## 留言板

 引用本文: 鲍四元, 沈峰. 分数阶常微分方程的改进精细积分法[J]. 应用数学和力学, 2019, 40(12): 1309-1320.
BAO Siyuan, SHEN Feng. An Improved Precise Integration Method for Fractional Ordinary Differential Equations[J]. Applied Mathematics and Mechanics, 2019, 40(12): 1309-1320. doi: 10.21656/1000-0887.390355
 Citation: BAO Siyuan, SHEN Feng. An Improved Precise Integration Method for Fractional Ordinary Differential Equations[J]. Applied Mathematics and Mechanics, 2019, 40(12): 1309-1320.

• 中图分类号: O175

## An Improved Precise Integration Method for Fractional Ordinary Differential Equations

Funds: The National Natural Science Foundation of China(11202146；51709194)
• 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.
•  [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.
##### 计量
• 文章访问数:  1539
• HTML全文浏览量:  296
• PDF下载量:  423
• 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

/

• 分享
• 用微信扫码二维码

分享至好友和朋友圈