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Variational method for a backward problem for a time-fractional diffusion equation. (English) Zbl 1471.65128

In this paper, the authors study an ill-posed backward problem of computing the initial condition from a given final time state in a time-fractional diffusion equation by using a variational method in terms of an appropriate Tikhonov regularization functional. A conjugate gradient algorithm is formulated for approximating solutions to this problem and the effectiveness of the scheme proposed is shown in four numerical simulations both in the one-dimensional and two-dimensional cases. From the theoretical point of view, the authors obtain a stronger regularity of the weak solution for the direct problem, and the existence and uniqueness of a weak solution for the adjoint problem are proved analytically.

MSC:

65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
35R11 Fractional partial differential equations
35A15 Variational methods applied to PDEs
35D30 Weak solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
49N45 Inverse problems in optimal control
49N60 Regularity of solutions in optimal control
65F10 Iterative numerical methods for linear systems
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
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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. 36 (2000) 149-C158.
[2] R. Courant and D. Hilbert, Methods of Mathematical Physics. Interscience, Vol. 1, New York (1953). · Zbl 0053.02805
[3] H.W. Engl, M. Hanke and A. Neubauer, Regularization of inverse problems. In: Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht (1996). · Zbl 0859.65054
[4] K.M. Hanke and L.P.C. Hansen, Regularization methods for large-scale problems. Surv. Math. Ind. 3 (1993) 253-315. · Zbl 0805.65058
[5] B.I. Henry, T.A. Langlands and S.L. Wearne, Fractional cable models for spiny neuronal dendrites. Phys. Rev. Lett. 100 (2008) 128103. · doi:10.1103/PhysRevLett.100.128103
[6] Y.J. Jiang and J.T. Ma, High-order finite element methods for time-fractional partial differential equations. Sci. China Math. 235 (2011) 3285-3290. · Zbl 1216.65130
[7] A.A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Vol. 204. Elsevier Science Limited (2006). · Zbl 1092.45003
[8] Y.M. Lin and C.J. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225 (2007) 1533-C1552. · Zbl 1126.65121
[9] J.J. Liu and M. Yamamoto, A backward problem for the time-fractional diffusion equation. Appl. Anal. 89 (2010) 1769-1788. · Zbl 1204.35177
[10] Y. Luchko, Maximum principle and its application for the time-fractional diffusion equations. Fract. Calc. Appl. Anal. 14 (2011) 110-124. · Zbl 1273.35297
[11] R. Metzler and J. Klafter, Boundary value problems for fractional diffusion equations. Phys. A Stat. Mech. Appl. 278 (2000) 107-125. · doi:10.1016/S0378-4371(99)00503-8
[12] R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000) 1-77. · Zbl 0984.82032
[13] R. Metzler and J. Klafter, Subdiffusive transport close to thermal equilibrium: from the langevin equation to fractional diffusion. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdisciplinary Top. 61 (2000) 6308-6311.
[14] V.A. Morozov, Methods for Solving Incorrectly Posed Problems. Springer-Verlag (1984). · Zbl 0549.65031 · doi:10.1007/978-1-4612-5280-1
[15] D.A. Murio, Implicit finite difference approximation for time fractional diffusion equations. Comput. Math. Appl. 56 (2008) 1138-1145. · Zbl 1155.65372
[16] I. Podlubny, Fractional differential equations. In: Mathematics in Science and Engineering (1999). · Zbl 0924.34008
[17] H. Pollard, The completely monotonic character of the Mittag-Leffler function E_α(−x). Bull. Am. Math. Soc. 54 (1948) 1115-1116. · Zbl 0033.35902 · doi:10.1090/S0002-9904-1948-09132-7
[18] M. Raberto, E. Scalas and F. Mainardi, Waiting-times and returns in high-frequency financial data: an empirical study. Phys. A Stat. Mech. Appl. 314 (2002) 749-755. · Zbl 1001.91033 · doi:10.1016/S0378-4371(02)01048-8
[19] C.X. Ren, X. Xu and S. Lu, Regularization by projection for a backward problem of the time-fractional diffusion equation. J. Inverse Ill-Posed Probl. 22 (2014) 121-139. · Zbl 1282.65114
[20] 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) 426C447. · Zbl 1219.35367
[21] S. Shen, F. Liu, V. Anh and I. Turner, Implicit difference approximation for the time fractional diffusion equation. J. Appl. Math. Comput. 22 (2006) 87-99. · Zbl 1140.65094
[22] I.M. Sokolov and J. Klafter, From diffusion to anomalous diffusion: a century after einstein’s brownian motion. Chaos 15 (2005) 26103. · Zbl 1080.82022
[23] L.L. Sun and T. Wei, Identification of the zeroth-order coefficient in a time fractional diffusion equation. Appl. Numer. Math. 111 (2017) 160-180. · Zbl 1353.65102
[24] J.G. Wang and T. Wei, An iterative method for backward time-fractional diffusion problem. Numer. Methods Part. Differ. Equ. 30 (2014) 2029-2041. · Zbl 1314.65120 · doi:10.1002/num.21887
[25] J.G. Wang, T. Wei and Y.B. Zhou, Tikhonov regularization method for a backward problem for the time-fractional diffusion equation. Appl. Math. Model. 37 (2013) 8518-8532. · Zbl 1427.65229
[26] L.Y. Wang and J.J. Liu, Data regularization for a backward time-fractional diffusion problem. Comput. Math. Appl. 64 (2012) 3613-3626. · Zbl 1268.65128
[27] T. Wei, X.L. Li and Y.S. Li, An inverse time-dependent source problem for a time-fractional diffusion equation. Inverse Probl. 32 (2016) 085003. · Zbl 1351.65072
[28] T. Wei and J.G. Wang, A modified quasi-boundary value method for the backward time-fractional diffusion problem. ESAIM: M2AN 48 (2014) 603-621. · Zbl 1295.35378 · doi:10.1051/m2an/2013107
[29] T. Wei and J.G. Wang, Determination of Robin coefficient in a fractional diffusion problem. Appl. Math. Model. Simul. Comput. Eng. Environ. Syst. 40 (2016) 7948-7961. · Zbl 1471.65127
[30] T. Wei and Z.Q. Zhang, Robin coefficient identification for a time-fractional diffusion equation. Inverse Probl. Sci. Eng. 24 (2016) 1-20.
[31] W. Wyss, The fractional diffusion equation. J. Math. Phys. 27 (1986) 2782-2785. · Zbl 0632.35031
[32] S.B. Yuste, K. Lindenberg and J.J. Ruiz-Lorenzo, Subdiffusion-limited reactions. Chem. Phys. 284 (2002) 169-180.
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