摘要
随着科学技术的不断进步,非线性科学得到了迅速的发展。在研究非线性科学的过程中不可避免会碰到各种各样的非线性方程,对于求解这些非线性方程成为研究非线性科学的关键,同时也是难点所在。一般而言,对非线性方程很难求得其精确解。因此,研究非线性方程的数值解具有很大理论和现实意义。其中,Adomian分解方法和同伦分析方法等是求非线性方程数值解的重要方法。
本文借助数学计算软件Maple,分别运用Adomian分解方法和同伦分析方法研究了一些具有重要物理意义的非线性微分差分方程,并获得了有用的数值解。
本文安排如下:
第一章系统地简要回顾了孤立子理论的历史和发展,介绍了Adomian分解方法和同伦分析方法的发展和研究现状,最后说明了研究非线性微分差分方程的意义。
第二章具体介绍了Adomian分解方法,把Adomian分解和Pad6近似结合起来,研究了相对Toda晶格方程和修正的Volterra晶格方程,获得了具有实际意义的数值解,并给出了相应图形分析。
第三章具体介绍了同伦分析方法,利用同伦分析方法研究了三场Blaszak-Marciniak晶格方程,得到了方程组的数值解,最后并画图模拟了其数值解。
第四章给出了本文的总结和展望。
With the development of science and technology, nonlinear science has developed rapidly. In the course of studying nonlinear science, we would encounter all kinds of nonlinear equations inevitably. Therefore how to solve these nonlinear equations not only becomes the key to scientific research, but also becomes the difficulty points. However, in general, it is still difficult to obtain exact solutions of the nonlinear equations. So the study of numerical solutions for nonlinear equations becomes very significant. Adomian decomposition method and homotopy analysis method are two important methods to solve nonlinear equations which can obtain numerical solutions.
In this paper, based on symbolic computation software Maple, we make use of Adomian decomposition method and homotopy analysis method to investigate some important differential-difference equations, then obtain the useful numerical solutions and give out figure anlysis.
It is organized as follows:
Chapter 1 briefly reviews the history and the progress of soliton theory. Adomian decomposition method and homotopy analysis method are introduced. The end of the chapter shows the significance of studying the nonlinear differential-difference equations. The corresponding figures anlysis is given.
Chapter 2 detailedly introduces the Adomian decomposition method. We combine Adomian decomposition method and Padéapproximants to solve the relativistic Toda lattice equation and the modified Volterra lattice equation. The practical significance of numerical solution is obtained.
Chapter 3 concretely explains Homotopy analysis method. We apply this method to Blaszak-Marciniak three-field lattice equations and obtain the approximate solutions. At last, we give the figures to simulate the approximate solutions.
Chapter 4 is the summary and outlook of the dissertation.
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