Simultaneous inversion of the source term and initial value of the time fractional diffusion equation
DOI: https://doi.org/10.3846/mma.2024.18133Abstract
In this paper, the problem we investigate is to simultaneously identify the source term and initial value of the time fractional diffusion equation. This problem is ill-posed, i.e., the solution (if exists) does not depend on the measurable data. We give the conditional stability result under the a-priori bound assumption for the exact solution. The modified Tikhonov regularization method is used to solve this problem, and under the a-priori and the a-posteriori selection rule for the regularization parameter, the convergence error estimations for this method are obtained. Finally, numerical example is given to prove the effectiveness of this regularization method.
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time fractional diffusion equation, source term and initial value, inverse problem, ill-posed, modified Tikhonov methodHow to Cite
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Copyright (c) 2024 The Author(s). Published by Vilnius Gediminas Technical University.
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References
J. Cheng, J. Nakagama, M. Yamamoto and T. Yamazaki. Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation. Inverse Problems, 25(11):115002, 2009. https://doi.org/10.1088/0266-5611/25/11/115002"> https://doi.org/10.1088/0266-5611/25/11/115002
M.I. Ismailov and M. Cicek. Inverse source problem for a time-fractional diffusion equation with nonlocal boundary conditions. Applied Mathematical Modelling, 40(7–8):4891–4899, 2016. https://doi.org/10.1016/j.apm.2015.12.020"> https://doi.org/10.1016/j.apm.2015.12.020
J. Janno and N. Kinash. Reconstruction of an order of derivative and a source term in a fractional diffusion equation from final measurements. Inverse Problems, 34(2):025007, 2018. https://doi.org/10.1088/1361-6420/aaa0f0"> https://doi.org/10.1088/1361-6420/aaa0f0
G.S. Li, D.L. Zhang, X.Z. Jia and M. Yamamoto. Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation. Inverse Problems, 29(6):065014, 2013. https://doi.org/10.1088/0266-5611/29/6/065014"> https://doi.org/10.1088/0266-5611/29/6/065014
K.F. Liao and T. Wei. Identifying a fractional order and a space source term in a time-fractional diffusion-wave equation simultaneously. Inverse Problems, 35(11):115002, 2019. https://doi.org/10.1088/1361-6420/ab383f"> https://doi.org/10.1088/1361-6420/ab383f
J.J. Liu and M. Yamamoto. A backward problem for the timefractional diffusion equation. Applicable Analysis, 89(11):1769–1788, 2010. https://doi.org/10.1080/00036810903479731"> https://doi.org/10.1080/00036810903479731
Y.K. Liu, W. Rundell and M. Yamamoto. Strong maximum principle for fractional diffusion equations and an application to an inverse source problem. Fractional Calculus and Applied Analysis, 19(4):888–906, 2016. https://doi.org/10.1515/fca-2016-0048"> https://doi.org/10.1515/fca-2016-0048
I. Podlubny. Fractional differential equations. Academic Press, New York, 1999.
S.F. Qiu, W. Zhang and J.M. Peng. Simultaneous determination of the space-dependent source and the initial distribution in a heat equation by regularizing Fourier coefficients of the given measurements. Advances in Mathematical Physics, 2018:8247584, 2018. https://doi.org/10.1155/2018/8427584"> https://doi.org/10.1155/2018/8427584
Z.S. Ruan, Z.J. Jerry and X.L. Lu. Tikhonov regularisation method for simultaneous inversion of the source term and initial data in a time-fractional diffusion equation. East Asian Journal on Applied Mathematics, 5(3):273–300, 2015. https://doi.org/10.4208/eajam.310315.030715a"> https://doi.org/10.4208/eajam.310315.030715a
J.G. Wang, T. Wei and Y.B. Zhou. Tikhonov regularization method for a backward problem for the time-fractional diffusion equation. Applied Mathematical Modelling, 37(18–19):8518–8532, 2013. https://doi.org/10.1016/j.apm.2013.03.071"> https://doi.org/10.1016/j.apm.2013.03.071
J.G. Wang, T. Wei and Y.B. Zhou. Optimal error bound and simplified Tikhonov regularization method for a backward problem for the time-fractional diffusion equation. Journal of computational and applied mathematics, 279(18–19):277– 292, 2015. https://doi.org/10.1016/j.cam.2014.11.026"> https://doi.org/10.1016/j.cam.2014.11.026
J.G. Wang, Y.B. Zhou and T. Wei. A posteriori regularization parameter choice rule for the quasi-boundary value method for the backward timefractional diffusion problem. Applied Mathematics Letters, 26(7):741–747, 2013. https://doi.org/10.1016/j.aml.2013.02.006"> https://doi.org/10.1016/j.aml.2013.02.006
T. Wei and J.G. Wang. A modified quasi-boundary value method for the backward time-fractional diffusion problem. ESAIM: Mathematical Modelling and Numerical Analysis, 48(2):603–621, 2014. https://doi.org/10.1051/m2an/2013107"> https://doi.org/10.1051/m2an/2013107
J. Wen, Z.X. Liu and S.S. Wang. Conjugate gradient method for simultaneous identification of the source term and initial data in a time-fractional diffusion equation. Applied Mathematics in Science and Engineering, 30(1):324–338, 2022. https://doi.org/10.1080/27690911.2022.2075358"> https://doi.org/10.1080/27690911.2022.2075358
J. Wen, Z.X. Liu and S.S. Wang. A non-stationary iterative Tikhonov regularization method for simultaneous inversion in a time-fractional diffusion equation. Journal of Computational and Applied Mathematics, 426:115094, 2023. https://doi.org/10.1016/j.cam.2023.115094"> https://doi.org/10.1016/j.cam.2023.115094
J. Wen, X.J. Ren and S.J. Wang. Simultaneous determination of source term and initial value in the heat conduction problem by modified quasi-reversibility regularization method. Numerical Heat Transfer, Part B: Fundamentals, 82(3–4):112–124, 2022. https://doi.org/10.1080/10407790.2022.2079281"> https://doi.org/10.1080/10407790.2022.2079281
J. Wen, X.J. Ren and S.J. Wang. Simultaneous determination of source term and the initial value in the space-fractional diffusion problem by a novel modified quasi-reversibility regularization method. Physica Scripta, 98(2):025201, 2023. https://doi.org/10.1088/1402-4896/acaa68"> https://doi.org/10.1088/1402-4896/acaa68
J. Wen, M. Yamamoto and T. Wei. Simultaneous determination of a timedependent heat source and the initial temperature in an inverse heat conduction problem. Inverse Problems in Science and Engineering, 21(3):485–499, 2013. https://doi.org/10.1080/17415977.2012.701626"> https://doi.org/10.1080/17415977.2012.701626
X.T. Xiong, H.B. Guo and X.H. Li. An inverse problem for a fractional diffusion equation. Journal of Computational and Applied Mathematics, 236(17):4474–4484, 2012. https://doi.org/10.1016/j.cam.2012.04.019"> https://doi.org/10.1016/j.cam.2012.04.019
X.T. Xiong, W.X. Shi and X.M. Xue. Determination of three parameters in a time-space fractional diffusion equation. AIMS Mathematics, 6(6):5909–5923, 2021. https://doi.org/10.3934/math.2021350"> https://doi.org/10.3934/math.2021350
F. Yang, Y.P. Ren and X.X. Li. The quasi-reversibility method for a final value problem of the time-fractional diffusion equation with inhomogeneous source. Mathematical Methods in the Applied Sciences, 41(5):1774–1795, 2018. https://doi.org/10.1002/mma.4705"> https://doi.org/10.1002/mma.4705
F. Yang, Y.P. Ren, X.X. Li and D.G. Li. Landweber iterative method for identifying a space-dependent source for the time-fractional diffusion equation.Boundary Value Problems,2017(1):1–19, 2017. https://doi.org/10.1186/s13661-017-0898-2"> https://doi.org/10.1186/s13661-017-0898-2
F. Yang, H.H. Wu and X.X. Li. Three regularization methods for identifying the initial value of time fractional advection–dispersion equation. Computational and Applied Mathematics, 41(1):1–38, 2022. https://doi.org/10.1007/s40314-022-01762-0"> https://doi.org/10.1007/s40314-022-01762-0
M. Yang and J.J. Liu. Solving a final value fractional diffusion problem by boundary condition regularization. Applied Numerical Mathematics, 66(1):45– 58, 2013. https://doi.org/10.1016/j.apnum.2012.11.009"> https://doi.org/10.1016/j.apnum.2012.11.009
S. Yu, Z.W. Wang and H.Q. Yang. Simultaneous inversion of the space-dependent source term and the initial value in a time-fractional diffusion equation. Computational Methods in Applied Mathematics, 2023. https://doi.org/10.1515/cmam-2022-0058"> https://doi.org/10.1515/cmam-2022-0058
Y. Zhang, T. Wei and Y.X. Zhang. Simultaneous inversion of two initial values for a time-fractional diffusion-wave equation. Numerical Methods for Partial Differential Equations, 37(1):24–43, 2021. https://doi.org/10.1002/num.22517"> https://doi.org/10.1002/num.22517
Z.Q. Zhang and T. Wei. Identifying an unknown source in time-fractional diffusion equation by a truncation method. Applied Mathematics and Computation, 219(11):5972–5983, 2013. https://doi.org/10.1016/j.amc.2012.12.024"> https://doi.org/10.1016/j.amc.2012.12.024
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