﻿ 一个非线性偏微分方程边值问题的对称约化及其数值解 Symmetry Reduction and Its Numerical Solution to the Boundary Value Problem of a Nonlinear Partial Differential Equation

Vol.05 No.03(2016), Article ID:18350,6 pages
10.12677/AAM.2016.53046

Symmetry Reduction and Its Numerical Solution to the Boundary Value Problem of a Nonlinear Partial Differential Equation

Yanqing Han, Bilige Sudao*

College of Science, Inner Mongolia University of Technology, Hohhot Inner Mongolia

Received: Jul. 27th, 2016; accepted: Aug. 15th, 2016; published: Aug. 18th, 2016

ABSTRACT

We study the applications of the symmetry method on the boundary value problem for nonlinear partial differential equation. Firstly, the multi-parameter symmetry of a given boundary value problem for nonlinear partial differential equation is determined based on differential characteristic set algorithm. Secondly, by using the symmetry, the boundary value problem for nonlinear partial differential equation is reduced to an initial value problem of the original differential equation. Finally, we numerically solve the initial value problem of the original differential equations by using Runge-Kutta method.

Keywords:Boundary Value Problem for Nonlinear Partial Differential Equation, Differential Characteristic Set Algorithm, Symmetry Method, Runge-Kutta Method

1. 引言

Lie对称是公认的普适性方法之一，它以诸多传统方法为其特例，如：分离变量法、行波变换、相似变换等 [1] 。对称理论在现代数学、物理和力学等科学中有重要的理论和实际意义，并且已有了广泛的应用 [2] [3] 。但是除了在文 [1] [4] [5] 中研究者做了一些对称群在边值问题的应用外，这方面的研究还很少，所以利用对称群研究偏微分方程(简记为PDEs)边值问题是对称理论应用新的研究领域。在诸多求解非线性的问题中，常常利用相似变换对PDEs进行约化或降阶。我们知道PDEs的Lie变换群可以产生更一般形式的相似变换，并且这些变换有更多的数学和物理意义，所以利用对称方法研究非线性PDEs边值问题比直接用相似变换更有其优越性。

2. 一个非线性PDEs边值问题的对称约化及其数值解

(1)

，若 (2)

，若 (3)

(4)

(5)

，若 (6)

，若 (7)

2.1. 边值问题的对称约化

1) 产生关于无穷小生成函数的确定方程组。

(8)

2) 确定多参数对称。

(9)

，得到不变量

(10)

(11)

，当 (12)

，当 (13)

，当 (14)

(15)

2.2. 边值问题的数值解

(16)

(17)

Figure 1. Nomerical solution of in

3. 结论

Symmetry Reduction and Its Numerical Solution to the Boundary Value Problem of a Nonlinear Partial Differential Equation[J]. 应用数学进展, 2016, 05(03): 375-380. http://dx.doi.org/10.12677/AAM.2016.53046

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*通讯作者。