Applications of Mathematics, Vol. 64, No. 4, pp. 469-484, 2019


Inverse source problem in a space fractional diffusion equation from the final overdetermination

Amir Hossein Salehi Shayegan, Reza Bayat Tajvar, Alireza Ghanbari, Ali Safaie

Received September 19, 2018.   Published online June 3, 2019.

Abstract:  We consider the problem of determining the unknown source term $ f=f(x,t) $ in a space fractional diffusion equation from the measured data at the final time $ u(x,T) = \psi(x) $. In this way, a methodology involving minimization of the cost functional $ J(f) = \int_0^l ( u(x,t;f )|_{t = T} - \psi(x)) ^2 {\rm d}x$ is applied and shown that this cost functional is Fréchet differentiable and its derivative can be formulated via the solution of an adjoint problem. In addition, Lipschitz continuity of the gradient is proved. These results help us to prove the monotonicity and convergence of the sequence $\{J'(f^{(n)}) \}$, where $ f^{(n)} $ is the $ n$th iteration of a gradient like method. At the end, the convexity of the Fréchet derivative is given.
Keywords:  inverse source problem; space fractional diffusion equation; weak solution theory; adjoint problem; Lipschitz continuity
Classification MSC:  65N21, 65N20
DOI:  10.21136/AM.2019.0251-18


References:
[1] I. Bushuyev: Global uniqueness for inverse parabolic problems with final observation. Inverse Probl. 11 (1995), L11-L16. DOI 10.1088/0266-5611/11/4/001 | MR 1345998 | Zbl 0840.35120
[2] M. Choulli: An inverse problem for a semilinear parabolic equation. Inverse Probl. 10 (1994), 1123-1132. DOI 10.1088/0266-5611/10/5/009 | MR 1296363 | Zbl 0807.35154
[3] M. Choulli, M. Yamamoto: Generic well-posedness of an inverse parabolic problem - the Hölder-space approach. Inverse Probl. 12 (1996), 195-205. DOI 10.1088/0266-5611/12/3/002 | MR 1391534 | Zbl 0851.35133
[4] L. B. Feng, P. Zhuang, F. Liu, I. Turner, Y. T. Gu: Finite element method for space-time fractional diffusion equation. Numer. Algorithms 72 (2016), 749-767. DOI 10.1007/s11075-015-0065-8 | MR 3514786 | Zbl 1343.65122
[5] N. J. Ford, J. Xiao, Y. Yan: A finite element method for time fractional partial differential equations. Fract. Calc. Appl. Anal. 14 (2011), 454-474. DOI 10.2478/s13540-011-0028-2 | MR 2837641 | Zbl 1273.65142
[6] A. Hasanov: Simultaneous determination of source terms in a linear parabolic problem from the final overdetermination: Weak solution approach. J. Math. Anal. Appl. 330 (2007), 766-779. DOI 10.1016/j.jmaa.2006.08.018 | MR 2308406 | Zbl 1120.35083
[7] A. Hasanov Hasanoğlu, V. G. Romanov: Introduction to Inverse Problems for Differential Equations. Springer, Cham (2017). DOI 10.1007/978-3-319-62797-7 | MR 3676924 | Zbl 1385.65053
[8] V. Isakov: Inverse parabolic problems with the final overdetermination. Commun. Pure Appl. Math. 54 (1991), 185-209. DOI 10.1002/cpa.3160440203 | MR 1085828 | Zbl 0729.35146
[9] M. Kaya: Determination of the unknown source term in a linear parabolic problem from the measured data at the final time. Appl. Math., Praha 59 (2014), 715-728. DOI 10.1007/s10492-014-0081-3 | MR 3277735 | Zbl 1340.35384
[10] X. Li, C. Xu: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47 (2009), 2108-2131. DOI 10.1137/080718942 | MR 2519596 | Zbl 1193.35243
[11] X. Li, C. Xu: Existence and uniqueness of the weak solution of the space-time fractional diffusion equation and a spectral method approximation. Commun. Comput. Phys. 8 (2010), 1016-1051. DOI 10.4208/cicp.020709.221209a | MR 2674276 | Zbl 1364.35424
[12] M. M. Meerschaert, C. Tadjeran: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172 (2004), 65-77. DOI 10.1016/j.cam.2004.01.033 | MR 2091131 | Zbl 1126.76346
[13] 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 | MR 1809268 | Zbl 0984.82032
[14] A. H. Salehi Shayegan, A. Zakeri: A numerical method for determining a quasi solution of a backward time-fractional diffusion equation. Inverse Probl. Sci. Eng. 26 (2018), 1130-1154. DOI 10.1080/17415977.2017.1384826 | MR 3802827 | Zbl 07039161
[15] W. Y. Tian, C. Li, W. Deng, Y. Wu: Regularization methods for unknown source in space fractional diffusion equation. Math. Comput. Simul. 85 (2012), 45-56. DOI 10.1016/j.matcom.2012.08.011 | MR 2999850 | Zbl 1260.35246
[16] A. Zeghal: Existence results for inverse problems associated with a nonlinear parabolic equation. J. Math. Anal. Appl. 272 (2002), 240-248. DOI 10.1016/s0022-247x(02)00155-5 | MR 1930713 | Zbl 1007.35104

Affiliations:   Amir Hossein Salehi Shayegan (corresponding author), Reza Bayat Tajvar, Alireza Ghanbari, Ali Safaie, Mathematics Department, Faculty of Basic Science, Khatam-ol-Anbia (PBU) University, Tehran, Iran, e-mail: ahsalehi.kau@gmail.com, r.bayat.tajvar@gmail.com, ARGhanbari@gmail.com, alisafaie459@gmail.com


 
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