对称分类在非线性偏微分方程组边值问题中的应用

苏道毕力格; 王晓民; 乌云莫日根

doi:10.7498/aps.63.040201

摘要

研究了微分方程对称分类在非线性偏微分方程组边值问题中的应用. 首先，利用偏微分方程（组）完全对称分类微分特征列集算法确定了给定非线性偏微分方程组边值问题的完全对称分类；其次，利用一个扩充对称将非线性偏微分方程组边值问题约化为常微分方程组初值问题；最后，利用龙格-库塔法求解了常微分方程组初值问题的数值解.

关键词:

In this paper, we study the application of the symmetry classification to the boundary value problem of nonli-near partial differential equations. Firstly, by using differential characteristic set algorithm for the complete symmetry classification of partial differential equations, the complete symmetry classification of a given boundary value problem of nonlinear partial differential equations is proposed. Secondly, by using an extended symmetry, the boundary value problem of nonlinear partial differential equations is reduced to an initial value problem of the original differential equations. Finally, we numerically solve the initial value problem of the original differential equations by using Runge-Kutta method.

Keywords:

symmetry classification /
differential characteristic set algorithm /
the boundary value problem of partial differential equations

作者及机构信息

1.
内蒙古工业大学理学院, 呼和浩特 010051

基金项目: 国家自然科学基金（批准号：11071159, 11261034）和内蒙古自治区高等学校科学技术研究项目（批准号：NJZY12056）资助的课题.

Authors and contacts

1.
College of Sciences, Inner Mongolia University of Technology, Hohhot 010051, China

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 11071159, 11261034) and the High Education Science Research Program of Inner Mongolia, China (Grant No. NJZY12056).

参考文献

[1]	Lie S 1881 Areh. Math. 6 328
[2]	Noether A E 1918 Nachr. Akad. Wiss Göttingen Math. Phys. KI 2 235
[3]	Bluman G W, Cole J D 1969 J. Math. Mech. 18 1025
[4]	Lian Z J, Lou S Y 2004 Chin. Phys. Lett. 21 219
[5]	Ma W X 1990 J. Phys. A: Math. Gen. 23 2707
[6]	Clarkson P A, Kruskal M D 1989 J. Math. Phys. 30 2201
[7]	Lou S Y, Tang X Y 2001 Chin. Phys. 10 897
[8]	Jiao X Y, Lou S Y 2009 Chin. Phys. B 18 3611
[9]	Jiao X Y 2011 Acta Phys. Sin. 60 120201 (in Chinese) [焦小玉 2011 60 120201]
[10]	Li X S 1988 Sci. Sin. Math. A 31 1 (in Chinese) [李诩神 1988 中国科学 A 31 1]
[11]	Cheng Y, Wang Q C 1991 Acta Math. Appl. Sin. 14 180 (in Chinese) [程艺, 汪启存 1991 应用数学学报 14 180]
[12]	Ma W X, Chen M 2009 Appl. Math. Comput. 215 2835
[13]	Ma W X 2013 Appl. Math. Comput. 220 117
[14]	Bluman G W, Kumei S 1989 Symmetries and Differential Equations (New York, Berlin: Spring-Verlag)
[15]	Bluman G W, Cheviakov A F, Anco S C 2009 Applications of Symmetry Methods to Partial Differential ( New York: Spring-Verlag)
[16]	Seshadri R, Na T Y 1985 Group Invarianace in Engineering Boundary Value Problems (New York, Berlin, Heidelberg, Tokyo: Springer-Verlag)
[17]	Yrsoy M, Pakdemirli M, Noyan Ö F 2001 Int. J. Nonlin. Mech. 36 955
[18]	Chao L 1999 Acta Math. Sci. 19 326 (in Chinese) [朝鲁 1999 数学 19 326]
[19]	Temuer C, Gao X S 2002 Acta Math. Sin. 45 1041 (in Chinese) [特木尔朝鲁, 高小山 2002 数学学报 45 1041]
[20]	Temuer C 2003 Adv. Math. 32 208
[21]	Bluman G W, Temuer C 2005 J. Math. Phys. 46 023505
[22]	Bluman G W, Temuer C 2006 J. Math. Anal. Appl. 322 233
[23]	Sudao B, Chao L 2006 J. Inner Mongolia Univ. (Nat. Sci. Ed.) 37 366 (in Chinese) [苏道毕力格, 朝鲁 2006 内蒙古大学学报 (自然科学版) 37 366]
[24]	Temuer C, Yin S 2007 Acta Math. Sin. 50 1017 (in Chinese) [特木尔朝鲁, 银山 2007 数学学报 50 1017]
[25]	Temuer C, EerDun B, Zheng L X 2007 Acta Math. Appl. Sin. 30 910 (in Chinese) [特木尔朝鲁, 额尔敦布和, 郑丽霞 2007 应用数学学报 30 910]
[26]	Temuer C, Bai Y S 2011 Chin. J. Eng. Math. 28 617
[27]	Temuer C, Yin S 2012 J. Sys. Sci. Math. Sci. 32 976 (in Chinese) [特木尔朝鲁, 银山 2012 系统科学与数学 32 976]
[28]	Wang X M, Sudao B, Temuer C 2013 J. Inner Mongolia Univ. (Nat. Sci. Ed.) 44 129 (in Chinese) [王晓民, 苏道毕力格, 特木尔朝鲁 2013 内蒙古大学学报 (自然科学版) 44 129]
[29]	Sudao B 2011 Ph. D. Dissertation (Hohhot: Inner Mongolia University of Technology) (in Chinese) [苏道毕力格 2011 博士学位论文 (呼和浩特: 内蒙古工业大学)]
[30]	Lu L, Temuer C 2011 Comput. Math. Appl. 61 2164
[31]	Lu L, Temuer C 2011 Int. J. Nonlinear Sci. Numer. Simul. 11 967
[32]	EerDun B, Temuer C 2012 Chin. Phys. B 21 035201
[33]	Temuer C, Bai Y S 2009 Appl. Math. Mech. Engl. Ed. 30 595
[34]	Temuer C, Pang J 2010 J. Eng. Math. 66 181
[35]	Temuer C, Bai Y S 2010 Sci. Sin. Math. A 40 1 (in Chinese) [特木尔朝鲁, 白玉山 2010 中国科学 A 40 1]
[36]	Temuer C, Zhang Z Y 2009 J. Sys. Sci. Math. Sci. 29 389 (in Chinese) [特木尔朝鲁, 张智勇 2009 系统科学与数学 29 389]

施引文献

[1]	Lie S 1881 Areh. Math. 6 328
[2]	Noether A E 1918 Nachr. Akad. Wiss Göttingen Math. Phys. KI 2 235
[3]	Bluman G W, Cole J D 1969 J. Math. Mech. 18 1025
[4]	Lian Z J, Lou S Y 2004 Chin. Phys. Lett. 21 219
[5]	Ma W X 1990 J. Phys. A: Math. Gen. 23 2707
[6]	Clarkson P A, Kruskal M D 1989 J. Math. Phys. 30 2201
[7]	Lou S Y, Tang X Y 2001 Chin. Phys. 10 897
[8]	Jiao X Y, Lou S Y 2009 Chin. Phys. B 18 3611
[9]	Jiao X Y 2011 Acta Phys. Sin. 60 120201 (in Chinese) [焦小玉 2011 60 120201]
[10]	Li X S 1988 Sci. Sin. Math. A 31 1 (in Chinese) [李诩神 1988 中国科学 A 31 1]
[11]	Cheng Y, Wang Q C 1991 Acta Math. Appl. Sin. 14 180 (in Chinese) [程艺, 汪启存 1991 应用数学学报 14 180]
[12]	Ma W X, Chen M 2009 Appl. Math. Comput. 215 2835
[13]	Ma W X 2013 Appl. Math. Comput. 220 117
[14]	Bluman G W, Kumei S 1989 Symmetries and Differential Equations (New York, Berlin: Spring-Verlag)
[15]	Bluman G W, Cheviakov A F, Anco S C 2009 Applications of Symmetry Methods to Partial Differential ( New York: Spring-Verlag)
[16]	Seshadri R, Na T Y 1985 Group Invarianace in Engineering Boundary Value Problems (New York, Berlin, Heidelberg, Tokyo: Springer-Verlag)
[17]	Yrsoy M, Pakdemirli M, Noyan Ö F 2001 Int. J. Nonlin. Mech. 36 955
[18]	Chao L 1999 Acta Math. Sci. 19 326 (in Chinese) [朝鲁 1999 数学 19 326]
[19]	Temuer C, Gao X S 2002 Acta Math. Sin. 45 1041 (in Chinese) [特木尔朝鲁, 高小山 2002 数学学报 45 1041]
[20]	Temuer C 2003 Adv. Math. 32 208
[21]	Bluman G W, Temuer C 2005 J. Math. Phys. 46 023505
[22]	Bluman G W, Temuer C 2006 J. Math. Anal. Appl. 322 233
[23]	Sudao B, Chao L 2006 J. Inner Mongolia Univ. (Nat. Sci. Ed.) 37 366 (in Chinese) [苏道毕力格, 朝鲁 2006 内蒙古大学学报 (自然科学版) 37 366]
[24]	Temuer C, Yin S 2007 Acta Math. Sin. 50 1017 (in Chinese) [特木尔朝鲁, 银山 2007 数学学报 50 1017]
[25]	Temuer C, EerDun B, Zheng L X 2007 Acta Math. Appl. Sin. 30 910 (in Chinese) [特木尔朝鲁, 额尔敦布和, 郑丽霞 2007 应用数学学报 30 910]
[26]	Temuer C, Bai Y S 2011 Chin. J. Eng. Math. 28 617
[27]	Temuer C, Yin S 2012 J. Sys. Sci. Math. Sci. 32 976 (in Chinese) [特木尔朝鲁, 银山 2012 系统科学与数学 32 976]
[28]	Wang X M, Sudao B, Temuer C 2013 J. Inner Mongolia Univ. (Nat. Sci. Ed.) 44 129 (in Chinese) [王晓民, 苏道毕力格, 特木尔朝鲁 2013 内蒙古大学学报 (自然科学版) 44 129]
[29]	Sudao B 2011 Ph. D. Dissertation (Hohhot: Inner Mongolia University of Technology) (in Chinese) [苏道毕力格 2011 博士学位论文 (呼和浩特: 内蒙古工业大学)]
[30]	Lu L, Temuer C 2011 Comput. Math. Appl. 61 2164
[31]	Lu L, Temuer C 2011 Int. J. Nonlinear Sci. Numer. Simul. 11 967
[32]	EerDun B, Temuer C 2012 Chin. Phys. B 21 035201
[33]	Temuer C, Bai Y S 2009 Appl. Math. Mech. Engl. Ed. 30 595
[34]	Temuer C, Pang J 2010 J. Eng. Math. 66 181
[35]	Temuer C, Bai Y S 2010 Sci. Sin. Math. A 40 1 (in Chinese) [特木尔朝鲁, 白玉山 2010 中国科学 A 40 1]
[36]	Temuer C, Zhang Z Y 2009 J. Sys. Sci. Math. Sci. 29 389 (in Chinese) [特木尔朝鲁, 张智勇 2009 系统科学与数学 29 389]

[1]	黄亮, 李建远. 基于单粒子模型与偏微分方程的锂离子电池建模与故障监测. , 2015, 64(10): 108202. doi: 10.7498/aps.64.108202
[2]	何郁波, 林晓艳, 董晓亮. 应用格子Boltzmann模型模拟一类二维偏微分方程. , 2013, 62(19): 194701. doi: 10.7498/aps.62.194701
[3]	侯祥林, 郑夕健, 张良, 刘铁林. 薄板弯曲大变形高阶非线性偏微分方程推导与优化算法研究. , 2012, 61(18): 180201. doi: 10.7498/aps.61.180201
[4]	侯祥林, 翟中海, 郑莉, 刘铁林. 一类非线性偏微分方程初边值问题的逐层优化算法. , 2012, 61(1): 010201. doi: 10.7498/aps.61.010201
[5]	侯祥林, 刘铁林, 翟中海. 非线性偏微分方程边值问题的优化算法研究与应用. , 2011, 60(9): 090202. doi: 10.7498/aps.60.090202
[6]	楼智美. 微扰Kepler系统轨道微分方程的近似Lie对称性与近似不变量. , 2010, 59(10): 6764-6769. doi: 10.7498/aps.59.6764
[7]	套格图桑, 斯仁道尔吉. Volterra差分微分方程和KdV差分微分方程新的精确解. , 2009, 58(9): 5887-5893. doi: 10.7498/aps.58.5887
[8]	郭美玉, 高洁. 变系数广义Gardner方程的微分不变量及群分类. , 2009, 58(10): 6686-6691. doi: 10.7498/aps.58.6686
[9]	何光, 梅凤翔. 三质点Toda晶格微分方程的积分. , 2008, 57(1): 18-20. doi: 10.7498/aps.57.18
[10]	张睿超, 王连海, 岳成庆. 微分方程的部分Hamilton化与积分. , 2007, 56(6): 3050-3053. doi: 10.7498/aps.56.3050
[11]	胡楚勒. 一类非完整系统运动微分方程的Lie对称性与Hojman型守恒量. , 2007, 56(7): 3675-3677. doi: 10.7498/aps.56.3675
[12]	吴惠彬, 张永发, 梅凤翔. 求解微分方程的Hojman方法. , 2006, 55(10): 4987-4990. doi: 10.7498/aps.55.4987
[13]	施勇, 马善钧. 利用杨辉三角形对称性推导高阶运动微分方程. , 2006, 55(10): 4991-4994. doi: 10.7498/aps.55.4991
[14]	杨鹏飞. 一类带限定变换的二阶耦合线性微分方程组的解析解. , 2006, 55(11): 5579-5584. doi: 10.7498/aps.55.5579
[15]	张相武. 完整力学系统的高阶运动微分方程. , 2005, 54(9): 3978-3982. doi: 10.7498/aps.54.3978
[16]	谢元喜, 唐驾时. 对“求一类非线性偏微分方程解析解的一种简洁方法”一文的一点注记. , 2005, 54(3): 1036-1038. doi: 10.7498/aps.54.1036
[17]	谢元喜, 唐驾时. 求一类非线性偏微分方程解析解的一种简洁方法. , 2004, 53(9): 2828-2830. doi: 10.7498/aps.53.2828
[18]	卢竞, 颜家壬. 非线性偏微分方程的多孤子解. , 2002, 51(7): 1428-1433. doi: 10.7498/aps.51.1428
[19]	李志斌, 姚若侠. 非线性耦合微分方程组的精确解析解. , 2001, 50(11): 2062-2067. doi: 10.7498/aps.50.2062
[20]	武宇. 边界形状的变化对偏微分方程本征值的影响. , 1963, 19(8): 538-540. doi: 10.7498/aps.19.538

计量

文章访问数: 6210
PDF下载量: 535
被引次数: 0

姓名
邮箱
手机号码
标题
留言内容
验证码

搜索

留言板