【作者】 曾昕；

【导师】 张鸿庆；

【作者基本信息】 大连理工大学 ， 计算数学， 2005， 硕士

【摘要】 本文根据数学机械化的思想,在导师张鸿庆教授“AC=BD”理论的指导下,研究在流体力学、空气动力学、等离子体物理、生物物理和化学物理等现代科学技术中引出的非线性偏微分方程的若干求精确解的方法。第一章介绍了数学机械化的思想与应用的情况;回顾了孤立子研究的历史与发展以及非线性偏微分方程精确解的若干构造性方法,同时介绍了一些关于该学科领域的国内外学者所取得的成果。第二章在“AC=BD”统一理论框架下考虑非线性偏微分方程(组)精确解的构造。给出了“AC=BD”理论的基本思想和应用,通过具体的变换给出了构造C-D对的算法。第三章主要介绍了我们推广的一种直接求解方法--广义代数方法。以(2+1) 维色散长波方程为例,说明了广义代数方法具体的应用。推广后的方法可以获得非线性偏微分方程(组)的更多类型的精确解(孤波解、类孤波解、周期解、类周期解、有理解)。第四章考虑非线性偏微分方程的Painleve性质和Backlund变换。介绍了Painleve奇性分析的一般原理,利用WTC方法证明了(2+1) 维Boussinesq方程具有Painleve性质,并经截断展开原理获得了方程的Backlund变换;对Backlund变换作了简单介绍,通过对(2+1) 维Boussinesq方程的种子解作适当的未知函数替换,进一步发展了Backlund变换,并得到了方程形式丰富的精确解(类孤子解,有理解)。更多还原

【Abstract】 In this dissertation, by applying the ideas of the mathematics mechanization, under the instruction of the AC=BD theory of Professor Zhang Hongqing, considers some methods seeking exact solutions for the nonlinear partial differential equation(s) arising from the fields of fluid mechanics, aerodynamics, plasma physics, biophysics and chemical physics.Chapter 1 of this dissertation is devoted to investigating the theory and application of mathematics mechanization; Reviewing the history and development of the soliton theory and the construction of the Nonlinear partial differential equation. In addition, some achievements on the subject domestic and abroad are presented.Chapter 2 concerns the construction of exact solutions of nonlinear partial differential equation(s) under the uniform frame work of AC=BD theory. The basic theory and application about AC=BD model and the construction of the operators of C and D are introduced.Chapter 3 is devoted mainly to generalized algebraic method, which is a direct method. the generalized algebraic method is shown to solve (2+l)-dimensional dispersive long wave equations, and which can obtain abundant exact solution(including solitary solutions, soliton-like solutions, periodic solutions, periodic-like solutions and rational solutions) of nonlinear partial differential equation(s).Chapter 4 mainly deals with the Painleve property and Backlund transformation of nonlinear partial differential equation. The general theory of Painleve singular analysis is discussed, (2+l)-dimensional Boussinesq equation’s Painleve property is shown to pass the Painleve test by using WTC method, and its Backlund transformation is obtained through Painleve truncating expansion; Backlund transformation is discussed simply and is further expanded, (2+1)-dimensional Boussinesq equation’s abundant exact solution(including soliton-like solutions, rational solutions) is obtained by replacing the seed solution with unknown function.更多还原