Unique recovery of the initial state of distributed order time fractional diffusion equation

Year 2024, Volume: 53 Issue: 6 , 1560 - 1574 , 28.12.2024

Hui Liu Zhiyuan Li , Zhousheng Ruan

https://doi.org/10.15672/hujms.1285069

https://izlik.org/JA74GM26GN

Abstract

We use Phragmén-Lindelöf-Liouville argument to prove the uniqueness for the determining the initial state of solution for the time fractional diffusion equation with distributed order derivative. Several numerical experiments are presented to show the accuracy and efficiency of the algorithm.

Keywords

ultraslow diffusion , inverse problem , regularization method

Supporting Institution

National Natural Science Foundation of China

Project Number

12271277; 12061008

References

• [1] D. Araya and C. Lizama, Almost automorphic mild solutions to fractional differential equations, Nonlinear Analysis Theory Methods and Applications, 69(11), 3692–3705, 2008.
• [2] J.M. Boggs and E.E. Adams, Field study of dispersion in a heterogeneous aquifer: 4. investigation of adsorption and sampling bias, Water Resources Research, 28(12), 3325–3336, 1992.
• [3] J.P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications, Physics reports, 195(4-5), 127–293, 1990.
• [4] M. Caputo, Mean fractional-order-derivatives differential equations and filters, Annali dellUniversita di Ferrara, 41(1), 73–84, 1995.
• [5] J.M. Carcione, F.J. Sanchez-Sesma, F. Luzón, and J.J. Perez Gavilán, Theory and simulation of time-fractional fluid diffusion in porous media, Journal of Physics A: Mathematical and Theoretical, 46(34), 345501, 2013.
• [6] A.V. Chechkin, R. Gorenflo, I.M. Sokolov, and V.Y. Gonchar, Distributed order time fractional diffusion equation, Fractional Calculus and Applied Analysis, 6(3), 259– 280, 2003.
• [7] J. Cheng, J. Nakagawa, M. Yamamoto, and T. Yamazaki, Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse problems, 25(11), 115002, 2009.
• [8] Y. Gong, P. Li, X. Wang and X. Xu, Numerical solution of an inverse random source problem for the time fractional diffusion equation via phaselift, Inverse Problems, 37(4), 045001, 2021.
• [9] Y. Hatano and N. Hatano, Dispersive transport of ions in column experiments: An explanation of long-tailed profiles, Water resources research, 34(5), 1027–1033, 1998.
• [10] X. Huang, Y. Kian, E. Soccorsi, and M. Yamamoto, Determination of source and initial values for acoustic equations with a time-fractional attenuation, arXiv preprint arXiv:2111.05240, 2021.
• [11] D. Jiang, Z. Li, Y. Liu, and M. Yamamoto, Weak unique continuation property and a related inverse source problem for time-fractional diffusion-advection equations, Inverse Problems, 33(5), 055013, 2017.
• [12] B. Jin and W. Rundell, A tutorial on inverse problems for anomalous diffusion processes, Inverse problems, 31(3), 035003, 2015.
• [13] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and applications of fractional differential equations, volume 204. elsevier, 2006.
• [14] M. Kirane and S.A. Malik, Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time, Applied Mathematics and Computation, 218(1), 163–170, 2011.
• [15] Y. Liu, G. Hu, and M. Yamamoto, Inverse moving source problem for time-fractional evolution equations: determination of profiles, Inverse Problems, 37(8), 084001, 2021.
• [16] Z. Li, Y. Kian and E. Soccorsi, Initial-boundary value problem for distributed order time-fractional diffusion equations, Asymptotic Analysis, 115(1-2), 95–126, 2019.
• [17] Z. Li, Y. Liu and M. Yamamoto, Inverse source problem for a one-dimensional timefractional diffusion equation and unique continuation for weak solutions, arXiv eprints arXiv:2112.01018, 2021.
• [18] X. Li and H. Rui, A block-centered finite difference method for the distributed-order time-fractional diffusion-wave equation, Applied Numerical Mathematics, 131, 123– 139, 2018.
• [19] J.J. Liu and M. Yamamoto, A backward problem for the time-fractional diffusion equation, Applicable Analysis, 89(11), 1769–1788, 2010.
• [20] Z. Li and M. Yamamoto, Uniqueness for inverse problems of determining orders of multi-term time-fractional derivatives of diffusion equation, Applicable Analysis, 94(3), 570–579, 2015.
• [21] Z. Li and M. Yamamoto, Unique continuation principle for the one-dimensional timefractional diffusion equation, Fractional Calculus and Applied Analysis, 22, 644–657, 2019.
• [22] Y. 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.
• [23] F. Mainardi, A. Mura, G. Pagnini and R. Gorenflo, Time-fractional diffusion of distributed order, Journal of Vibration and Control, 14(9-10), 1267–1290, 2008.
• [24] K. Sakamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, Journal of Mathematical Analysis and Applications, 382(1), 426–447, 2011.
• [25] K. Sakamoto and M. Yamamoto, Inverse source problem with a final overdetermination for a fractional diffusion equation, Math. Control Relat. Fields, 1(4), 509–518, 2011.
• [26] E.M. Stein and R. Shakarchi, Complex analysis, volume 2. Princeton University Press, 2010.
• [27] N.H. Tuan, L.N. Huynh, T.B. Ngoc and Y. Zhou, On a backward problem for nonlinear fractional diffusion equations, Applied Mathematics Letters, 92, 76–84, 2019.
• [28] N.H. Tuan, L.D. Long and S. Tatar, Tikhonov regularization method for a backward problem for the inhomogeneous time-fractional diffusion equation, Applicable Analysis, 97(5), 842–863, 2018.
• [29] L. Wang and J. Liu, Total variation regularization for a backward time-fractional diffusion problem, Inverse problems, 29(11), 115013, 2013.
• [30] 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.
• [31] Z. Zhang, An undetermined coefficient problem for a fractional diffusion equation, Inverse Problems, 32(1), 015011, 2015.
• [32] Y. Zhang and X. Xu, Inverse source problem for a fractional diffusion equation, Inverse problems, 27(3), 035010, 2011.