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一类非线性时间分数阶扩散方程反问题的变分型正则化

柳冕 程浩 石成鑫

柳冕,程浩,石成鑫. 一类非线性时间分数阶扩散方程反问题的变分型正则化 [J]. 应用数学和力学,2022,43(X):1-12 doi: 10.21656/1000-0887.420168
引用本文: 柳冕,程浩,石成鑫. 一类非线性时间分数阶扩散方程反问题的变分型正则化 [J]. 应用数学和力学,2022,43(X):1-12 doi: 10.21656/1000-0887.420168
LIU Mian, CHENG Hao, SHI Chengxin. Variational Regularization For The Inverse Problem of a Nonlinear Time-Fractional Diffusion Equations[J]. Applied Mathematics and Mechanics. doi: 10.21656/1000-0887.420168
Citation: LIU Mian, CHENG Hao, SHI Chengxin. Variational Regularization For The Inverse Problem of a Nonlinear Time-Fractional Diffusion Equations[J]. Applied Mathematics and Mechanics. doi: 10.21656/1000-0887.420168

一类非线性时间分数阶扩散方程反问题的变分型正则化

doi: 10.21656/1000-0887.420168
基金项目: 国家自然科学基金(11426117)
详细信息
    作者简介:

    柳冕(1997—),男,硕士生(E-mali:819340002@qq.com

    程浩(1983—),男,副教授,硕士生导师(通讯作者。E-mail:chenghao@jiangnan.edu.cn

    石成鑫(1997—),男,硕士生(E-mail:1772065320@qq.com

  • 中图分类号: O241.8

Variational Regularization For The Inverse Problem of a Nonlinear Time-Fractional Diffusion Equations

  • 摘要: 考虑了一类二维非线性时间分数阶扩散方程,并从最终位置获取的测量数据来反演物质在$ u(0,y,t) $处的物理信息。这个问题是严重不适定的,即问题的解并不连续依赖于测量数据,因此提出了变分型正则化方法来稳定求解该问题。给出了精确解与正则近似解之间的误差估计,数值算例验证了该方法的有效性。
  • 图  1  $ {x}=0 $时的精确解和不同误差水平下正则近似解$ {{u}}^{{\alpha },{\varepsilon }} $

    Figure  1.  The exact solution $ u $ at $ x=0 $ and the regularized solution $ {u}^{\alpha ,\varepsilon } $ corresponding to different error levels

    图  2  正则化解和精确解在不同x的取值下的比较(y=0)

    Figure  2.  The exact solution and its regularized solution corresponding to different x (y=0)

    表  1  不同误差水平下的相对误差$ {E}_{{\rm{r}}} $

    Table  1.   Relative error corresponding to different error levels

    $ {{E}}_{\rm{r}} $
    $ {x} $$ {\varepsilon }=1{\rm{E}}-1 $$ {\varepsilon }=1{\rm{E}}-2 $$ {\varepsilon }=1{\rm{E}}-3 $
    02.749E−12.685E−12.682E−1
    0.12.639E−12.522E−12.520E−1
    0.22.409E−12.367E−12.348E−1
    0.32.251E−12.219E−12.227E−1
    0.42.047E−12.017E−12.026E−1
    0.51.848E−11.709E−11.713E−1
    0.61.551E−11.366E−11.345E−1
    0.71.213E−11.030E−11.026E−1
    0.81.003E−18.010E−27.600E−2
    0.99.800E−29.750E−29.470E−2
    下载: 导出CSV
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出版历程
  • 收稿日期:  2021-06-17
  • 修回日期:  2021-08-26
  • 网络出版日期:  2022-01-08

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