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分数阶Cable方程的有限点法分析

陈虹伶 李小林

陈虹伶,李小林. 分数阶Cable方程的有限点法分析 [J]. 应用数学和力学,2022,43(6):700-706 doi: 10.21656/1000-0887.420183
引用本文: 陈虹伶,李小林. 分数阶Cable方程的有限点法分析 [J]. 应用数学和力学,2022,43(6):700-706 doi: 10.21656/1000-0887.420183
CHEN Hongling, LI Xiaolin. Analysis of the Finite Point Method for Fractional Cable Equations[J]. Applied Mathematics and Mechanics, 2022, 43(6): 700-706. doi: 10.21656/1000-0887.420183
Citation: CHEN Hongling, LI Xiaolin. Analysis of the Finite Point Method for Fractional Cable Equations[J]. Applied Mathematics and Mechanics, 2022, 43(6): 700-706. doi: 10.21656/1000-0887.420183

分数阶Cable方程的有限点法分析

doi: 10.21656/1000-0887.420183
基金项目: 国家自然科学基金(面上项目)(11971085);重庆市高校创新研究群体(CXQT19018);重庆市教委科学技术研究项目(重大项目)(KJZD-M201800501);重庆市研究生教育教学改革研究项目(yjg203063)
详细信息
    作者简介:

    陈虹伶(1996—),女,硕士 ( E-mail:913626174@qq.com)

    李小林(1983—),男,教授, 博士 (通讯作者. E-mail:lxlmath@163.com)

  • 中图分类号: O241.82

Analysis of the Finite Point Method for Fractional Cable Equations

  • 摘要:

    通过采用中心差分格式离散Riemann-Liouville时间分数阶导数和用有限点法建立离散代数系统,提出了数值求解分数阶Cable方程的无网格有限点法,详细推导了该方法的理论误差估计。数值算例证实了该方法的有效性和收敛性,并验证了理论分析结果。

  • 图  1  算例在$ \alpha = 0.2 $$ \beta = 0.8 $$ T = 5 $$ h = {1 /{20}} $$ \tau = 1/20 $时的数值解和误差:(a) 数值解;(b) 误差

    Figure  1.  Numerical solution results and errors gained with $ \alpha = 0.2 $, $ \beta = 0.8 $, $ T = 5 $$ h = 1/20 $ and $ \tau = 1/20 $: (a) numerical solution results; (b) errors

    图  2  h=0.01,T=1时误差与时间步长$ \tau $的关系:(a) 相对误差;(b) $ {L^\infty } $误差

    Figure  2.  The relationship between relative errors and $ {L^\infty } $ errors obtained for h=0.01 and T=1 with respect to time-step size $ \tau $: (a) relative errors ; (b) $ {L^\infty } $ errors

    图  3  $\tau = 0.000\;1,\;T=1$时误差与节点间距$ h $的关系:(a) 相对误差;(b) $ {L^\infty } $误差

    Figure  3.  The relationship between relative errors and $ {L^\infty } $ errors obtained for $\tau = 0.000\;1\; {\rm{and}} \;\;T=1$ with respect to nodal spacing $ h $: (a) relative errors ; (b) $ {L^\infty } $ errors

    图  4  算例在$ \gamma = 0.4 $$ T = 5 $$ h = {1 / {20}} $$ \tau = 1/20 $时的数值解和误差:(a) 数值解;(b) 误差

    Figure  4.  Numerical solution results and errors gained with $ \gamma = 0.4 $, $ T = 5 $, $ h = 1/20 $ and $ \tau = 1/20 $: (a) numerical solution results; (b) errors

    图  5  误差与时间步长$ \tau $和节点间距$ h $的关系:(a) 时间步长$ \tau $;(b) 节点间距$ h $

    Figure  5.  The relationship between the errors and time-step size $ \tau $ as well as nodal spacing $ h $: (a) for time-step size $ \tau $; (b) for nodal spacing $ h $

    表  1  有限点法和径向基函数法在$h = 0.1 ,\;T=1$时的$ {L^\infty } $误差

    Table  1.   The $ {L^\infty } $-errors of the finite point method and the radial basis function method gained with $h = 0.1,\; T=1$

    $ \tau $$ \gamma = 0.25 $$ \gamma = 0.3 $
    the finite point methodthe radial basis function method[8]the finite point methodthe radial basis function method[8]
    1/101.670 9E−54.515 2E−42.256 6E−53.831 0E−4
    1/201.110 9E−52.542 5E−41.409 1E−51.926 8E−4
    1/407.983 2E−61.557 3E−49.429 7E−69.759 9E−4
    1/806.275 8E−61.065 3E−46.903 4E−65.011 6E−5
    下载: 导出CSV
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出版历程
  • 收稿日期:  2021-07-02
  • 修回日期:  2021-09-19
  • 网络出版日期:  2022-06-02
  • 刊出日期:  2022-06-30

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