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非线性分数阶常微分方程的分段线性插值多项式方法

高兴华 李宏 刘洋

高兴华, 李宏, 刘洋. 非线性分数阶常微分方程的分段线性插值多项式方法[J]. 应用数学和力学, 2021, 42(5): 531-540. doi: 10.21656/1000-0887.410149
引用本文: 高兴华, 李宏, 刘洋. 非线性分数阶常微分方程的分段线性插值多项式方法[J]. 应用数学和力学, 2021, 42(5): 531-540. doi: 10.21656/1000-0887.410149
GAO Xinghua, LI Hong, LIU Yang. A Piecewise Linear Interpolation Polynomial Method for Nonlinear Fractional Ordinary Differential Equations[J]. Applied Mathematics and Mechanics, 2021, 42(5): 531-540. doi: 10.21656/1000-0887.410149
Citation: GAO Xinghua, LI Hong, LIU Yang. A Piecewise Linear Interpolation Polynomial Method for Nonlinear Fractional Ordinary Differential Equations[J]. Applied Mathematics and Mechanics, 2021, 42(5): 531-540. doi: 10.21656/1000-0887.410149

非线性分数阶常微分方程的分段线性插值多项式方法

doi: 10.21656/1000-0887.410149
基金项目: 国家自然科学基金(11761053;11661058);内蒙古自然科学基金(2017MS0107;2020MS01003);内蒙古自治区青年科技英才支持计划(NJYT-17-A07)
详细信息
    作者简介:

    高兴华(1991—),女,博士生(E-mail: gaoxinghua1991@126.com);李宏(1973—),女,教授,博士生导师(通讯作者. E-mail: smslh@imu.edu.cn).

  • 中图分类号: O241.82

A Piecewise Linear Interpolation Polynomial Method for Nonlinear Fractional Ordinary Differential Equations

Funds: The National Natural Science Foundation of China(11761053;11661058)
  • 摘要: 通过分段线性插值多项式方法构造了一类含有Hadamard有限部分积分的非线性常微分方程的数值离散格式.在时间方向上, 利用分段线性插值多项式方法对分数阶导数项进行近似, 并通过二阶向后差分格式来离散整数阶导数项.经过详细的证明, 得到了收敛精度为O(τmin{1+α,1+β})的误差估计结果.最后,通过数值算例和理论结果的对比直观地说明了理论分析的正确性.
  • [1] LIU Y, DU Y W, LI H, et al. Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction-diffusion problem[J]. Computers and Mathematics With Applications,2015,70(4): 573-591.
    [2] CHEN H Z, WANG H. Numerical simulation for conservative fractional diffusion equations by an expanded mixed formulation[J]. Journal of Computational and Applied Mathematics,2016,296: 480-498.
    [3] SHI D Y, YANG H J. Superconvergence analysis of a new low order nonconforming MFEM for time-fractional diffusion equation[J]. Applied Numerical Mathematics,2018,131: 109-122.
    [4] LIU Y, DU Y W, LI H, et al. A two-grid finite element approximation for a nonlinear time-fractional Cable equation[J]. Nonlinear Dynamics,2016,85: 2535-2548.
    [5] HENRY B I, LANGLANDS T A M, WEARNE S L. Fractional cable models for spiny neuronal dendrites[J]. Physical Review Letters,2008,100(12): 128103.
    [6] GOUFO E F D. Stability and convergence analysis of a variable order replicator-mutator process in a moving medium[J]. Journal of Theoretical Biology,2016,403: 178-187.
    [7] 杨旭, 梁英杰, 孙洪广, 等. 空间分数阶非Newton流体本构及圆管流动规律研究[J]. 应用数学和力学, 2018,39(11): 1213-1226.(YANG Xu, LIANG Yingjie, SUN Hongguang, et al. A study on the constitutive relation and the flow of spatial fractional non-Newtonian fluid in circular pipes[J]. Applied Mathematics and Mechanics,2018,39(11): 1213-1226.(in Chinese))
    [8] KOH C G, KELLY J M. Application of fractional derivatives to seismic analysis of base-isolated models[J]. Earthquake Engineering and Structural Dynamics,1990,19(2): 229-241.
    [9] KLAS A, MIKAEL E, STIG L. Adaptive discretization of fractional order viscoelasticity using sparse time history[J]. Computer Methods in Applied Mechanics and Engineering,2004,193(42/44): 4567-4590.
    [10] LI C, ZHAO S. Efficient numerical schemes for fractional water wave models[J]. Computers and Mathematics With Applications,2016,71(1): 238-254.
    [11] LIU Y, YU Z D, LI H, et al. Time two-mesh algorithm combined with finite element method for time fractional water wave model[J]. International Journal of Heat and Mass Transfer,2018,120: 1132-1145.
    [12] ZHANG J, XU C J. Finite difference/spectral approximations to a water wave model with a nonlocal viscous term[J]. Applied Mathematical Modelling,2014,38(19): 4912-4925.
    [13] 孙志忠, 高广花. 分数阶微分方程的有限差分方法[M]. 北京: 科学出版社, 2015.(SUN Zhizhong, GAO Guanghua. Finite Difference Method for Fractional Differential Equations [M]. Beijing: Science Press, 2015.(in Chinese))
    [14] DIETHELM K. An algorithm for the numerical solution of differential equations of fractional order[J]. Electronic Transactions on Numerical Analysis,1997,5: 1-6.
    [15] DIETHELM K, FORD N J, FREED A D, et al. Algorithms for the fractional calculus: a selection of numerical methods[J]. Computer Methods in Applied Mechanics and Engineering,2005,194(6/8): 743-773.
    [16] LI Z Q, YAN Y B, FORD N J. Error estimates of a high order numerical method for solving linear fractional differential equations[J].Applied Numerical Mathematics,2017,114: 201-220.
    [17] HADAMARD J, MORSE P M. Lectures on Cauchy’s Problem in Linear Partial Differential Equations [M]. New York: Dover Publications, 1953.
    [18] MONEGATO G. Definitions, properties and applications of finite-part integrals[J]. Journal of Computational and Applied Mathematics,2008,229(2): 425-439.
    [19] MA L, LI C P. On finite part integrals and Hadamard-type fractional derivatives[J]. Journal of Computational and Nonlinear Dynamics,2018,13(9): 090905.
    [20] GALAPON E A. The Cauchy principal value and the Hadamard finite part integral as values of absolutely convergent integrals[J]. Journal of Mathematical Physics,2016,57(3): 033502.
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出版历程
  • 收稿日期:  2020-05-26
  • 修回日期:  2020-11-16
  • 刊出日期:  2021-05-01

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