Tikhonov-type regularization method for a sideways problem of the time-fractional diffusion equation

Hongwu Zhang; Xiaoju Zhang; Hongwu Zhang; Xiaoju Zhang

doi:10.3934/math.2021007

AIMS Mathematics

2021, Volume 6, Issue 1: 90-101. doi: 10.3934/math.2021007

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Research article

Tikhonov-type regularization method for a sideways problem of the time-fractional diffusion equation

Hongwu Zhang ^{1
,
,},
Xiaoju Zhang ²

1.
School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China
2.
Center for Faculty Development, North Minzu University, Yinchuan 750021, China

Received: 22 July 2020 Accepted: 24 September 2020 Published: 29 September 2020
MSC : 65N20, 65N21

A sideways problem of the time-fractional diffusion equation is investigated. The solution of this problem does not depend on the given data. In view of this, this article uses a Tikhonov-type regularized method to construct an approximate solution and overcome the ill-posedness of considered problem. The a-posteriori convergence estimates of logarithmic and double logarithmic types for the regularized method are derived. Finally, for smooth and non-smooth cases we respectively verify the effectiveness of proposed method by doing the coresponding numerical experiments.
- sideways problem,
- time-fractional diffusion equation,
- Tikhonov-type regularization method,
- a-posteriori convergence estimate,
- numerical implementation
Citation: Hongwu Zhang, Xiaoju Zhang. Tikhonov-type regularization method for a sideways problem of the time-fractional diffusion equation[J]. AIMS Mathematics, 2021, 6(1): 90-101. doi: 10.3934/math.2021007

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Abstract

A sideways problem of the time-fractional diffusion equation is investigated. The solution of this problem does not depend on the given data. In view of this, this article uses a Tikhonov-type regularized method to construct an approximate solution and overcome the ill-posedness of considered problem. The a-posteriori convergence estimates of logarithmic and double logarithmic types for the regularized method are derived. Finally, for smooth and non-smooth cases we respectively verify the effectiveness of proposed method by doing the coresponding numerical experiments.

References

[1]	S. Das, I. Pan, Fractional order signal processing, Heidelberg: Springer, 2012.
[2]	F. Mainardi, Fractional calculus and waves in linear viscoelasticity, London: Imperial College Press, 2010.
[3]	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
[4]	I. Podlubny, Fractional differential equations, San Diego: Academic Press Inc., 1999.
[5]	Y. Povstenko, Fractional thermoelasticity, Cham: Springer, 2015.
[6]	J. Sabatier, P. Lanusse, P. Melchior, A. Oustaloup, Fractional order differentiation and robust control design, Dordrecht: Springer, 2015.
[7]	E. Scalas, R. Gorenflo, F. Mainardi, Fractional calculus and continuous-time finance, Physica A, 284 (2000), 376-384. doi: 10.1016/S0378-4371(00)00255-7
[8]	V. V. Uchaikin, Fractional derivatives for physicists and engineers, Heidelberg: Springer, 2013.
[9]	D. A. Murio, Stable numerical solution of a fractional-diffusion inverse heat conduction problem, Comput. Math. Appl., 53 (2007), 1492-1501. doi: 10.1016/j.camwa.2006.05.027
[10]	D. A. Murio, Time fractional IHCP with Caputo fractional derivatives, Comput. Math. Appl., 56 (2008), 2371-2381. doi: 10.1016/j.camwa.2008.05.015
[11]	G. H. Zheng, T. Wei, Spectral regularization method for the time fractional inverse advectiondispersion equation, Math. Comput. Simulat., 81 (2010), 37-51. doi: 10.1016/j.matcom.2010.06.017
[12]	G. H. Zheng, T. Wei, Spectral regularization method for solving a time-fractional inverse diffusion problem, Appl. Math. Comput., 218 (2011), 396-405.
[13]	M. Li, X. X. Xi, X. T. Xiong, Regularization for a fractional sideways heat equation, J. Comput. Appl. Math., 255 (2014), 28-43. doi: 10.1016/j.cam.2013.04.035
[14]	X. T. Xiong, Q. Zhou, Y. C. Hon, An inverse problem for fractional diffusion equation in 2- dimensional case: Stability analysis and regularization, J. Math. Anal. Appl., 393 (2012), 185-199. doi: 10.1016/j.jmaa.2012.03.013
[15]	X. T. Xiong, H. B. Guo, X. H. Liu, An inverse problem for a fractional diffusion equation, J. Comput. Appl. Math., 236 (2012), 4474-4484. doi: 10.1016/j.cam.2012.04.019
[16]	G. H. Zheng, T. Wei, A new regularization method for the time fractional inverse advectiondispersion problem, SIAM J. Numer. Anal., 49 (2011), 1972-1990. doi: 10.1137/100783042
[17]	G. H. Zheng, T. Wei, A new regularization method for solving a time-fractional inverse diffusion problem, J. Math. Anal. Appl., 378 (2011), 418-431. doi: 10.1016/j.jmaa.2011.01.067
[18]	U. Tautenhahn, U. Hämarik, B. Hofmann, Y. Shao, Conditional stability estimates for ill-posed pdeproblems by using interpolation, Numer. Funct. Anal. Optim., 34 (2013), 1370-1417. doi: 10.1080/01630563.2013.819515
[19]	C. Shi, C. Wang, G. H. Zheng, T. Wei, A new a posteriori parameter choice strategy for the convolution regularization of the space-fractional backward diffusion problem, J. Comput. Appl. Math., 279 (2015), 233-248. doi: 10.1016/j.cam.2014.11.013
[20]	V. A. Morozov, Z. Nashed, A. B. Aries, Methods for solving incorrectly posed problems, New York: Springer, 1984.

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