【作者】 王琪；

【导师】 张鸿庆；

【作者基本信息】 大连理工大学 ， 计算数学， 2006， 博士

【摘要】 本文根据数学机械化思想,以计算机符号数值计算软件为工具,研究了孤立子理论,分数微分方程和混沌系统中的若干问题：1.构造非线性发展方程的精确解及其机械化实现；2.构造非线性分数微分方程的数值解；3.混沌同步及其自动推理算法.第一章介绍了数学机械化,孤立子理论,分数阶微积分和混沌同步研究的历史发展及现状,同时介绍了国内外学者在这些学科领域所取得的成果.第二章介绍了Wu-Ritt微分消元理论,分数阶微积分理论中的几个基本的定义及其性质,AC=BD的基本理论及在这一统一理论框架下考虑非线性发展方程(组)精确解的构造问题.第三章基于将非线性发展方程精确求解代数化,算法化,机械化的指导思想,首次提出了求解非线性发展方程精确解的有理形式展开法,然后以符号计算软件为工具,运用吴消元法,研究了有理形式展开法的各种具体形式及其应用：(1)利用一个新的辅助方程研究了高维耦合Burgers方程,得到了许多新的complexiton解；(2)基于椭圆函数,提出椭圆函数有理展开法,得到了(2+1)-维色散长波方程许多新的有理形式椭圆函数解；(3)基于Riccati方程,提出Riccati方程有理展开法,获得了(2+1)-维Broer-Kaup-Kupershmidt方程许多新的精确解；(4)提出广义椭圆方程有理展开法并研究了Whitham-Broer-Kaup方程,得到了许多新的有理形式精确解；(5)推广了Riccati方程有理展开法,并应用其求解(2+1)-维Burgers方程；(6)提出了多辅助方程有理展开法,应用其构造新型complexiton解；(7)扩展了Riccati方程有理展开法,求得了一类非线性发展方程的非行波解；(8)针对随机微分方程,改进了Riccati方程有理展开法,从而得到了许多新的随机精确解；(9)将有理展开法推广到微分-差分情形,并用其求解了两类Toda方程许多新的精确解.第四章将原本用于求解整数阶微分方程精确解及数值解的Adomian分解法和同伦扰动法推广到分数微分的情形,运用它们研究了一些有重要物理意义的非线性分数阶偏微分方程如分数阶KdV-mKdV方程,分数阶Boussinesq方程,分数阶KdV-Burgers方程,分数阶Kuramoto-Sivashinsky方程,首次获得了一些有实际物理意义的数值解,且给出了一个简单的收敛性判定定理.第五章首先改进了混沌和超混沌系统的广义Q-s(滞后,完全,提前)同步机理和自动推理算法.在符号数值计算软件的帮助下,获得了两个初值不同的Chua系统,超混沌Rossler系统和超混沌Tamasevicius-Namajunas-Cenys系统,以及两个初值不同的广义Henon映射,Henon-like映射和广义Henon映射之间的广义Q-s同步,并使用数值模拟说明了该算法的有效性.然后给出了双向广义部分(滞后,完全,提前)同步定义,并给出了获得该同步的一个机械化算法,以Rossler系统,统一的Lorenz-Chen-Lii系统,超混沌Tamasevicius-Namajunas-Cenys系统为例,通过数值模拟说明了该算法的有效性.更多还原

【Abstract】 In this dissertation, under the guidance of mathematics mechanization and by means of symbolic-numeric computation software, some problems in the theory of soliton, fractional dif-ferential equation and chaotic system are discussed as follows:1. the realization of mechanization for constructing the exact solutions for nonlinear evo-lution equations;2. constructing the numerical solutions for nonlinear fractional differential equations;3. chaos synchronization and automatic reasoning scheme developed for them.Chapter 1 is devoted to reviewing the history and development of the mathematics mech-anization, soliton theory, fractional calculous and chaos synchronization, with an emphasis on some achievements on the subjects involved in this dissertation are presented at home and abroad.Chapter 2 introduces Wu-Ritt differential elimination theory, some basic notations and properties on fractional calculous, basic theories on AC=BD as well as the construction of exact solutions of nonlinear evolution equation(s) under the instruction of this theory.Based on the idea of solving nonlinear evolution equations, algebraic method, algorithm reality, mechanization, Chapter 3 firstly presents the rational expansion method to uniformly construct the exact solutions for nonlinear evolution equations, and then by means of Wu elimina-tion theory and symbolic computation software, considers some concrete forms and applications of the rational expansion method:(1) the high dimension coupled Burgers equation is considered by a new auxiliary equation method and some new complexiton solutions are found;(2) based on the elliptic functions, the elliptic function rational expansion method is pre-sented and some new rational formal elliptic function solutions of (2+1)-dimensional dispersive long wave equation are found;(3) Riccati equation rational expansion method is presented based on the Riccati equa-tion, and some new exact solutions of (2+1)-dimensional Broer-Kaup-Kupershmidt equation are obtained;(4) Whitham-Broer-Kaup equation is studied by using the generalized elliptic rational expansion method and some new rational formal exact solutions are found; (5) the Riccati equation rational expansion method is extended to study the (2+1)-dimensiona Burgers equation;(6) multiple auxiliary equations rational expansion method and its application to construct new complexiton solutions;(7) generalized Riccati equation rational expansion method and some non-travelling wave solutions of nonlinear evolution equations;(8) for stochastic differential equations, the Riccati equation rational expansion method is further improved and some new stochastic exact solutions are obtained;(9) the rational expansion method is further improved for differential-difference equations and some new exact solutions of two Toda equations are found.In chapter 4, the Adomian decomposition method and homotopy perturbation method which traditionally developed for differential equations of integer order are directly extended to derive numerical solutions of the nonlinear fractional differential equations. Some nonlinear fractional differential equations with physical significance, such as fractional KdV-mKdV equa-tion, fractional Boussinesq equation, fractional KdV-Burgers equation and fractional Kuramoto-Sivashinsky equation are investigated. Some available numerical solutions are firstly obtained. A simple theorem is also given to determine convergence of these methods.Chapter 5 first improves the automatic reasoning scheme for generalized Q-S (lag, com-plete and anticipated) synchronization of chaotic and hyperchaotic system. Based on the symbolic-numeric computation software, the generalized Q-S synchronization between two iden-tical Chua’s circuit with different initial values, hyperchaotic Rossler system and hyperchaotic Tamasevicius-Namajunas-Cenys system, two identical generalized Henon map with different ini-tial values, Henon-like map and generalized Henon map are obtained. Numerical simulations verify the effectiveness of the proposed scheme. Then we present the definition of bidirectional partial generalized (lag, complete and anticipated) synchronization and an automatic reason-ing scheme to obtain it. The Rossler system, a new unified Lorenz-Chen-Lu system as well as the hyperchaotic Tamasevicius-Namajunas-Cenys system are chosen to illustrate the proposed scheme. Numerical simulations are used to verify the effectiveness of the proposed scheme.更多还原