FILTER REGULARIZATION FOR AN INVERSE SOURCE PROBLEM OF THE TIME-FRACTIONAL DIFFUSION EQUATION

Wan-Xia Shi; Xiang-Tuan Xiong

doi:10.11948/20210295

2023 Volume 13 Issue 4

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Wan-Xia Shi, Xiang-Tuan Xiong. FILTER REGULARIZATION FOR AN INVERSE SOURCE PROBLEM OF THE TIME-FRACTIONAL DIFFUSION EQUATION[J]. Journal of Applied Analysis & Computation, 2023, 13(4): 1702-1719. doi: 10.11948/20210295

Citation:

Wan-Xia Shi, Xiang-Tuan Xiong. FILTER REGULARIZATION FOR AN INVERSE SOURCE PROBLEM OF THE TIME-FRACTIONAL DIFFUSION EQUATION[J]. Journal of Applied Analysis & Computation, 2023, 13(4): 1702-1719. doi: 10.11948/20210295

FILTER REGULARIZATION FOR AN INVERSE SOURCE PROBLEM OF THE TIME-FRACTIONAL DIFFUSION EQUATION

Wan-Xia Shi^1, ,,
Xiang-Tuan Xiong^2,

1.
School of Science, Lanzhou University of Technology, Langongping, 730050 Lanzhou, China
2.
School of Mathematics and Statistics, Northwest Normal University, Anning East Road, 730070 Lanzhou, China

Author Bio: Email: xiongxt@gmail.com(X. T. Xiong)
Corresponding author: Email: shi_wan_xia@163.com(W. X. Shi)
Fund Project: The authors were supported by Natural Science Foundation of China (11661072) and Science and Technology Plan Foundation of Gansu Province of China (22JR5RA302)

Abstract

Abstract

In this paper, we focus on a problem of identifying the unknown source of time-fractional diffusion equation. It is known that such problem is ill-posed in the sense that reconstructed solution does not depend continuously on the observation data. Based on this fact, a GFR (general filter regularized method) is proposed. We further give the error convergence estimates under deterministic case and random noise, respectively. Lastly, some special cases and numerical examples are presented to illustrate the efficiency of our method.
- Inverse source problem /
- time-fractional diffusion equation /
- ill-posedness /
- general filter regularization /
- error estimates
MSC: 35R30, 65N20, 65M12

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References

[1]	B. Berkowitz, H. Scher and S. E. Silliman, Anomalous transport in laboratory-scale heterogeneous porous media, Water Resour. Res., 2000, 36(1), 149–158. doi: 10.1029/1999WR900295 CrossRef Google Scholar
[2]	L. Cavalier, Inverse problems in statistics, Heidelberg, April, 2007. Google Scholar
[3]	J. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki, Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Probl., 2009, 25(11), 115002. doi: 10.1088/0266-5611/25/11/115002 CrossRef Google Scholar
[4]	G. Chi, G. Li and X. Jia, Numerical inversions of a source term in the FADE with a Dirichlet boundary condition using final observations, Comput. Math. Appl., 2011, 62(4), 1619–1626. Google Scholar
[5]	B. Jin and W. Rundell, An inverse problem for a one-dimensional time-fractional diffusion problem, Inverse Prob., 2012, 28, 075010. doi: 10.1088/0266-5611/28/7/075010 CrossRef Google Scholar
[6]	J. Liu and M. Yamamoto, A backward problem for the time-fractional diffusion equation, Appl. Anal., 2010, 89(11), 1769–1788. doi: 10.1080/00036810903479731 CrossRef Google Scholar
[7]	R. Metzler and J. Klafter, Subdiffusive transport close to thermal equilibrium: from the Langevin equation to fractional diffusion, Phys. Rev. E., 2000, 61, 6308–6311. doi: 10.1103/PhysRevE.61.6308 CrossRef Google Scholar
[8]	D. Murio, Stable numerical solution of a fractional-diffusion inverse heat conduction problem, Comput. Math. Appl., 2007, 53, 1492–1501. Google Scholar
[9]	D. Murio and C. E. Mejia, Source terms identification for time fractional diffusion equation, Rev. Colomb. Mat., 2008, 42(1), 25–46. Google Scholar
[10]	D. Murio, Stable numerical solution of a fractional-diffusion inverse heat conduction problem, Comput. Math. Appl., 2007, 53, 1492–1501. Google Scholar
[11]	H. Qin and T. Wei, Some filter regularization methods for a backward heat conduction problem, Appl. Math. Comput., 2011, 217(24), 10317–10327. Google Scholar
[12]	C. Ren, X. Xu and S. Lu, Regularization by projection for a backward problem of the time fractional diffusion equation, J. Inverse Ill-Posed Probl., 2014, 22, 121–139. doi: 10.1515/jip-2011-0021 CrossRef Google Scholar
[13]	K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 2011, 382(1), 426–447. doi: 10.1016/j.jmaa.2011.04.058 CrossRef Google Scholar
[14]	I. Sokolov and J. Klafter, From diffusion to anomalous diffusion: a century after Einsteins Brownian motion, Chaos., 2005, 15(2), 1–7. Google Scholar
[15]	S. Tatar, R. Tinaztepe and S. Ulusoy, Determination of an unknown source term in a space-time fractional diffusion equation, J. Frac. Calc. Appl., 2015, 6(2), 94–101. Google Scholar
[16]	N. Tuan, M. Kirane, B. Bin-Mohsin and P. Tam, Filter regularization for final value fractional diffusion problem with deterministic and random noise, Comput. Math. Appl., 2017, 74, 1340–1361. Google Scholar
[17]	N. Tuan, L. Long, N. Thinh and T. Tran, On a final value problem for the time-fractional diffusion equation with inhomogeneous source, Inverse Probl. Sci. Eng., 2017, 25(9), 1367–1395. doi: 10.1080/17415977.2016.1259316 CrossRef Google Scholar
[18]	J. Wang, Y. Zhou and T. Wei, Two regularization methods to identify a space-dependent source for the time-fractional diffusion equation, Appl. Numer. Math., 2013, 68, 39–57. doi: 10.1016/j.apnum.2013.01.001 CrossRef Google Scholar
[19]	J. Wang and T. Wei, Quasi-reversibility method to identify a space-dependent source for the time-fractional diffusion equation, Appl. Math. Model., 2015, 39, 6139–6149. doi: 10.1016/j.apm.2015.01.019 CrossRef Google Scholar
[20]	X. Xiong and X. Xue, A fractional Tikhonov regularization method for identifying a space-dependent source in the time-fractional diffusion equation, Appl. Math. Comput., 2019, 349, 292–303. doi: 10.1016/j.cam.2018.06.011 CrossRef Google Scholar
[21]	F. Yang and C. Fu, The quasi-reversibility regularization method for identifying the unknown source for time fractional diffusion equation, Appl. Math. Model., 2015, 39, 1500–1512. doi: 10.1016/j.apm.2014.08.010 CrossRef Google Scholar
[22]	F. Yang, C. Fu and X. Li, The inverse source problem for time fractional diffusion equation: stability analysis and regularization, Inverse Probl. Sci. Eng., 2015, 23(6), 969–996. doi: 10.1080/17415977.2014.968148 CrossRef Google Scholar
[23]	F. Yang, X. Liu, X. Li and C. Ma, Landweber iterative regularization method for identifying the unknown source of the time-fractional diffusion equation, Adv. Differ. Equ., 2017, 388. Google Scholar
[24]	S. Yeganeh, R. Mokhtari and J. Hesthaven, Space-dependent source determination in a time-fractional diffusion equation using a local discontinuous Galerkin method, BIT Numer. Math., 2017, 57, 685–707. doi: 10.1007/s10543-017-0648-y CrossRef Google Scholar
[25]	Y. Zhang and X. Xu, Inverse source problem for a fractional diffusion equation, Inverse Probl., 2011, 27, 035010. doi: 10.1088/0266-5611/27/3/035010 CrossRef Google Scholar