【作者】 张善卿；

【导师】 李志斌；

【作者基本信息】 华东师范大学 ， 系统分析与集成， 2004， 博士

【摘要】 以物理学中的问题为背景的非线性微分方程的研究是当代非线性科学的一个重要方面。创造和发展非线性微分方程新的求解方法是非线性物理最前沿的研究课题之一。目前，已经存在许多的获得非线性微分方程的精确解的方法。本文对一些求解方法进行了研究，特别是Lie对称方法，分析并改进了前人的理论和算法，并在计算机符号系统Maple上给出了相应的实现。这些理论、算法、实现对非线性微分方程的精确解构造是十分有益的。 我们对已有的求解方法，如双曲正切法、Jacobi椭圆函数展开法、假设法等进行了改进和推广，并利用改进后的方法结合我国著名数学家吴文俊的数学机械化思想，针对一些微分方程获得了一些新的精确解。这些解的发现将有助于弄清物质在非线性作用下的运动规律，对相应物理现象的科学解释起到重要作用。 但是上述方法比较分散、不系统。众所周知，Lie群方法将求解特定类型的微分方程的分散的积分方法统一到共同的概念之下。实际上，Lie无穷小变换方法为寻找常微分方程的闭合形式的解提供了广泛的应用技巧。应用到偏微分方程，Lie方法能够导出对称。找到偏微分方程的对称，可以由此获得其精确解。 目前，对称的概念在数学和物理的研究和发展中扮演着关键的角色。但具体到应用时，Lie群的方法涉及到大量的繁冗计算，因此，设计相关的计算机符号软件包非常有必要。 我们讨论了经典Lie对称和非经典Lie对称计算中的有关理论和算法，分别给出了产生经典Lie对称和非经典Lie对称决定方程组的软件包GDS和NGDS，发现了现行Maple系统上软件包liesymm的一些漏洞。 由于决定方程组是超定的、线性的或非线性的偏微分方程组，完全求解它们非常困难。通过引入对合除法的概念，将它们完备化为内嵌所有可积性条件的一种特殊形式-对合形式，这样有助于求解决定方程组。 对经典对称情形，我们分析和重新描述了计算线性偏微分方程组的最小对合基算法和Janet对合基算法，并给出了各自实现的软件包MiniIB和Janet。将软件包GDS和Janet相结合研究了广义Burgers方程的势对称，得到了其无穷参数的势对称，并利用此无穷参数势对称获得了广义Burgers方程一个新的精确解。 对于非经典对称情形，我们描述和改进了完备化非线性代数偏微分方程组到被动的对合形式的对合特征集算法。这个算法包含了已有的乘子变量法，例如基于Janet除法的Ritt算法和基于Thomas除法的Wu微分特征列算法。最近一些新的对合除法以及算法的相继提出，可明显减少Wu-Ritt特征列算法的计算步骤。基于对合特征集算法，我们给出了具体的实现软件包ICS。通过大量的计算试验，我们分析了此算法对不同等对合除法以及项序的依赖关系，获得了一些试验性的结论，这些结论对今后代数偏微分方程组的对合特征集的计算具有一定的指导和借鉴意义。更多还原

【Abstract】 The nonlinear differential equation (NDE) based on physics is one of the important aspects in the contemporary study of nonlinear science. Exploring and developing new method to solve the NDE is one of the forefront topics in the studies of nonlinear physics. Now there are many methods for finding the exact solutions of NDE. In this paper, not only some methods are studied, especially Lie symmetry method, but also the related theory and algorithms are improved. With computer symbolic system Maple, some packages to implement the important algorithms are presented. The theory, algorithms and implementations are very instructive for constructing the exact solution of NDE.We present and extend some methods such as the tanh-function method, the Jacobi elliptic function method, the ansatz method and so on. Based on the above methods and the theory of mathematics mechanization proposed by famous mathematician Wu Wentsiin, some type of particular exact solutions to nonlinear equations are obtained, which are helpful in clarifying the movement of matter under the nonlinear interactivities and play an important role in scientifically explaining of the corresponding physical phenomenon.However these methods have some common shortcomings: dispersive, unsystematically. It is well known that Lie groups techniques brought diverse integration methods for solving special classes of differential equations under a common concept. Indeed, Lie’s infinitesimal transformation method provides a widely applicable technique to find closed form solutions of ordinary differential equations (ODEs). Applied to partial differential equations (PDEs), Lie’s method can lead to symmetries. Exploiting the symmetries of PDEs, new solutions can be derived.Nowadays, the concept of symmetry plays a key role in the study and development of mathematics and physics. But the application of Lie group method to concrete physical systems involves tedious computations. So it is necessary to design some symbolic packages for it.We discuss methods and algorithms used in the computation of Lie nonclassical symmetries as well as classical symmetries, and provide the symbolic packages GDS and NGDS, which are used to generate the determining system of differential equation’s classical symmetry and nonclassical symmetry respectively. For GDS, some bugs are found in the package liesymm on Maple.Because the determining systems are a linear or nonlinear overdetermined PDEs, it is very hard to solve them completely. After introducing the concept of involutive division, we can complete them to involutive forms which include all integrable conditions and maybe solve them more easily.For the case of classical symmetry, the minimal involutive base algorithm and Janet base algorithm are analyzed and described, and the associated implementations which are named MinilB and Janet are presented and some examples are tested. With GDS and Janet, we study thepotential symmetry of a generalized Burgers equation, for which an infinite parameter potential symmetry and a new exact solution are obtained.For the case of nonclassical symmetry, an involutive characteristic set algorithm(ICS) which reduces a nonlinear algebraic partial differential equation system to passive involution is described and improved. This algorithm converts all the existed methods using the multiplier variable approach such as Ritt’s algorithm based on Janet division and Wu’s algorithm based on Thomas division. Recently some new involutive divisions and algorithms are proposed, which can significantly reduce the computational steps in Wu-Ritt’s characteristic set method for nonlinear algebraic PDEs. Based on the algorithm ICS, a package ICS is designed in Maple for computing involutive characteristic set. By testing many examples, we analysis the dependency of the algorithm ICS for the various orderings and involutive divisions. Some experimental results are obtained which may show a hint for computing involutive characteristic set of arbitrary algebraic partial differential equation systems there更多还原