A modified quasi-boundary value method for the backward time-fractional diffusion problem
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 2, p. 603-621

In this paper, we consider a backward problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain. That is to determine the initial data from a noisy final data. Based on a series expression of the solution, a conditional stability for the initial data is given. Further, we propose a modified quasi-boundary value regularization method to deal with the backward problem and obtain two kinds of convergence rates by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed methods.

DOI : https://doi.org/10.1051/m2an/2013107
Classification:  35R11,  35R30
Keywords: backward problem, fractional diffusion equation, modified quasi-boundary value method, convergence analysis, a priori parameter choice, morozov's discrepancy principle
@article{M2AN_2014__48_2_603_0,
author = {Wei, Ting and Wang, Jun-Gang},
title = {A modified quasi-boundary value method for the backward time-fractional diffusion problem},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {48},
number = {2},
year = {2014},
pages = {603-621},
doi = {10.1051/m2an/2013107},
zbl = {1295.35378},
mrnumber = {3177859},
language = {en},
url = {http://www.numdam.org/item/M2AN_2014__48_2_603_0}
}

Wei, Ting; Wang, Jun-Gang. A modified quasi-boundary value method for the backward time-fractional diffusion problem. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 2, pp. 603-621. doi : 10.1051/m2an/2013107. http://www.numdam.org/item/M2AN_2014__48_2_603_0/

[1] K.A. Ames and J.F. Epperson, A kernel-based method for the approximate solution of backward parabolic problems. SIAM J. Numer. Anal. (1997) 1357-1390. | MR 1461788 | Zbl 0889.65100

[2] K.A. Ames and L.E. Payne, Asymptotic behavior for two regularizations of the Cauchy problem for the backward heat equation. Math. Models Methods Appl. Sci. 8 (1998) 187. | MR 1612020 | Zbl 0982.35119

[3] B. Berkowitz, H. Scher and S.E. Silliman, Anomalous transport in laboratory-scale, heterogeneous porous media. Water Resour. Res. 36 (2000) 149-158.

[4] J. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki, Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation. Inverse Problems 25 (2009) 115002. | MR 2545997 | Zbl 1181.35322

[5] 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. 62 (2011) 1619-1626. | MR 2834147 | Zbl 1231.65158

[6] G.W. Clark and S.F. Oppenheimer, Quasireversibility methods for non-well-posed problems Electron. J. Differ. equ. (1994) 1-9. | MR 1302574 | Zbl 0811.35157

[7] M. Denche and K. Bessila, A modified quasi-boundary value method for ill-posed problems. J. Math. Anal. Appl. 301 (2005) 419-426. | MR 2105682 | Zbl 1084.34536

[8] X.L. Feng, L. Eldén and C.L. Fu, A quasi-boundary-value method for the cauchy problem for elliptic equations with nonhomogeneous neumann data. J. Inverse Ill-Posed Probl. 18 (2010) 617-645. | MR 2746685 | Zbl 1279.65129

[9] D.N. Hào, N.V. Duc and D. Lesnic, A non-local boundary value problem method for the Cauchy problem for elliptic equations. Inverse Probl. 25 (2009) 055002. | MR 2501020 | Zbl 1170.35555

[10] D.N. Hào, N.V. Duc and D. Lesnic, Regularization of parabolic equations backward in time by a non-local boundary value problem method. IMA J. Appl. Math. 75 (2010) 291-315. | MR 2606977 | Zbl 1194.35501

[11] D.N. Hào, N.V. Duc and H. Sahli, A non-local boundary value problem method for parabolic equations backward in time. J. Math. Anal. Appl. 345 (2008) 805-815. | MR 2429181 | Zbl 1160.35070

[12] Y.J. Jiang and J.T. Ma, High-order finite element methods for time-fractional partial differential equations. J. Comput. Appl. Math. 235 (2011) 3285-3290. | MR 2784043 | Zbl 1216.65130

[13] B.T. Jin and W. Rundell, An inverse problem for a one-dimensional time-fractional diffusion problem. Inverse Probl. 28 (2012). | MR 2946798 | Zbl 1247.35203

[14] S.M. Kirkup and M. Wadsworth, Solution of inverse diffusion problems by operator-splitting methods. Appl. Math. Modelling 26 (2002) 1003-1018. | Zbl 1014.65095

[15] J.J. Liu and M. Yamamoto, A backward problem for the time-fractional diffusion equation. Appl. Anal. 89 (2010) 1769-1788. | MR 2683680 | Zbl 1204.35177

[16] Y. Luchko, Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation. Comput. Math. Appl. 59 (2010) 1766-1772. | MR 2595950 | Zbl 1189.35360

[17] Y. Luchko, Maximum principle and its application for the time-fractional diffusion equations. Fract. Calc. Appl. Anal. 14 (2011) 110-124. | MR 2782248 | Zbl 1273.35297

[18] F. Mainardi, G. Pagnini and R. Gorenflo, Some aspects of fractional diffusion equations of single and distributed order. Appl. Math. Comput. 187 (2007) 295-305. | MR 2323582 | Zbl 1122.26004

[19] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000) 1-77. | MR 1809268 | Zbl 0984.82032

[20] R. Metzler and J. Klafter, Subdiffusive transport close to thermal equilibrium: From the Langevin equation to fractional diffusion. Phys. Rev. E 61 (2000) 6308-6311.

[21] D.A. Murio, Stable numerical solution of a fractional-diffusion inverse heat conduction problem. Comput. Math. Appl. 53 (2007) 1492-1501. | MR 2332298 | Zbl 1152.65463

[22] D.A. Murio, Implicit finite difference approximation for time fractional diffusion equations. Comput. Math. Appl. 56 (2008) 1138-1145. | MR 2437884 | Zbl 1155.65372

[23] D.A. Murio, Time fractional IHCP with Caputo fractional derivatives. Comput. Math. Appl. 56 (2008) 2371-2381. | MR 2466760 | Zbl 1165.65386

[24] D.A. Murio, Stable numerical evaluation of Grünwald-Letnikov fractional derivatives applied to a fractional ihcp. Inverse Probl. Sci. Engrg. 17 (2009) 229-243. | MR 2588316 | Zbl 1159.65313

[25] D.A. Murio and C.E. Mejía, Source terms identification for time fractional diffusion equation. Revista Colombiana de Matemáticas 42 (2008) 25-46. | MR 2581255 | Zbl 1189.65199

[26] D.A. Murio, Implicit finite difference approximation for time fractional diffusion equations. Comput. Math. Appl. 56 (2008) 1138-1145. | MR 2437884 | Zbl 1155.65372

[27] I. Podlubny, Fractional differential equations, in vol. 198 of Math. Sci. Eng. Academic Press Inc., San Diego, CA (1999). | MR 1658022 | Zbl 0924.34008

[28] I. Podlubny and M. Kacenak, Mittag-leffler function. The MATLAB routine, available at http://www.mathworks.com/matlabcentral/fileexchange (2006).

[29] H. Pollard, The completely monotonic character of the mittag-leffler function Eα( − x). Bull. Amer. Math. Soc. 54 (1948) 1115-1116. | MR 27375 | Zbl 0033.35902

[30] Z. Qian, Optimal modified method for a fractional-diffusion inverse heat conduction problem. Inverse Probl. Sci. Engrg. 18 (2010) 521-533. | MR 2674752 | Zbl 1195.65125

[31] W. Rundell, X. Xu and L. H. Zuo, The determination of an unknown boundary condition in a fractional diffusion equation. Appl. Anal. http://dx.doi.org/10.1080/00036811.2012.686605. | MR 3169116 | Zbl pre06199418

[32] 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. 382 (2011) 426-447. | MR 2805524 | Zbl 1219.35367

[33] E. Scalas, R. Gorenflo and F. Mainardi, Fractional calculus and continuous-time finance. Physica A 284 (2000) 376-384. | MR 1773804 | Zbl 1138.91444

[34] R. Scherer, S.L. Kalla, L. Boyadjiev and B. Al-Saqabi, Numerical treatment of fractional heat equations. Appl. Numer. Math. 58 (2008) 1212-1223. | MR 2428973 | Zbl 1143.65105

[35] R.E. Showalter, The final value problem for evolution equations. J. Math. Anal. Appl. 47 (1974) 563-572. | MR 352644 | Zbl 0296.34059

[36] R.E. Showalter, Cauchy problem for hyper-parabolic partial differential equations. North-Holland Math. Stud. 110 (1985) 421-425. | MR 817519 | Zbl 0587.35044

[37] I.M. Sokolov and J. Klafter, From diffusion to anomalous diffusion: A century after Einsteins Brownian motion. Chaos 15 (2005) 1-7. | MR 2150232 | Zbl 1080.82022

[38] H. Wei, W. Chen, H.G. Sun and X.C. Li, A coupled method for inverse source problem of spatial fractional anomalous diffusion equations. Inverse Probl. Sci. Engrg. 18 (2010) 945-956. | MR 2743232 | Zbl 1204.65116

[39] W. Wyss, The fractional diffusion equation. J. Math. Phys. 27 (1986) 2782-2785. | MR 861345 | Zbl 0632.35031

[40] M. Yang and J.J. Liu, Solving a final value fractional diffusion problem by boundary condition regularization. Appl. Numer. Math. 66 (2013) 45-58. | MR 3018647 | Zbl 1269.65093

[41] P. Zhang and F. Liu. Implicit difference approximation for the time fractional diffusion equation. J. Appl. Math. Comput. 22 (2006) 87-99. | MR 2323899 | Zbl 1140.65094

[42] Y. Zhang and X. Xu, Inverse source problem for a fractional diffusion equation. Inverse Probl. 27 (2011) 035010. | MR 2772529 | Zbl 1211.35280

[43] G.H. Zheng and T. Wei, Spectral regularization method for a Cauchy problem of the time fractional advection-dispersion equation. J. Comput. Appl. Math. 233 (2010) 2631-2640. | MR 2577848 | Zbl 1186.65128

[44] G.H. Zheng and T. Wei, A new regularization method for a Cauchy problem of the time fractional diffusion equation. Advances Comput. Math. 36 (2012) 377-398. | MR 2886197 | Zbl 1245.35145

[45] G.H. Zheng and T. Wei, Two regularization methods for solving a riesz-feller space-fractional backward diffusion problem. Inverse Probl. 26 (2010) 115017. | MR 2732909 | Zbl 1206.65226

[46] P. Zhuang and F. Liu, Implicit difference approximation for the time fractional diffusion equation. J. Appl. Math. Comput. 22 (2006) 87-99. | MR 2323899 | Zbl 1140.65094