一类非线性偏微分方程初边值问题的逐层优化算法

侯祥林; 翟中海; 郑莉; 刘铁林

doi:10.7498/aps.61.010201

摘要

针对非线性偏微分方程初边值问题,基于差分法和动态设计变量优化算法原理, 以时间计算层上离散节点的未知函数值为设计变量,以离散节点的差分方程组构造程式化的目标函数, 提出了离散节点处未知函数值的逐层高精度优化算法.编制通用程序求解具体典型算例. 并通过与解析解对比,表明了求解方法的正确性和有效性,为广泛的工程应用提供条件.

关键词:

For non-linear partial differential equations with initial-boundary value problems, based on the difference method and the optimization method with dynamic design variables, using unknown function values on discrete node points on time layer as design variables, the difference equations sets of all the discrete node points are constructed as stylized objective function. A layered accurate optimization algorithm about computing unknown function value on discrete node point is proposed. Universal computing program is designed, and practical examples are analyzed. Through comparing computation results with exact results, the effectiveness and the feasibility of proposed method are verified. The method can provide the condition for engineering application.

Keywords:

nonlinear partial differential equation /
Initial-boundary value problem /
optimization method with dynamic design variables /
program design

作者及机构信息

1.
沈阳建筑大学理学院, 沈阳 110168;

2.
沈阳建筑大学土木工程学院, 沈阳 110168

基金项目: 国家自然科学基金 (批准号:10972144)、辽宁省自然科学基金(批准号:201102181)和辽宁省教育厅 (批准号:L2010445)资助的课题.

Authors and contacts

1.
Shenyang Jianzhu University, School of Science, Shenyang 110168;

2.
Shenyang Jianzhu University, School of Civil Engineering, Shenyang 110168, China

Funds: Project supported by the National Natural Science Foundation of China (Grant No.10972144), the Natural Science Foundation of Liaoning Province, China (Grant No. 201102181), and the Education Department of Liaoning Province, China (Grant No. L2010445).

参考文献

[1]	Gu C H, Li D Q, Shen W X 1994 Appliced Partial Differential Equations (Beijing: Higher Education Press) pp161–192 (in Chinese) [谷超豪,李大潜,沈玮熙 1994 应用偏微分方程(北京:高等教育出版社) 第161-192页]
[2]	Liu S K, Liu S D 2000 Nolinear Equations in Physics (Beijing: Peking University Press) pp7–15 (in Chinese) [刘式适,刘式达 2000 物理学中的非线性方程 (北京:北京大学出版社)第7-15页]
[3]	Taogetusang, Sirendaoerji 2010 Acta Phys. Sin. 59 5194 (in Chinese) [套格图桑,斯仁道尔吉 2010 59 5194]
[4]	Sun Z Z 2005 Numerical Method about Partial Differential Equations (Beijing: Science Press)pp73–80, 112,131–135 (in Chinese) [孙志忠 2005 偏微分方程数值解法 (北京:科学出版社)第73-80,112,131-135页]
[5]	Lu J, Yan J R 2002 Acta Phys. Sin. 51 1428 (in Chinese) [卢竞,颜家壬 2002 51 1428]
[6]	Xie Y X, Tang J S 2004 Acta Phys. Sin. 53 2828 (in Chinese) [谢元喜,唐驾时 2004 53 2828]
[7]	Wang M L, Li X Z, Zhang J L 2008 Phys. Lett. A 372 417
[8]	Taogetusang, Sirendaoerji 2009 Acta Phys. Sin. 58 5887 (in Chinese) [套格图桑,斯仁道尔吉 2009 58 5887]
[9]	Mo J Q 2011 Acta Phys.Sin. 60 020202(in Chinese) [莫嘉琪 2011 60 020202]
[10]	Cheng X P, Lin J, Yao J M 2009 Chin. Phys. B 18 391
[11]	Li Y Z, Feng W G, Li K M, Lin C 2007 Chin. Phys. 16 2510
[12]	Zheng L C, Feng Z F, Zhang X X 2007 Acta Phys. Sin. 56 1550 (in Chinese) [郑连存,冯志丰,张欣欣 2007 56 1550]
[13]	Guo Y C 2007 Introduction about Partial Differential Equations (Beijing:Tsinghua University Press) pp186–226 (in Chinese) [郭玉翠 2007 非线性偏微分方程引论(北京: 清华大学出版社)第186-226页]
[14]	Ma W X, Gu X, Gao L 2009 Adv. Appl. Math. Mech. 1 573
[15]	Chen L J, Ma C F 2010 Chin. Phys. B 19 010504
[16]	Ma W X, Huang T W, Zhang Y 2010 Phys. Scr. 82 065003
[17]	Hirota R 2004 The Direct Method in Soliton Theory (Cambridge: Cambridge University Press)
[18]	Hietarinta J 2005 Phys. AUC 15 31
[19]	W X Ma, Fan E G 2011 Comput. Math. Appl. 61 950
[20]	Hou X L, Qian Y, Wu H T 2010 Acta Math. Eng. 27 663 (in Chinese) [侯祥林, 钱颖, 吴海涛 2010 工程数学学报 27 663]
[21]	Hou X L, Liu T L, Zhai Z H 2011 Acta Phys. Sin. 60 090202 (in Chinese) [侯祥林,刘铁林,翟中海 2011 60 090202]
[22]	Polyanin A D, Zaitsev V F 2004 Handbook of Nonlinear Partial Differential Equations (Boca Raton, London, NewYork: Chapman & Hall/CRC) pp1–2

施引文献

[1]	Gu C H, Li D Q, Shen W X 1994 Appliced Partial Differential Equations (Beijing: Higher Education Press) pp161–192 (in Chinese) [谷超豪,李大潜,沈玮熙 1994 应用偏微分方程(北京:高等教育出版社) 第161-192页]
[2]	Liu S K, Liu S D 2000 Nolinear Equations in Physics (Beijing: Peking University Press) pp7–15 (in Chinese) [刘式适,刘式达 2000 物理学中的非线性方程 (北京:北京大学出版社)第7-15页]
[3]	Taogetusang, Sirendaoerji 2010 Acta Phys. Sin. 59 5194 (in Chinese) [套格图桑,斯仁道尔吉 2010 59 5194]
[4]	Sun Z Z 2005 Numerical Method about Partial Differential Equations (Beijing: Science Press)pp73–80, 112,131–135 (in Chinese) [孙志忠 2005 偏微分方程数值解法 (北京:科学出版社)第73-80,112,131-135页]
[5]	Lu J, Yan J R 2002 Acta Phys. Sin. 51 1428 (in Chinese) [卢竞,颜家壬 2002 51 1428]
[6]	Xie Y X, Tang J S 2004 Acta Phys. Sin. 53 2828 (in Chinese) [谢元喜,唐驾时 2004 53 2828]
[7]	Wang M L, Li X Z, Zhang J L 2008 Phys. Lett. A 372 417
[8]	Taogetusang, Sirendaoerji 2009 Acta Phys. Sin. 58 5887 (in Chinese) [套格图桑,斯仁道尔吉 2009 58 5887]
[9]	Mo J Q 2011 Acta Phys.Sin. 60 020202(in Chinese) [莫嘉琪 2011 60 020202]
[10]	Cheng X P, Lin J, Yao J M 2009 Chin. Phys. B 18 391
[11]	Li Y Z, Feng W G, Li K M, Lin C 2007 Chin. Phys. 16 2510
[12]	Zheng L C, Feng Z F, Zhang X X 2007 Acta Phys. Sin. 56 1550 (in Chinese) [郑连存,冯志丰,张欣欣 2007 56 1550]
[13]	Guo Y C 2007 Introduction about Partial Differential Equations (Beijing:Tsinghua University Press) pp186–226 (in Chinese) [郭玉翠 2007 非线性偏微分方程引论(北京: 清华大学出版社)第186-226页]
[14]	Ma W X, Gu X, Gao L 2009 Adv. Appl. Math. Mech. 1 573
[15]	Chen L J, Ma C F 2010 Chin. Phys. B 19 010504
[16]	Ma W X, Huang T W, Zhang Y 2010 Phys. Scr. 82 065003
[17]	Hirota R 2004 The Direct Method in Soliton Theory (Cambridge: Cambridge University Press)
[18]	Hietarinta J 2005 Phys. AUC 15 31
[19]	W X Ma, Fan E G 2011 Comput. Math. Appl. 61 950
[20]	Hou X L, Qian Y, Wu H T 2010 Acta Math. Eng. 27 663 (in Chinese) [侯祥林, 钱颖, 吴海涛 2010 工程数学学报 27 663]
[21]	Hou X L, Liu T L, Zhai Z H 2011 Acta Phys. Sin. 60 090202 (in Chinese) [侯祥林,刘铁林,翟中海 2011 60 090202]
[22]	Polyanin A D, Zaitsev V F 2004 Handbook of Nonlinear Partial Differential Equations (Boca Raton, London, NewYork: Chapman & Hall/CRC) pp1–2

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