非线性波、符号积分及其应用
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摘要
本文以数学机械化思想和导师张鸿庆教授提出的AC＝BD理论为指导,借助于符号计算软件Maple,研究了符号积分和微分方程求解中的一些问题:精确波解、有理积分、微分扩张、初等积分、Liouville定理以及Order函数及其在微分方程中的应用.
第一章介绍数学物理机械化和符号积分相关方面的发展,重点介绍微分方程与计算机代数之间的关系.
第二章和第三章主要考虑了非线性偏微分方程的精确解.首先介绍了AC＝BD模式和C-D对理论.第三章具体研究了这一模式的应用:引入两个相互独立的且含有不同变量的辅助方程,推广了有理展开法.最后举例来展示该方法的有效性,获得了大量的complexiton解.
第四章介绍符号积分相关的理论,无平方因子分解、Hermite约化、Rothstein-Trager算法以及微分扩张.
第五章研究Order函数的理论及其应用.利用Order函数求偏微分方程的精确波解.同时,利用Order函数研究了常微分方程的特殊函数解.最后介绍了Liouville定理及其证明.
In this dissertation, under the guidance of mathematical mechanization and the AC=BD theory put forward by Prof. Zhang Hongqing, and by means of symbolic computation software Maple, some topics on symbolic integration and differential equations are studied, including exact solutions, rational integration, differential extension, elementary integration, Liouville Theorem, Order function and its applications to differential equations.
Chapter 1 is to introduce the related development of mathematical physics mechanization and symbolic integration, emphasizing on the relation between differential equations and computer algebra.
Chapter 2 and 3 are devoted to investigating exact solutions of nonlinear partial differential equations. Firstly, the basic theories of AC=BD model and C-D pairs are introduced. Then, we illustrate them in Chapter 3. We introduce two independent sub-equations with different variables, and study extended rational expansion method. Finally, illustrative examples of complexiton solutions are exhibited.
Chapter 4 is to introduce the relative theory of the symbolic integration, including squarefree factorization, Hermite reduction, Rothstein-Trager algorithm and differential extensions.
In the last chapter, we study basic theory of Order function and its applications. We apply Order function to nonlinear partial differential equations for obtaining many exact solutions, and study the solutions of ordinary differential equations in terms of special functions by Order function. Finally, we introduce the Liouville Theorem and its proof in the paper.
第一章介绍数学物理机械化和符号积分相关方面的发展,重点介绍微分方程与计算机代数之间的关系.
第二章和第三章主要考虑了非线性偏微分方程的精确解.首先介绍了AC＝BD模式和C-D对理论.第三章具体研究了这一模式的应用:引入两个相互独立的且含有不同变量的辅助方程,推广了有理展开法.最后举例来展示该方法的有效性,获得了大量的complexiton解.
第四章介绍符号积分相关的理论,无平方因子分解、Hermite约化、Rothstein-Trager算法以及微分扩张.
第五章研究Order函数的理论及其应用.利用Order函数求偏微分方程的精确波解.同时,利用Order函数研究了常微分方程的特殊函数解.最后介绍了Liouville定理及其证明.
In this dissertation, under the guidance of mathematical mechanization and the AC=BD theory put forward by Prof. Zhang Hongqing, and by means of symbolic computation software Maple, some topics on symbolic integration and differential equations are studied, including exact solutions, rational integration, differential extension, elementary integration, Liouville Theorem, Order function and its applications to differential equations.
Chapter 1 is to introduce the related development of mathematical physics mechanization and symbolic integration, emphasizing on the relation between differential equations and computer algebra.
Chapter 2 and 3 are devoted to investigating exact solutions of nonlinear partial differential equations. Firstly, the basic theories of AC=BD model and C-D pairs are introduced. Then, we illustrate them in Chapter 3. We introduce two independent sub-equations with different variables, and study extended rational expansion method. Finally, illustrative examples of complexiton solutions are exhibited.
Chapter 4 is to introduce the relative theory of the symbolic integration, including squarefree factorization, Hermite reduction, Rothstein-Trager algorithm and differential extensions.
In the last chapter, we study basic theory of Order function and its applications. We apply Order function to nonlinear partial differential equations for obtaining many exact solutions, and study the solutions of ordinary differential equations in terms of special functions by Order function. Finally, we introduce the Liouville Theorem and its proof in the paper.
引文
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[9]Chou S C.Automated reasoning in differential geometry and mechanics using the characteristic set method.I.An improved version of Ritt-Wu's decomposition algorithm.J.Auto.Reasoning,1993,10:161-172.
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