【作者】 梁小磊；

【导师】 林京；

【作者基本信息】 合肥工业大学 ， 计算数学， 2010， 硕士

【摘要】 本文的研究对象是非线性偏微分方程,由于这些偏微分方程来源于物理和其它应用学科,具有鲜明的物理意义,因此又称为非线性数学物理方程。本文讨论几个经典的非线性偏微分方程及他们的孤立波解,特别是较为详细地介绍了反演散射方法,以及利用这一方法来求KdV方程的单孤立波解和多孤立波解。反演散射法是解非线性偏微分方程的最常用,也是最普遍的方法,许多方程都可以利用这种方法来求解,目前也取得了一些结果。本文概述了非线性偏微分方程的一种数值解法——Adomian分解方法(ADM法),包括基本原理,Adomian多项式,噪声现象和收敛性分析。这种方法是比较简单实用的,它对方程和解法的要求都不高,但是它的缺点也是明显的,就是收敛区间比较小,我们通过对ADM法解出的级数解使用Padé逼近,有效地改进ADM法的这一缺陷,取得了良好的效果。通过对形变Boussinesq方程的实验,我们验证了ADM方法的应用,同时,通过这一例子,也说明了Padé逼近对ADM法的改进效果是非常明显的。更多还原

【Abstract】 In this thesis,the main research is about nonlinear partial differential equations.As these equations originate from physics and other applied subjects,with obvious physical meaning,they are also called nonlinear mathematical physics equations.We mainly summarizes some classical nonlinear partial differential equations and their solitary wave solutions,especially emphasizes the inverse scattering transform method(IST).Using this method,we get the single soliton solution and multiple soliton solutions of KdV equation.IST is the most common method in solving nonlinear partial differential equations.We also consider a numerical method of nonlinear partial differential equations-------- Adomian decomposition method(ADM).The main points of the decomposition method is:the principle,the Adomian polynomials,the noise terms and the convergence results.This method is very easy and practical,but the disadvantage of this method is very obvious-----the convergent domain is very narrow.Using ADM,we get a series solution.In this paper,we used the Padéapproximant with ADM in overcoming this drawback.The ADM method together with Padéapproximant extends the domain of solution and gives better accuracy and better convergence than using ADM alone.更多还原