Determination of three parameters in a time-space fractional diffusion equation

Xiangtuan Xiong; Wanxia Shi; Xuemin Xue; Xiangtuan Xiong; Wanxia Shi; Xuemin Xue

doi:10.3934/math.2021350

AIMS Mathematics

2021, Volume 6, Issue 6: 5909-5923. doi: 10.3934/math.2021350

Previous Article Next Article

Research article

Determination of three parameters in a time-space fractional diffusion equation

Department of Mathematics, Northwest Normal University, Lanzhou, Gansu 730070, China

Received: 23 November 2020 Accepted: 17 March 2021 Published: 29 March 2021
MSC : 34A55, 35R30, 65F22, 65L12

In this paper, we consider a nonlinear inverse problem of recovering two fractional orders and a diffusion coefficient in a one-dimensional time-space fractional diffusion equation. The uniqueness of fractional orders and the diffusion coefficient, characterizing slow diffusion, can be obtained from the accessible boundary data. Two computational methods, Tikhonov method and Levenberg-Marquardt method, are proposed to solving this problem. Finally, an example is presented to illustrate the efficiency of the two numerical algorithm.
- ill-posedness,
- regularization,
- uniqueness,
- fractional diffusion equation
Citation: Xiangtuan Xiong, Wanxia Shi, Xuemin Xue. Determination of three parameters in a time-space fractional diffusion equation[J]. AIMS Mathematics, 2021, 6(6): 5909-5923. doi: 10.3934/math.2021350

Related Papers:

Abstract

In this paper, we consider a nonlinear inverse problem of recovering two fractional orders and a diffusion coefficient in a one-dimensional time-space fractional diffusion equation. The uniqueness of fractional orders and the diffusion coefficient, characterizing slow diffusion, can be obtained from the accessible boundary data. Two computational methods, Tikhonov method and Levenberg-Marquardt method, are proposed to solving this problem. Finally, an example is presented to illustrate the efficiency of the two numerical algorithm.

References

[1]	B. Berkowitz, H. Scher, S. Silliman, Anomalous transport in laboratory-scale, heterogeneous porous media, Water. Resour. Res., 36 (2000), 149–158. doi: 10.1029/1999WR900295
[2]	J. Cheng, J. Nakagawa, M. Yamamoto, T. Yamazaki, Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Probl., 25 (2009), 115002. doi: 10.1088/0266-5611/25/11/115002
[3]	R. Gorenflo, F. Mainardi, Random walk models for space fractional diffusion processes, Fract. Calculus Appl. Anal., 1 (1998), 167–191.
[4]	Y. Hatano, J. Nakagawa, S. Wang, M. Yamamoto, Determination of order in fractional diffusion equation, J. Math. Ind., 5 (2013), 51–57.
[5]	B. Henry, T. Langlands, S. Wearne, Fractional cable models for spiny neuronal dendrites, Phys. Rev. Lett., 100 (2008), 128103. doi: 10.1103/PhysRevLett.100.128103
[6]	M. Ilić, F. Liu, I. Turner, V. Anh, Numerical approximation of a fractional-in-space diffusion equation, Fract. Calc. Appl. Anal., 8 (2005), 323–341.
[7]	M. Ilić, F. Liu, I. Turner, V. Anh, Numerical approximation of a fractional-in-space diffusion equation (Ⅱ)–with nonhomogeneous boundary conditions, Fract. Calculus Appl. Anal., 9 (2006), 333–349.
[8]	B. Jin, W. Rundell, An inverse problem for a one-dimensional time-fractional diffusion problem, Inverse Probl., 28 (2012), 075010. doi: 10.1088/0266-5611/28/7/075010
[9]	B. Jin, W. Rundell, A tutorial on inverse problems for anomalous diffusion processes, Inverse Probl., 31 (2015), 035003. doi: 10.1088/0266-5611/31/3/035003
[10]	G. Li, D. Zhang, X. Jia, M. Yamamoto, Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation, Inverse Probl., 29 (2013), 065014. doi: 10.1088/0266-5611/29/6/065014
[11]	J. Liu, M. Yamamoto, A backward problem for the time-fractional diffusion equation, Appl. Anal., 89 (2010), 1769–1788. doi: 10.1080/00036810903479731
[12]	J. Liu, M. Yamamoto, L. Yan, On the reconstruction of unknown time-dependent boundary sources for time fractional diffusion process by distributing measurement, Inverse Probl., 32 (2016), 015009. doi: 10.1088/0266-5611/32/1/015009
[13]	Y. Luchko, W. Rundell, M. Yamamoto, L. Zuo, Uniqueness and reconstruction of an unknown semilinear term in a time-fractional reaction-diffusion equation, Inverse Probl., 29 (2013), 065019. doi: 10.1088/0266-5611/29/6/065019
[14]	R. Metzler, J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep., 339 (2000), 1–77. doi: 10.1016/S0370-1573(00)00070-3
[15]	R. Metzler, J. Klafter, Boundary value problems for fractional diffusion equations, Phys. A, 278 (2000), 107–125. doi: 10.1016/S0378-4371(99)00503-8
[16]	L. Miller, M. Yamamoto, Coefficient inverse problem for a fractional diffusion equation, Inverse Probl., 29 (2013), 075013. doi: 10.1088/0266-5611/29/7/075013
[17]	D. Murio, On the stable numerical evaluation of Caputo fractional derivatives, Comput. Math. Appl., 51 (2006), 1539–1550. doi: 10.1016/j.camwa.2005.11.037
[18]	L. Pandolfi, A Lavrentiev-type approach to the online computation of Caputo-fractional derivatives, Inverse Probl., 24 (2008), 015014. doi: 10.1088/0266-5611/24/1/015014
[19]	I. Podlubny, Fractional Differential Equations, 2 Eds., San Diego: Academic Press, 1999.
[20]	Mittag-Leffler function, The MATLAB routine, Mathworks, 2012. Available from: http://www.mathworks.com/matlabcentral/fileexchange.
[21]	M. Raberto, E. Scales, F. Mainardi, Waiting-times and returns in high-frequency financial data: An empirical study, Phys. A, 314 (2002), 749–755. doi: 10.1016/S0378-4371(02)01048-8
[22]	K. Sakamoto, M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426–447. doi: 10.1016/j.jmaa.2011.04.058
[23]	L. Sun, T. Wei, Identification of the zeroth-order coefficient in a time fractional diffusion equation, Appl. Numer. Math., 111 (2017), 160–180. doi: 10.1016/j.apnum.2016.09.005
[24]	S. Tatar, R. Tinaztepe, S. Ulusoy, Simultaneous inversion for the exponents of the fractional time and space derivatives in the space-time fractional diffusion equation, Appl. Anal., 95 (2016), 1–23. doi: 10.1080/00036811.2014.984291
[25]	L. Wang, J. Liu, Total variation regularization for a backward time-fractional diffusion problem, Inverse Probl., 29 (2013), 115013. doi: 10.1088/0266-5611/29/11/115013
[26]	T. Wei, X. Li, Y. Li, An inverse time-dependent source problem for a time-fractional diffusion equation, Inverse Probl., 32 (2016), 085003. doi: 10.1088/0266-5611/32/8/085003
[27]	T. Wei, J. Wang, Determination of Robin coefficient in a fractional diffusion problem, Appl. Math. Model., 40 (2016), 7948–7961. doi: 10.1016/j.apm.2016.03.046
[28]	M. Yang, J. Liu, Solving a final value fractional diffusion problem by boundary condition regularization, Appl. Numer. Math., 66 (2013), 45–58. doi: 10.1016/j.apnum.2012.11.009
[29]	Q. Yang, Novel Analytical and Numerical Methods for Solving Fractional Dynamical Systems, Doctor thesis, Queensland University of Technology, 2010.
[30]	Y. Zhang, X. Xu, Inverse source problem for a fractional diffusion equation, Inverse Probl., 27 (2011), 035010. doi: 10.1088/0266-5611/27/3/035010

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)