【作者】 宋丽娜；

【导师】 张鸿庆；

【作者基本信息】 大连理工大学 ， 应用数学， 2009， 博士

【摘要】 非线性微分方程一直以来都是备受关注的研究对象.本文基于数学机械化思想,以计算机符号计算软件为工具,围绕非线性微分方程这一课题开展了三个方面的研究:非线性发展方程的精确解;非线性分数阶微分方程的近似解;非线性微分方程的群分析.本文由以下五章组成:第一章介绍本文所涉及的学科:数学机械化、孤立子理论、分数微积分和分数阶微分方程、微分方程群理论分析的历史发展及现状,同时介绍了国内外学者在这些领域所取得的成果.最后,介绍了本文的主要工作.第二章介绍本文的理论基础:AC=BD理论和微分伪带余除法,并且阐述在AC=BD这一理论框架下的微分方程精确解的构造问题.第三章是在第二章理论的指导下,基于将非线性发展方程精确求解代数化、算法化、机械化的思想,提出两种求解非线性发展方程精确解的有效算法:第一种是广义的有理形式展开法,该方法是对已有的有理形式展开法的扩展,本章以高维耦合Burgers方程为例,验证了该方法的有效性;第二种是基于变系数Korteweg-de Vries（KdV）方程的子方程法,该方法用变系数KdV方程取代常微分方程作为辅助方程,本章应用该方法获得3+1维potential-YTSF方程的精确解.第四章是关于非线性分数阶微分方程的近似求解.首先介绍了本章涉及到的分数微积分的定义及其性质,然后介绍了非线性分数阶微分方程近似求解的方法及其应用.本章应用变分迭代法、Adomian分解法和同伦摄动法获得非线性分数阶Sharma-Tasso-Olever（STO）方程的有理近似解,并且通过对具体的数值、绝对误差和图形的分析,阐述这三种方法的自身特征、有效性和可靠性.本章将用于求解整数阶微分方程精确解和数值解的同伦分析法应用到非线性分数阶Benjamin-Bona-Mahony-Burgers（BBM-Burgers）方程中,获得其近似解.第五章主要研究非线性微分方程的群分类和守恒律分类.首先介绍标准型和符号计算软件Maple中的软件包K_IFS_IMP,此软件包可以化超定的微分方程组为标准型.本章成功地利用软件包R_IFS_IMP获得非线性变系数电报方程,f（x）u_tt=（H（u）u_x）_x+h（x）K（u）u_x的群分类,并且给出在等价变换群下该方程的守恒律分类.更多还原

【Abstract】 The nonlinear differential equation remains a major concern all the time. In the dissertation, under the guidance of mathematics mechanization and by means of symbolic-numeric computation software, the following problems around the nonlinear differential equation are discussed as follows: the exact solutions of nonlinear evolution equations; the approximation solutions of nonlinear fractional differential equations; the group analysis of nonlinear differential equations.The paper consists of the following chapters.Chapter 1 is devoted to reviewing the history and development of the mathematics mechanization, soliton theory, fractional calculous, fractional differential equation, and group analysis of differential equations. Some works and achievements on these subjects involved in this dissertation are presented at home and abroad. Finally, our main works are listed.Chapter 2 introduces the basic theories of AC = BD and pseudo-differential division with remainder, as well as the construction of exact solutions of differential equations under the instruction of the theory of AC = BD.Based on the theories in Chapter 2 and the idea of algebraic method、algorithm reality and mechanization for solving nonlinear evolution equations, Chapter 3 presents two kinds methods for obtaining the exact solutions of nonlinear evolution equations: the one is the generalized rational expansion method, which extended the known rational expansion method, the chapter takes the Burgers equation for example to illuminate the effectiveness of the method; the other is the variable coefficient Korteweg-de Vries （KdV） equation-based sub-equation method, which takes the variable coefficient KdV equation substituting ordinary differential equation as the sub-equation, and the exact solutions of the 3+1-dimensional potential-YTSF equation are obtained by the method.Chapter 4 is on the approximation solutions of nonlinear fractional differential equations.The first section is to introduce some corresponding basic definiens and properties on the fractional calculous. The rest of the chapter is to present solving methods for the approximation solutions of the nonlinear fractional differential equations and their application.The chapter applies the variational iteration method、the Adomian decomposition method and the homotopy perturbation method to obtain the rational approximation solutions of the nonlinear fractional Sharma-Tasso-Olever （STO） equation, and demonstratesthe significant features, efficiency and reliability of three methods by investigating some numerical results, absolute error and figures. The homotopy analysis which traditionally developed for differential equations of integer order are directly extended to derive approximation solutions of the nonlinear fractional Benjamin-Bona-Mahony-Burgers （BBM-Burgers） equation in the chapter.Chapter 5 is to study the group classification and the classification of conservation law for the nonlinear differential equations. Here, standard form and the package R_IFS_IMP in the symbolic computation software Maple are firstly introduced.. The package is used to reduce over-determined differential equations to standard form. The chapter applies successfullythe package R_IFS_IMP to obtain the group classification of the nonlinear variable coefficient telegraph equations f（x）u_tt = （H（u）u_x）_x + h（x）K（u）u_x, and gives the classification of conservation law with respect to the group of equivalence transformations.更多还原