般表达式。 第四章中列出了编制的程序代码以及其应用方法。 第五章中对算法中遇到的问题进行总结的同时提出了新的研究任务。
In this dissertation, by applying the ideas of generalized separation of variables pro-posed by Titov and Galaktionov and Wu"s method proposed by famous mathematicianWu Wentsun, we considers the calculation of exact solutions to nonlinear evolution arisingfrom the field of mechanics and physics and the algorithm is implemented in symboliccomputation software Mathematica. In chapter 1, as a preparative knowledge, the main results about symmetry theoryand Wu"s method are introduced, in addition some fundamental definitions, such as trans-formation group, infinitesimal operator, prolongation, first integral, infinitesimal criterionfbr invariance of an ODE, characteristic set are recalled. In chapter 2, as the main result obtained in this paper, a generalized algorithm ofseparation variables is given and as an application of the method, some two and three-order nonlinear evolution equations are solved and some new and general solutions areobtained. Consequently the form of the equations suited to the algorithm is extended. In chapter 3, the problems of constructing finite-dimensional functional linear invari-ant spaces under a given differential operator as well as an inverse problem concerning thedescription of all operators possessing a given invariant space are investigated. In chapter 4, the original code of implemented program and the operating instructionare given. In chapter 5, a conclusion remark and many new tasks are presented.
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