扩展的F-展式法及其应用 Alternative Title Extended F-expansion method and its applications to nonlinear evolution equations 冯胜章 Thesis Advisor 周宇斌 2005-05-20 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 硕士 Keyword F-展式法 精确解 周期波解 Abstract 非线性偏微分方程的求解问题，特别是求非线性偏微分方程的精确解，是现代数学和物理学科中一类重要的问题．齐次平衡原则和在其基础上发展起来的F-展式法是最近出现的求非线性发展方程精确解的新方法．该方法易于求得方程的周期波解与孤立波解，是齐次平衡原则的新应用，概括了Jacobi椭圆函数、三角函数、双曲函数展式法等多种方法，易于在计算机上实现，具有广泛的适用性． 本文首先在原来的F-展式法的基础上进行扩展，思想上利用Jacobi函数之间的关系，在解的形式中加入另一个Jacobi函数G，来求解非线性偏微分方程Hamiltonian振幅方程、（2+1）维的MKdV方程，不仅可以得到原来F-展式法求得的解，而且还可以求得许多不同的新解．其次在原来F-展式法的基础上进行修正，在解的形式中 加入函数的导数，来求解一种高次KdV型方程和耦合的非线性Ito方程组，也可以得到原来F-展式法求得的解以及一些新解．扩展的以及修正的F-展式法可以求得方程更一般形式的解，这是以前F-展式法所不能得到的．" Other Abstract It is a class of important problem in mathematics and physics for solving nonlinear PDEs, especially the exact solutions of them. It is a new method to derive the exact solutions of the nonlinear equations by using the homogeneous balance principle and the F-expansion method for recent years. By virtue of this method, the periodic wave solutions and solitary wave solutions can be easily obtained. It is a new application of the homogeneous balance principle, which has included the Jacobi elliptic function, trigonometric function, hyperbolic function expanded method etc. and it is easy to compute by programming. In this paper, by the relations between Jacobi functions, we extend the F-expansion method, namely, add another Jacobi function G to the form of solution. Appling this method to nonlinear evolution equations, such as Hamiltonian amplitude equation and (2+1) dimension MKdV equation, some new solutions can be obtained, including the ones by F-expansion. Second, we modify and extend this kind of method, the varius combined Jacobi function solutions are obtained, which include the existed ones and some new ones." URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/224933 Collection 数学与统计学院 Recommended CitationGB/T 7714 冯胜章. 扩展的F-展式法及其应用[D]. 兰州. 兰州大学,2005.
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