## 留言板 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 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. ## 分数阶Cable方程的有限点法分析

##### doi: 10.21656/1000-0887.420183

###### 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 method the radial basis function method the finite point method the radial basis function method 1/10 1.670 9E−5 4.515 2E−4 2.256 6E−5 3.831 0E−4 1/20 1.110 9E−5 2.542 5E−4 1.409 1E−5 1.926 8E−4 1/40 7.983 2E−6 1.557 3E−4 9.429 7E−6 9.759 9E−4 1/80 6.275 8E−6 1.065 3E−4 6.903 4E−6 5.011 6E−5
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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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