Close Search Advanced
Journals
Resources
About Us
Open Access

Tikhonov Regularisation Method for Simultaneous Inversion of the Source Term and Initial Data in a Time-Fractional Diffusion Equation

Tikhonov Regularisation Method for Simultaneous Inversion of the Source Term and Initial Data in a Time-Fractional Diffusion Equation
Preview
Add to basket

Year:    2015

East Asian Journal on Applied Mathematics, Vol. 5 (2015), Iss. 3 : pp. 273–300

Abstract

The inverse problem of identifying the time-independent source term and initial value simultaneously for a time-fractional diffusion equation is investigated. This inverse problem is reformulated into an operator equation based on the Fourier method. Under a certain smoothness assumption, conditional stability is established. A standard Tikhonov regularisation method is proposed to solve the inverse problem. Furthermore, the convergence rate is given for an a priori and a posteriori regularisation parameter choice rule, respectively. Several numerical examples, including one-dimensional and two-dimensional cases, show the efficiency of our proposed method.

Submit Article

You do not have full access to this article.

Already a Subscriber? Sign in as an individual or via your institution

Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/eajam.310315.030715a

East Asian Journal on Applied Mathematics, Vol. 5 (2015), Iss. 3 : pp. 273–300

Published online:    2015-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    28

Keywords:    Time-fractional diffusion equation conditional stability Tikhonov regularisation Morozov discrepancy principle convergence rate.

  1. Simultaneous determination of the space-dependent source and initial value for a two-dimensional heat conduction equation

    Qiao, Yu | Xiong, Xiangtuan

    Computers & Mathematics with Applications, Vol. 147 (2023), Iss. P.25

    https://doi.org/10.1016/j.camwa.2023.07.009 [Citations: 8]

  2. Simultaneous Inversion of the Space-Dependent Source Term and the Initial Value in a Time-Fractional Diffusion Equation

    Yu, Shuang | Wang, Zewen | Yang, Hongqi

    Computational Methods in Applied Mathematics, Vol. 23 (2023), Iss. 3 P.767

    https://doi.org/10.1515/cmam-2022-0058 [Citations: 2]

  3. Simultaneous determination of source term and the initial value in the space-fractional diffusion problem by a novel modified quasi-reversibility regularization method

    Wen, Jin | Ren, Xue-Juan | Wang, Shi-Juan

    Physica Scripta, Vol. 98 (2023), Iss. 2 P.025201

    https://doi.org/10.1088/1402-4896/acaa68 [Citations: 3]

  4. Simultaneous inversion of the fractional order and the space-dependent source term for the time-fractional diffusion equation

    Ruan, Zhousheng | Zhang, Wen | Wang, Zewen

    Applied Mathematics and Computation, Vol. 328 (2018), Iss. P.365

    https://doi.org/10.1016/j.amc.2018.01.025 [Citations: 11]

  5. SIMULTANEOUS INVERSION OF THE SOURCE TERM AND INITIAL VALUE OF THE TIME FRACTIONAL DIFFUSION EQUATION

    Yang, Fan | Xu, Jian-ming | Li, Xiao-xiao

    Mathematical Modelling and Analysis, Vol. 29 (2024), Iss. 2 P.193

    https://doi.org/10.3846/mma.2024.18133 [Citations: 0]

  6. Simultaneous inversion of two initial values for a time‐fractional diffusion‐wave equation

    Zhang, Yun | Wei, Ting | Zhang, Yuan‐Xiang

    Numerical Methods for Partial Differential Equations, Vol. 37 (2021), Iss. 1 P.24

    https://doi.org/10.1002/num.22517 [Citations: 12]

  7. A non-iterative method for recovering the space-dependent source and the initial value simultaneously in a parabolic equation

    Wang, Zewen | Chen, Shuli | Qiu, Shufang | Wu, Bin

    Journal of Inverse and Ill-posed Problems, Vol. 28 (2020), Iss. 4 P.499

    https://doi.org/10.1515/jiip-2019-0017 [Citations: 7]

  8. The backward problem for a time-fractional diffusion-wave equation in a bounded domain

    Wei, Ting | Zhang, Yun

    Computers & Mathematics with Applications, Vol. 75 (2018), Iss. 10 P.3632

    https://doi.org/10.1016/j.camwa.2018.02.022 [Citations: 69]

  9. Iterated fractional Tikhonov method for recovering the source term and initial data simultaneously in a two-dimensional diffusion equation

    Qiao, Yu | Xiong, Xiangtuan

    Journal of Computational and Applied Mathematics, Vol. 451 (2024), Iss. P.116062

    https://doi.org/10.1016/j.cam.2024.116062 [Citations: 0]

  10. Regularization with two differential operators and its application to inverse problems

    Yu, Shuang | Yang, Hongqi

    Applied Numerical Mathematics, Vol. 201 (2024), Iss. P.129

    https://doi.org/10.1016/j.apnum.2024.02.017 [Citations: 0]

  11. A nonstationary iterated quasi-boundary value method for reconstructing the source term in a time–space fractional diffusion equation

    Zhang, Yun | Feng, Xiaoli

    Journal of Computational and Applied Mathematics, Vol. 440 (2024), Iss. P.115612

    https://doi.org/10.1016/j.cam.2023.115612 [Citations: 0]

  12. A FRACTIONAL LANDWEBER ITERATION METHOD FOR SIMULTANEOUS INVERSION IN A TIME-FRACTIONAL DIFFUSION EQUATION

    Wen, Jin | Yue, Chong-Wang | Liu, Zhuan-Xia | O'Regan, Donal

    Journal of Applied Analysis & Computation, Vol. 13 (2023), Iss. 6 P.3374

    https://doi.org/10.11948/20230051 [Citations: 0]

  13. A mollifier approach to the simultaneous identification of the unknown source and initial distribution in a space-fractional diffusion equation

    Qiao, Yu | Xiong, Xiangtuan

    Applied Mathematics and Computation, Vol. 489 (2025), Iss. P.129175

    https://doi.org/10.1016/j.amc.2024.129175 [Citations: 0]

  14. FRACTIONAL TIKHONOV REGULARIZATION METHOD FOR SIMULTANEOUS INVERSION OF THE SOURCE TERM AND INITIAL DATA IN A TIME-FRACTIONAL DIFFUSION EQUATION

    Wen, Jin | Yue, Chong-Wang | Liu, Zhuan-Xia | Wang, Shi-Juan

    Rocky Mountain Journal of Mathematics, Vol. 53 (2023), Iss. 1

    https://doi.org/10.1216/rmj.2023.53.249 [Citations: 2]

  15. Identifying source term and initial value simultaneously for the time-fractional diffusion equation with Caputo-like hyper-Bessel operator

    Yang, Fan | Cao, Ying | Li, XiaoXiao

    Fractional Calculus and Applied Analysis, Vol. 27 (2024), Iss. 5 P.2359

    https://doi.org/10.1007/s13540-024-00304-1 [Citations: 0]

  16. Simultaneous identification of the unknown source term and initial value for the time fractional diffusion equation with local and nonlocal operators

    Qiao, Li | Yang, Fan | Li, Xiaoxiao

    Chaos, Solitons & Fractals, Vol. 189 (2024), Iss. P.115601

    https://doi.org/10.1016/j.chaos.2024.115601 [Citations: 0]

  17. Three Landweber iterative methods for solving the initial value problem of time-fractional diffusion-wave equation on spherically symmetric domain

    Yang, Fan | Sun, Qiao-Xi | Li, Xiao-Xiao

    Inverse Problems in Science and Engineering, Vol. 29 (2021), Iss. 12 P.2306

    https://doi.org/10.1080/17415977.2021.1914603 [Citations: 5]

  18. Conjugate gradient method for simultaneous identification of the source term and initial data in a time-fractional diffusion equation

    Wen, Jin | Liu, Zhuan-Xia | Wang, Shan-Shan

    Applied Mathematics in Science and Engineering, Vol. 30 (2022), Iss. 1 P.324

    https://doi.org/10.1080/27690911.2022.2075358 [Citations: 8]

  19. On the simultaneous reconstruction of the initial diffusion time and source term for the time-fractional diffusion equation

    Ruan, Zhousheng | Chen, Zhenxing | Luo, Min | Zhang, Wen

    International Journal of Computer Mathematics, Vol. 100 (2023), Iss. 11 P.2077

    https://doi.org/10.1080/00207160.2023.2260011 [Citations: 0]

  20. A non-stationary iterative Tikhonov regularization method for simultaneous inversion in a time-fractional diffusion equation

    Wen, Jin | Liu, Zhuan-Xia | Wang, Shan-Shan

    Journal of Computational and Applied Mathematics, Vol. 426 (2023), Iss. P.115094

    https://doi.org/10.1016/j.cam.2023.115094 [Citations: 9]

  21. The fractional Tikhonov regularization methods for identifying the initial value problem for a time-fractional diffusion equation

    Yang, Fan | Pu, Qu | Li, Xiao-Xiao

    Journal of Computational and Applied Mathematics, Vol. 380 (2020), Iss. P.112998

    https://doi.org/10.1016/j.cam.2020.112998 [Citations: 24]

  22. Well-posedness of the stochastic time-fractional diffusion and wave equations and inverse random source problems

    Lassas, Matti | Li, Zhiyuan | Zhang, Zhidong

    Inverse Problems, Vol. 39 (2023), Iss. 8 P.084001

    https://doi.org/10.1088/1361-6420/acdab9 [Citations: 1]

  23. Landweber iteration method for simultaneous inversion of the source term and initial data in a time-fractional diffusion equation

    Wen, Jin | Liu, Zhuan-Xia | Yue, Chong-Wang | Wang, Shi-Juan

    Journal of Applied Mathematics and Computing, Vol. 68 (2022), Iss. 5 P.3219

    https://doi.org/10.1007/s12190-021-01656-0 [Citations: 15]