多项时间分数阶抛物型方程反源问题的拟逆方法
A Quasi-Inverse Method for Inverse Problems of Parabolic Equationsof the Time Fractional Order
摘要: 本文利用分数阶拟逆方法解决多项时间分数阶抛物型方程的反源问题，该反问题是不适定的。 首先给出了反问题的条件稳定性，然后提出分数阶拟逆方法，即在原方程中引入了与椭圆微分算子 有关的新的扰动项，最后基于多项Mittag-Leffler函数的一些性质，在理论上我们给出了正则化解在先验正则化参数选择规则下相应的收敛速度。
Abstract: In this paper, the fractional quasi-inverse method is used to solve the inverse source problem of polynomial time fractional parabolic equations, which is ill-posed. First- ly, the conditional stability of the inverse problem is given, and then the fractional quasi-inverse method is proposed, that is, the perturbation term related to elliptic differential operator is introduced into the original equation. Finally, based on some properties of Mittag-Leffler function, the corresponding convergence rate of the regu- lar solution under the prior selection rule is given in theory.
文章引用：王雨欣. 多项时间分数阶抛物型方程反源问题的拟逆方法[J]. 应用数学进展, 2023, 12(6): 2861-2875. https://doi.org/10.12677/AAM.2023.126288

参考文献

[1] Ionescu, C.-M., Lopes, A., Copot, D., Machado, J.A.T. and Bates, J.H.T. (2017) The Role of Fractional Calculus in Modelling Biological Phenomena. Communications in Nonlinear Science and Numerical Simulation, 51, 141-159.
https://doi.org/10.1016/j.cnsns.2017.04.001
[2] Hilfer, R. (2000) Applications of Fractional Calculus in Physics. World Scientific, Singapore.
https://doi.org/10.1142/3779
[3] Laskin, N. (2017) Time Fractional Quantum Mechanics. Chaos, Solitons & Fractals, 102, 16-28.
https://doi.org/10.1016/j.chaos.2017.04.010
[4] Tarasov, V.E. (2010) Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer-Verlag, Beijing.
[5] Machado, J.A.T. and Lopes, A.M. (2016) Relative Fractional Dynamics of Stock Markets. Nonlinear Dynamics, 29, 1613-1619.
https://doi.org/10.1007/s11071-016-2980-1
[6] Jin, B. and Rundell, W. (2015) A Tutorial on Inverse Problems for Anomalous Diffusion Pro- cesses. Inverse Problems, 31, Article 035003.
https://doi.org/10.1088/0266-5611/31/3/035003
[7] Caputo, M., Carcione, J. and Botelho, M. (2015) Modeling Extreme-Event Precursors with the Fractional Diffusion Equation. Fractional Calculus and Applied Analysis, 18, 208-222.
https://doi.org/10.1515/fca-2015-0014
[8] Weiss, G. (1994) Aspects and Applications of the Random Walk. North-Holland, Amsterdam.
[9] Podlubny, I. (1999) Fractional Differential Equations: An Introduction to Fractional Deriva- tives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Ap- plications. In: Mathematics in Science and Engineering, Vol. 198, Academic Press, Cambridge, MA.
[10] Zhang, Y. and Xu, X. (2011) Inverse Source Problem for a Fractional Diffusion Equation. Inverse Problems, 27, Article 035010.
https://doi.org/10.1088/0266-5611/27/3/035010
[11] Wei, T., Sun, L.L. and Li, Y.S. (2016) Uniqueness for an Inverse Space-Dependent Source Term in a Multi-Dimensional Time-Fractional Diffusion Equation. Applied Mathematics Letters, 61, 108-113.
https://doi.org/10.1016/j.aml.2016.05.004
[12] Wei, T. and Wang, J.G. (2014) A Modified Quasi-Boundary Value Method for the Backward Time-Fractional Diffusion Problem. ESAIM: Mathematical Modelling and Numerical Analysis, 78, 95-111.
https://doi.org/10.1016/j.apnum.2013.12.002
[13] Yang, F., Fu, J.L., Fan, P. and Li, X.X. (2021) Fractional Landweber Iterative Regularization Method for Identifying the Unknown Source of the Time-Fractional Diffusion Problem. Acta Applicandae Mathematicae, 175, Article No. 13.
https://doi.org/10.1007/s10440-021-00442-1
[14] Sun, C.L. and Liu, J.J. (2020) An Inverse Source Problem for Distributed Order Time- Fractional Diffusion Equation. Inverse Problems, 36, Article 055008.
https://doi.org/10.1088/1361-6420/ab762c
[15] Zhang, M.M. and Liu, J.J. (2021) On the Simultaneous Reconstruction of Boundary Robin Coefficient and Internal Source in a Slow Diffusion System. Inverse Problems, 37, Article 075008.
[16] Liu, J.J. and Yamamoto, M. (2010) A Backward Problem for the Time-Fractional Diffusion Equation. Applicable Analysis, 89, 1769-1788.
https://doi.org/10.1080/00036810903479731
[17] Cheng, J., Nakagawa, J., Yamamoto, M. and Yamazaki, T. (2009) Uniqueness in an Inverse Problem for a One-Dimensional Fractional Diffusion Equation. Inverse Problems, 25, Article 115002.
https://doi.org/10.1088/0266-5611/25/11/115002
[18] Li, Z.Y., Liu, Y.K. and Yamamoto, M. (2015) Initial-Boundary Value Problems for Multi-Term Time-Fractional Diffusion Equations with Positive Constant Coefficients. Applied Mathematics and Computation, 257, 381-397.
https://doi.org/10.1016/j.amc.2014.11.073
[19] Wang, J.-G. and Wei, T. (2014) An Iterative Method for Backward Time-Fractional Diffusion Problem. Numerical Methods for Partial Differential Equations, 30, 2029-2041.
https://doi.org/10.1002/num.21887
[20] Wang, J.G., Zhou, Y.B. and Wei, T. (2013) A Posteriori Regularization Parameter Choice Rule for the Quasi-Boundary Value Method for the Backward Time-Fractional Diffusion Problem. Applied Mathematics Letters, 26, 741-747.
https://doi.org/10.1016/j.aml.2013.02.006
[21] 刘继军. 不适定问题的正则化方法及应用[M]. 北京: 科学出版社, 2005.
[22] Bazhlekova, E. (2013) Properties of the Fundamental and the Impulse-Response Solutions of Multi-Term Fractional Differential Equations. Complex Analysis and Applications’13, Sofia, 31 October-2 November 2013, 55-64.
[23] Sun, L.L., Li, Y.S. and Zhang, Y. (2021) Simultaneous Inversion of the Potential Term and the Fractional Orders in a Multi-Term Time-Fractional Diffusion Equation. Inverse Problems, 37, Article 055007.
https://doi.org/10.1088/1361-6420/abf162

为你推荐