AAM  >> Vol. 5 No. 3 (August 2016)

    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

  • 全文下载: PDF(439KB) HTML   XML   PP.375-380   DOI: 10.12677/AAM.2016.53046  
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作者:  

韩雁清,苏道毕力格:内蒙古工业大学理学院,内蒙古 呼和浩特

关键词:
非线性偏微分方程边值问题微分特征列集算法对称方法龙格–库塔法Boundary Value Problem for Nonlinear Partial Differential Equation Differential Characteristic Set Algorithm Symmetry Method Runge-Kutta Method

摘要:
本文研究了微分方程对称方法在非线性偏微分方程边值问题中的应用。首先,基于微分特征列集算法确定了给定非线性偏微分方程边值问题的多参数对称;其次,利用对称将非线性偏微分方程边值问题化为常微分方程初值问题;最后,利用龙格-库塔法求解了常微分方程初值问题的数值解。

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 characte-ristic 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 equ-ation. Finally, we numerically solve the initial value problem of the original differential equations by using Runge-Kutta method.

文章引用:
韩雁清, 苏道毕力格. 一个非线性偏微分方程边值问题的对称约化及其数值解[J]. 应用数学进展, 2016, 5(3): 375-380. http://dx.doi.org/10.12677/AAM.2016.53046

参考文献

[1] Bluman, G.W. and Kumei, S. (1989) Symmetries and Differential Equations. Spring-Verlag, New York, Berlin.
[2] Bluman, G., Cheviakov, A.F. and Anco, S.C. (2010) Applications of Symmetry Methods to Partial Differential. Spring-Verlag, New York.
[3] Ibragimov, N.H. and Ibragimov, R.N. (2012) Applications of Lie Group Analysis to Mathematical Modelling in Natural Sciences. Mathematical Modelling of Natural Phenomena, 7, 52-65.
http://dx.doi.org/10.1051/mmnp/20127205
[4] Seshadri, R. and Na, T.Y. (1985) Group Invariance in Engineering Boundary Value Problems. Springer-Verlag, New York-Berlin-Heidelberg Tokyo.
[5] Yürüsoy M,Pakdemirli M,Noyan O F. (2001) Lie Group Analysis of Creeping Flow of a Second Grade Fluid. Interna-tional Journal of Non-Linear Mechanics, 36, 955-960.
http://dx.doi.org/10.1016/S0020-7462(00)00060-3
[6] 朝鲁. 微分方程(组)对称向量的吴–微分特征列集算法及其应用[J]. 数学物理学报, 1999, 19(3): 326-332.
[7] 特木尔朝鲁, 白玉山. 基于吴方法的确定和分类(偏)微分方程古典和非古典对称新算法理论[J]. 中国科学: A辑, 2010, 40(4): 1-18.
[8] 苏道毕力格. 一些求解偏微分方程解析解方法的研究[D]: [博士学位论文]. 呼和浩特: 内蒙古工业大学, 2011.
[9] 王晓民, 苏道毕力格, 特木尔朝鲁. 对称方法在非线性偏微分方程边值问题中的应用[J]. 内蒙古大学学报(自然科学版), 2013,44(2): 129-132.
[10] 苏道毕力格, 王晓民, 乌云莫日根. 对称分类在非线性偏微分方程组边值问题中的应用[J]. 物理学报, 2014, 63(4): 040201.
[11] 苏道毕力格, 王晓民, 鲍春玲. 利用对称方法求解非线性偏微分方程组边值问题的数值解[J]. 应用数学, 2014, 27(4): 10-15.
[12] Lu, L. and Temuer, C. (2011) A New Method for Solving Boundary Value Problems for Partial Differential Equations. Computers and Mathematics with Applications, 61, 2164-2167.
http://dx.doi.org/10.1016/j.camwa.2010.09.002
[13] Eerdunbuhe and Temuerchaolu (2012) Approximate Solution of the Magneto-Hydrodynamic Flow over a Nonlinear Stretching Sheet. Chinese Physics B, 21, 035201.
http://dx.doi.org/10.1088/1674-1056/21/3/035201