Simultaneous inversion of the source term and initial value of the time fractional diffusion equation
Abstract
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.
Keyword : time fractional diffusion equation, source term and initial value, inverse problem, ill-posed, modified Tikhonov method
How to Cite
Yang, F., Xu, J.- ming, & Li, X.- xiao. (2024). Simultaneous inversion of the source term and initial value of the time fractional diffusion equation. Mathematical Modelling and Analysis, 29(2), 193–214. https://doi.org/10.3846/mma.2024.18133
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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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
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
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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