|
|
时间分数阶扩散方程逆向问题的迭代分数次Tikhonov方法
|
Abstract:
研究了一个在一般有界域中的具有可变系数的时间分数阶扩散方程的逆向问题。提出了一种迭代的分数次Tikhonov正则化方法去解决这个逆向问题。此外，通过先验正则化参数选取规则和后验正则化参数选取规则，证明了正则化解的收敛率。迭代的分数次Tikhonov正则化方法超越了经典Tikonov正则化方法的饱和结果，在先验参数选取规则下，迭代的分数次Tikhonov正则化方法优于经典迭代Tikonov正则化方法。
The backward problem of a time-fractional diffusion equation with variable coefficients in a general bounded domain is studied. An iterative fractional Tikhonov regularization method was proposed to solve the backward problem. In addition, the convergence rates for the regularized solution can be proved by using an a priori regularization parameter choice rule and an a posteriori regulariza-tion parameter choice rule. The iterative fractional Tikhonov regularization method surpasses the saturation result of classical Tikhonov regularization method, and iterative fractional Tikhonov regularization method is superior to classical iterative Tikhonov regularization method under the a-priori parameter choice rule.
| [1] | 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 |
| [2] | Wang, J.G., Zhou, Y.B. and Wei, T. (2013) A Posteriori Regular-ization 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 |
| [3] | Han, Y., Xiong, X. and Xue, X. (2019) A Fractional Landweber Method for Solving Backward Time-Fractional Diffusion Problem. Computers & Mathematics with Applications, 78, 81-91. https://doi.org/10.1016/j.camwa.2019.02.017 |
| [4] | Bianchi, D., Buccini, A., Donatelli, M., et al. (2015) Iterated Frac-tional Tikhonov Regularization. Inverse Problems, 31, 055005. https://doi.org/10.1088/0266-5611/31/5/055005 |
| [5] | Podlubny, L. (1999) Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering 198. Academic Press, San Diego. |
| [6] | Wang, J.G. and Wei, T. (2014) An Iterative Method for Backward Time-Fractional Diffusion Problem. Numerical Methods for Partial Differ-ential Equations, 30, 2029-2041. https://doi.org/10.1002/num.21887 |
| [7] | Klann, E. and Ramlau, R. (2008) Regularization by Fractional Filter Methods and Data Smoothing. Inverse Problems, 24, 045005. https://doi.org/10.1088/0266-5611/24/2/025018 |
| [8] | Sakamoto, K. and Yamamoto, M. (2011) Initial Value/Boundary Value Problems for Fractional Diffusion-Wave Equations and Applications to Some Inverse Problems. Journal of Mathematical Analysis and Applications, 382, 426-447. https://doi.org/10.1016/j.jmaa.2011.04.058 |
| [9] | Wei, T. and Wang, J.G. (2014) A Modified Qua-si-Boundary Value Method for the Backward Time-Fractional Diffusion Problem. ESAIM: Mathematical Modelling and Numerical Analysis, 48, 603-621.
https://doi.org/10.1051/m2an/2013107 |
| [10] | Yang, S., Xiong, X. and Nie, Y. (2021) Iterated Fractional Tikhonov Regularization Method for Solving the Spherically Symmetric Backward Time-Fractional Diffusion Equation. Applied Numerical Mathematics, 160, 217-241.
https://doi.org/10.1016/j.apnum.2020.10.008 |
