JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2024, Vol. 59 ›› Issue (8): 77-83.doi: 10.6040/j.issn.1671-9352.0.2022.626

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Iterated fractional Tikhonov method for simultaneous inversion of the source term and initial data in time-fractional diffusion equations

Wenhui DU(),Xiangtuan XIONG*()   

  1. College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China
  • Received:2022-11-28 Online:2024-08-20 Published:2024-07-31
  • Contact: Xiangtuan XIONG E-mail:duwenhui1227@163.com;xiongxt@fudan.edu.com

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Abstract:

The inverse problem of identifying the space-dependent source term and the initial value simultaneously for a time-fractional diffusion equation is investigated. This inverse problem is reformulated on the basis of Fourier method as operator equations of the first kind. An iterative fractional Tikhonov regularization method is proposed to solve this inverse problem. In addition, a prior regularization parameter choice rule is given and the corresponding convergence estimation is proved.

Key words: time-fractional diffusion equation, conditional stability, iterative fractional Tikhonov method, error estimation

CLC Number: 

  • O241.8
1 LIUJ J,YAMAMOTOM.A backward problem for the time-fractional diffusion equation[J].Applicable Analysis,2010,89(11):1769-1788.
doi: 10.1080/00036810903479731
2 WANGL Y,LIUJ J.Data regularization for a backward time-fractional diffusion problem[J].Computers & Mathematics with Applications,2012,64(11):3613-3626.
3 XIONGX T,XUEX M.A fractional Tikhonov regularization method for identifying a space-dependent source in the time-fractional diffusion equation[J].Applied Mathematics and Computation,2019,349,292-303.
doi: 10.1016/j.amc.2018.12.063
4 WENJ,LIUZ X,WANGS S.Conjugate gradient method for simultaneous identification of the source term and initial data in a time-fractional diffusion equation[J].Applied Mathematics in Science and Engineering,2022,30(1):324-338.
doi: 10.1080/27690911.2022.2075358
5 PODLUBNYI.Fractional differential equations[J].Mathematics in Science and Engineering,1999,198,41-119.
doi: 10.1016/S0076-5392(99)80021-6
6 RUANZ S,YANGJ Z,LUX L.Tikhonov regularisation method for simultaneous inversion of the source term and initial data in a time-fractional diffusion equation[J].East Asian Journal on Applied Mathematics,2015,5(3):273-300.
doi: 10.4208/eajam.310315.030715a
7 SAOULIN,ZOUYEDF.A modified Tikhonov regularization method for a class of inverse parabolic problems[J].Analeleştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică,2020,28(1):181-504.
doi: 10.2478/auom-2020-0013
8 BIANCHID,BUCCINIA,DONATELLIM,et al.Iterated fractional Tikhonov regularization[J].Inverse Problems,2015,31(5):055005.
doi: 10.1088/0266-5611/31/5/055005
9 YANGShuping,XIONGXiangtuan,NIEYan.Iterated fractional Tikhonov regularization method for solving the spherically symmetric backward time-fractional diffusion equation[J].Applied Numerical Mathematics,2021,160,217-241.
doi: 10.1016/j.apnum.2020.10.008
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