分数阶常微分方程的改进精细积分法

鲍四元; 沈峰

doi:10.21656/1000-0887.390355

分数阶常微分方程的改进精细积分法

doi: 10.21656/1000-0887.390355

鲍四元^1,2,
沈峰²

¹苏州科技大学江苏省结构工程重点实验室,江苏苏州 215011;²苏州科技大学土木工程学院工程力学系,江苏苏州 215011

基金项目: 国家自然科学基金(11202146；51709194)

详细信息

作者简介:
鲍四元（1980—），男，副教授（通讯作者. E-mail: bsiyuan@126.com）.

中图分类号: O175
计量
- 文章访问数: 1319
- HTML全文浏览量: 230
- PDF下载量: 422
- 被引次数: 0
出版历程
- 收稿日期: 2018-12-24
- 修回日期: 2019-04-09
- 刊出日期: 2019-12-01

An Improved Precise Integration Method for Fractional Ordinary Differential Equations

BAO Siyuan^1,2,
SHEN Feng²

¹Key Laboratory of Structural Engineering of Jiangsu Province,Suzhou University of Science and Technology, Suzhou, Jiangsu 215011,P.R.China;²Department of Engineering Mechanics, School of Civil Engineering,Suzhou University of Science and Technology, Suzhou, Jiangsu 215011, P.R.China

Funds: The National Natural Science Foundation of China(11202146；51709194)

摘要

摘要: 基于Mittag-Leffler函数的定义式，构造Mittag-Leffler矩阵函数的精细迭代计算格式.与常规指数函数的迭代格式相比，迭代递推中多了修正项，其表达式与分数阶导数的阶次有关.对于以Caputo分数导数定义的动力学分数阶常微分方程，使用基于Mittag-Leffler函数的精细积分法可计算方程解在各时间段端点对应函数值.算例表明了所提计算方法的有效性，其精度可由所增加修正项的阶次控制.
- Mittag-Leffler函数 /
- 精细迭代格式 /
- 修正项 /
- 分数阶常微分方程 /
- Caputo分数阶导数
Abstract: Based on the definition of the Mittag-Leffler function, the precise iteration computation scheme for the Mittag-Leffler matrix function was constructed. Compared with the normal iteration scheme for exponential functions, the constructed scheme has additional correction items. The expression of the correction item is related to the order of the fractional derivative. For dynamic fractional ordinary differential equation D^(α)v=Hv with the Caputo fractional definition, the solution function value at the endpoint of the time phase can be obtained with the precise iteration method. The numerical examples demonstrated effectiveness and efficiency of the presented method.
- Mittag-Leffler function /
- precise iteration scheme /
- correction item /
- ordinary fractional differential equation /
- Caputo derivative

HTML全文

参考文献(31)

[1]	钟万勰. 暂态历程的精细计算方法[J]. 计算结构力学及其应用, 1995,12(1): 1-6.(ZHONG Wanxie. Precise integration for transient analysis[J]. Computational Structure Mechanics and Applications， 1995,12(1): 1-6.(in Chinese))
[2]	ZHONG W X. On precise integration method[J]. Journal of Computational and Applied Mathematics,2004,163(1): 59-78.
[3]	钟万勰, 高强. 辛破茧: 辛拓展新层次[M]. 大连: 大连理工大学出版社, 2011.(ZHONG Wanxie, GAO Qiang. Broken Cocoon: New Steps for Symplectic Extension [M]. Dalian: Dalian University of Technology Press, 2011.(in Chinese))
[4]	钟万勰. 应用力学的辛数学方法[M]. 北京: 高等教育出版社, 2005.(ZHONG Wanxie. Symplectic Solution Methodology in Applied Mechanics [M]. Beijing: Higher Education Press, 2005.(in Chinese))
[5]	顾元宪, 陈飚松, 张洪武. 结构动力方程的增维精细积分法[J]. 力学学报, 2000,32(4): 447-456.(GU Yuanxian, CHEN Biaosong, ZHANG Hongwu. Precise time-integration with dimension expanding method[J]. Acta Mechanica Sinica,2000,32(4): 447-456.(in Chinese))
[6]	CHEN B S, TONG L Y, GU Y X. Precise time integration for linear two-point boundary value problems[J]. Applied Mathematics Computation,2006,〖STHZ〗 175(1): 182-211.
[7]	向宇, 袁丽芸, 邹时智, 等. 求解非线性动力方程的一种齐次扩容精细积分法[J]. 华中科技大学学报(自然科学版), 2007,35(8): 109-111.(XIANG Yu, YUAN Liyun, ZOU Shizhi, et al. An extended homogeneous capacity integration method with high precision to solve nonlinear dynamic equation[J]. Journal of Huazhong University of Science and Technology(Nature Science),2007,35(8):109-111.(in Chinese))
[8]	张素英, 邓子辰. 非线性动力方程的增维精细积分法[J]. 计算力学学报, 2003,〖STHZ〗 20(4): 423-426.(ZHANG Suying, DENG Zichen. Increment-dimensional precise integration method for nonlinear dynamic equation[J]. Chinese Journal of Computational Mechanics,2003,20(4): 423-426.(in Chinese))
[9]	张继锋, 邓子辰. 结构动力方程的增维分块精细积分法[J]. 振动与冲击, 2008,27(12): 88-90.(ZHANG Jifeng, DENG Zichen. Dimensional increment and partitioning precise integration method for structural dynamic equation[J]. Journal of Vibration and Shock,2008,〖STHZ〗 27(12): 88-90.(in Chinese))
[10]	张森文, 曹开彬. 计算结构动力响应的状态方程直接积分法[J]. 计算力学学报, 2000,17(1):94-97.(ZHANG Senwen, CAO Kaibin. Direct integration of state equation method for dynamic response of structure[J]. Chinese Journal of Computational Mechanics,2000,17(1): 94-97.(in Chinese))
[11]	汪梦甫, 周锡元. 结构动力方程的更新精细积分方法[J]. 力学学报, 2004,36(2): 191-195.(WANG Mengfu, ZHOU Xiyuan. Gauss precise time integration of structural dynamic analysis[J]. Engineering Mechanics,2004,36(2): 191-195.(in Chinese))
[12]	谭述君, 钟万勰. 非齐次动力方程Duhamel项的精细积分[J]. 力学学报, 2007,〖STHZ〗 39(3): 374-381.(TAN Shujun, ZHONG Wanxie. Precise integration method for Duhamel terms arising from non-homogenous dynamic systems[J]. Chinese Journal of Theoretical and Applied Mechanic s, 2007,39(3): 374-381.(in Chinese))
[13]	富明慧, 刘祚秋, 林敬华. 一种广义精细积分法[J]. 力学学报, 2007,39(5): 672-677.(FU Minghui, LIU Zuoqiu, LIN Jinghua. A generalized precise time step integration method[J]. Chinese Journal of Theoretical and Applied Mechanics,2007,39(5): 672-677.(in Chinese))
[14]	高小科, 邓子辰, 黄永安. 基于三次样条插值的精细积分法[J]. 振动与冲击, 2007,26(9): 75-77, 82.(GAO Xiaoke, DENG Zichen, HUANG Yongan. A high precise direct integration based on cubic spline interpolation[J]. Journal of Vibration and Shock,2007,26(9): 75-77, 82.(in Chinese))
[15]	富明慧, 梁华力. 一种改进的精细-龙格库塔法[J]. 中山大学学报(自然科学版), 2009,48(5): 1-5.(FU Minghui, LIANG Huali. An improved precise Runge-Kutta integration[J]. Acta Scientiarum Naturalium Universitatis Sunyatseni,2009,48(5): 1-5.(in Chinese))
[16]	张文志, 富明慧, 刘祚秋. 结构动力方程的精细积分-FFT方法[J]. 中山大学学报(自然科学版), 2008,〖STHZ〗 47(6): 12-15.(ZHANG Wenzhi, FU Minghui, LIU Zuoqiu. Precise integration-FFT method for structural dynamics[J]. Acta Scientiarum Naturalium Universitatis Sunyatseni,2008,47(6): 12-15.(in Chinese))
[17]	富明慧, 廖子菊, 刘祚秋. 结构动力方程的样条精细积分法[J]. 计算力学学报, 2009,26(3): 379-384.(FU Minghui, LIAO Ziju, LIU Zuoqiu. Spline precise time-integration of structural dynamic analysis[J]. Chinese Journal of Computational Mechanics,2009,26(3): 379-384.(in Chinese))
[18]	张文志, 富明慧, 蓝林华. 两端边值问题的通用精细积分法[J]. 中山大学学报(自然科学版), 2010,49(6): 15-19.(ZHANG Wenzhi, FU Minghui, LAN Linhua. General precise integration method for two-point boundary value problems[J]. Acta Scientiarum Naturalium University Sunyatseni,2010,49(6): 15-19.(in Chinese))
[19]	鲍四元, 邓子辰. 结构撞击响应的一种弹性模型及其精细求解[J]. 工程力学, 2008,25(6): 14-17.(BAO Siyuan, DENG Zichen. An elastic modeland its precise solution for structural impact response[J]. Enginerrng Mechanics,2008,25(6): 14-17.(in Chinese))
[20]	吕和祥, 于洪洁, 裘春航. 精细积分的非线性动力学积分方程及其解法[J]. 固体力学学报, 2001,22(3): 303-308.(L Hexiang, YU Hongjie. An integral equation of non-linear dynamics and its solution method[J]. Acta Mechanica Solid Sinica,2001,22(3): 303-308.(in Chinese))
[21]	张洵安, 姜节胜. 结构非线性动力方程的精细积分算法[J]. 应用力学学报, 2000,〖STHZ〗 17(4): 164-168.(ZHANG Xun’an, JIANG Jiesheng. The precise integration algorithm for nonlinear dynamic equations of structures[J]. Chinese Journal of Applied Mechanics,2000,17(4): 164-168.(in Chinese))
[22]	梅树立, 张森文. 基于精细积分技术的非线性动力学方程的同伦摄动法[J]. 计算力学学报, 2005,22(6): 666-670.(MEI Shuli, ZHANG Senwen. Homotopy perturbation method for nonlinear dynamic equations based on precise integration technology[J]. Chinese Journal of Computational Mechanics,2005,22(6): 666-670.(in Chinese))
[23]	吴泽艳, 王立峰, 武哲. 大规模动力系统高精度増维精细积分方法快速算法[J]. 振动与冲击, 2014,33(2): 188-192.(WU Zeyan, WANG Lifeng, WU Zhe. Fast algorithm for precise integration with high accuracydimension expanding method with for large-scale dynamic systems[J]. Journal of Vibration and Shock,2014,33(2): 188-192.(in Chinese))
[24]	张洪武. 参变量变分原理与材料和结构力学分析[M]. 北京: 科学出版社, 2010.(ZHANG Hongwu. The Parameter Variation Principle and Materials and Structural Mechanics Analysis [M]. Beijing: Science Press, 2010.(in Chinese))
[25]	钟万勰, 姚征. 椭圆函数的精细积分算法: 应用力学进展[M]. 北京: 科学出版社, 2004.(ZHONG Wanxie, YAO Zheng. The Precise Integration Method for Jacobi Elliptic Functions and Application: Computation Method [M]. Beijing: Science Press, 2010.(in Chinese))
[26]	陈文, 孙洪广. 分数阶微分方程的数值算法: 现状和问题[J]. 计算机辅助工程, 2010,19(2): 1-2.(CHEN Wen, SUN Hongguang. Status and problems of numerical algorithm on fractional differential equations [J]. Computer Aided Engineering,2010,19(2): 1-2.(in Chinese))
[27]	银花, 陈宁, 赵尘, 等. 分数导数型粘弹性结构动力学方程的数值计算[J]. 南京林业大学学报, 2010,34(2): 115-118.(YIN Hua, CHEN Ning, ZHAO Chen, et al. A numerical algorithm of the dynamics equation of the fractional derivative viscoelasticity structure[J]. Journal of Nanjing Forestry University(Natrual Science Edition),2010,34(2): 115-118.(in Chinese))
[28]	薛齐文, 魏伟. 含分数阶导数微分方程的数值求解[J]. 大连交通大学学报, 2009,30(5): 88-92.(XUE Qiwen, WEI Wei. Numerical solution for differectial equations of fractional order[J]. Journal of Dalan Jiaotong University,2009,30(5): 88-92.(in Chinese))
[29]	王振滨, 曹广益. 分数微积分的两种系统建模方法[J]. 系统仿真学报, 2004,16(4): 810-812.(WANG Zhenbin, CAO Guangyi. Two system modeling methods using fractional calculus[J]. Journal of System Simulation,2004,16(4): 810-812.(in Chinese))
[30]	沈淑君, 刘发旺. 解分数阶Bagley-Torvik方程的一种计算有效的数值方法[J]. 厦门大学学报(自然科学版), 2004,43(3): 306-311.(SHEN Shujun, LIU Fawang. A computat ionally effective numerical method for the fractional order Bagley-Torvik equation[J]. Journal of Xiamen University(Natural Science),2004,43(3): 306-311.(in Chinese))
[31]	BAGLEY R L, TORVIK P J. On the appearance of the fractional derivative in the behavior of real materials[J]. Journal of Applied Mechanics,1984,51(2): 294-298.