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针对椭圆类非线性偏微分方程边值问题,以差分法和动态设计变量优化算法为基础,以离散网格点未知函数值为设计变量,以离散网格点的差分方程组构建为复杂程式化形式的目标函数.提出一种求解离散网格点处未知函数值的优化算法.编制了求解未知离散点函数值的通用程序.求解了具体算例.通过与解析解对比,表明了本文提出求解算法的有效性和精确性,将为更复杂工程问题分析提供良好的解决方法.
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关键词:
- 非线性偏微分方程 /
- 边值问题 /
- 动态设计变量优化算法 /
- 程序设计
For elliptic nonlinear partial differential equations with boundary value problem, based on difference method and dynamic design variable optimization method, by taking unknown function value on discrete net point as design variables, difference equation of all the discrete net points is constructed as an objective function. A kind of optimization algorithm about solving unknown function value on discrete net point is proposed. Universal computing program is designed. Practical example is analyzed. By comparing the computing result with the analytical solution, effectiveness and feasibility are verified. Thus complicated nonlinear mathematical physics equations can be solved by the numerical calculation method.-
Keywords:
- nonlinear partial differential equations /
- boundary value problem /
- dynamic design variables optimization method /
- program design
[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: Beijing 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] Li W, Xie F D 2004 Chin. Phys. 13 1639
[4] Zhang W G 1998 Acta Math. Appl. 21 249 (in Chinese)[张卫 国 1998 应用数学学报 21 249]
[5] Ma W X, Lee J H 2009 Chaos, Solitons & Fractals 42 1356 Wang M L, Li X Z, Zhang J L 2008 Phys. Lett. A 372 417 Ma W X, Huang T W, Zhang Y 2010 Phys. Scr. 82 065003 Li Y Z, Feng W G, Li K M, Lin C 2007 Chin. Phys. 16 2510
[6] Ma W X, Gu X, Gao L 2009 Adv. Appl. Math. & Mech. 1 573
[7] Chen L J, Ma C F 2010 Chin. Phys. B 19 010504 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]
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[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: Beijing 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] Li W, Xie F D 2004 Chin. Phys. 13 1639
[4] Zhang W G 1998 Acta Math. Appl. 21 249 (in Chinese)[张卫 国 1998 应用数学学报 21 249]
[5] Ma W X, Lee J H 2009 Chaos, Solitons & Fractals 42 1356 Wang M L, Li X Z, Zhang J L 2008 Phys. Lett. A 372 417 Ma W X, Huang T W, Zhang Y 2010 Phys. Scr. 82 065003 Li Y Z, Feng W G, Li K M, Lin C 2007 Chin. Phys. 16 2510
[6] Ma W X, Gu X, Gao L 2009 Adv. Appl. Math. & Mech. 1 573
[7] Chen L J, Ma C F 2010 Chin. Phys. B 19 010504 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]
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