数学物理学报(英文版) ›› 2022, Vol. 42 ›› Issue (4): 1485-1518.doi: 10.1007/s10473-022-0412-5

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TWO REGULARIZATION METHODS FOR IDENTIFYING THE SOURCE TERM PROBLEM ON THE TIME-FRACTIONAL DIFFUSION EQUATION WITH A HYPER-BESSEL OPERATOR

杨帆, 孙乔夕, 李晓晓   

  1. Department of Mathematics, Lanzhou University of Technology, Lanzhou, 730000, China
  • 收稿日期:2021-02-23 修回日期:2021-08-19 出版日期:2022-08-25 发布日期:2022-08-23
  • 通讯作者: Fan YANG,E-mail:yfggd114@163.com E-mail:yfggd114@163.com
  • 基金资助:
    The project is supported by the National Natural Science Foundation of China (11961044), the Doctor Fund of Lan Zhou University of Technology, and the Natural Science Foundation of Gansu Provice (21JR7RA214).

TWO REGULARIZATION METHODS FOR IDENTIFYING THE SOURCE TERM PROBLEM ON THE TIME-FRACTIONAL DIFFUSION EQUATION WITH A HYPER-BESSEL OPERATOR

Fan YANG, Qiaoxi SUN, Xiaoxiao LI   

  1. Department of Mathematics, Lanzhou University of Technology, Lanzhou, 730000, China
  • Received:2021-02-23 Revised:2021-08-19 Online:2022-08-25 Published:2022-08-23
  • Contact: Fan YANG,E-mail:yfggd114@163.com E-mail:yfggd114@163.com
  • Supported by:
    The project is supported by the National Natural Science Foundation of China (11961044), the Doctor Fund of Lan Zhou University of Technology, and the Natural Science Foundation of Gansu Provice (21JR7RA214).

摘要: In this paper, we consider the inverse problem for identifying the source term of the time-fractional equation with a hyper-Bessel operator. First, we prove that this inverse problem is ill-posed, and give the conditional stability. Then, we give the optimal error bound for this inverse problem. Next, we use the fractional Tikhonov regularization method and the fractional Landweber iterative regularization method to restore the stability of the ill-posed problem, and give corresponding error estimates under different regularization parameter selection rules. Finally, we verify the effectiveness of the method through numerical examples.

关键词: Time-fractional diffusion equation, source term problem, fractional Landweber regularization method, Hyper-Bessel operator, fractional Tikhonov regularization method

Abstract: In this paper, we consider the inverse problem for identifying the source term of the time-fractional equation with a hyper-Bessel operator. First, we prove that this inverse problem is ill-posed, and give the conditional stability. Then, we give the optimal error bound for this inverse problem. Next, we use the fractional Tikhonov regularization method and the fractional Landweber iterative regularization method to restore the stability of the ill-posed problem, and give corresponding error estimates under different regularization parameter selection rules. Finally, we verify the effectiveness of the method through numerical examples.

Key words: Time-fractional diffusion equation, source term problem, fractional Landweber regularization method, Hyper-Bessel operator, fractional Tikhonov regularization method

中图分类号: 

  • 35R25