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An inverse source problem of space-fractional diffusion equation. (English) Zbl 1476.35333

Summary: This paper is devoted to an inverse space-dependent source problem for space-fractional diffusion equation. Furthermore, we show that this problem is ill-posed in the sense of Hadamard, i.e., the solution (if it exists) does not depend continuously on the data. In addition, we propose a simplified generalized Tikhonov regularization method and prove the corresponding convergence estimates by using a priori regularization parameter choice rule and a posteriori parameter choice rule, respectively. Finally, numerical examples are carried to support the theoretical results and illustrate the effectiveness of the proposed method.

MSC:

35R30 Inverse problems for PDEs
35K15 Initial value problems for second-order parabolic equations
35R11 Fractional partial differential equations
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[1] Sabatelle, L.; Keating, S.; Dubley, J.; Richmond, P., Waiting time distributions in financial markets, Eur. Phys. J. B, 24, 273-275 (2002)
[2] Metzler, R.; Klafter, J., The restaurant at the end of the random walks: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A Math. Gen., 37, R161-R208 (2004) · Zbl 1075.82018
[3] Hall, MG; Barrick, TR, From diffusion-weighted MRI to anomalous diffusion imaging, Magn. Reson. Med., 59, 447-455 (2008)
[4] Jiang, H.; Liu, F.; Turner, I.; Burrage, K., Analytical solutions for the multi-term time-fractional diffusion-wave/diffusion equations in a finite domain, Comput. Math. Appl., 64, 3377-3388 (2012) · Zbl 1268.35124
[5] Jin, B.; Lazarov, R.; Zhou, Z., Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM J. Numer. Anal., 51, 445-466 (2013) · Zbl 1268.65126
[6] Li, Z.; Liu, Y.; Yamamoto, M., Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficient, Appl. Math. Comput., 257, 381-397 (2015) · Zbl 1338.35471
[7] Luchko, Y., Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl., 351, 218-223 (2009) · Zbl 1172.35341
[8] Murio, DA, Implicit finite difference approximation for time fractional diffusion equations, Comput. Math. Appl., 56, 1138-1145 (2008) · Zbl 1155.65372
[9] Sakamoto, K.; Yamamoto, M., Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382, 426-447 (2011) · Zbl 1219.35367
[10] Cheng, X.; Li, ZY; Yamamoto, M., Asymptotic behavior of solutions to space-time fractional diffusion-reaction equations, Math. Methods Appl. Sci., 40, 1019-1031 (2017) · Zbl 1372.35333
[11] Kemppainen, J.; Siljander, J.; Zacher, R., Representation of solutions and large-time behavior for fully nonlocal diffusion equations, J. Differ. Equ., 263, 149-201 (2017) · Zbl 1366.35218
[12] Cheng, J.; Nakagawa, J.; Yamamoto, M., Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Prob., 25, 115002 (2009) · Zbl 1181.35322
[13] Liu, JJ; Mamamoto, Y., A backward problem for the time-fractional diffusion equation, Appl. Anal., 89, 1769-1788 (2010) · Zbl 1204.35177
[14] Zheng, GH; Wei, T., Two regularization methods for solving a Riesz-Feller space-fractional backward diffusion problem, Inverse Prob., 26, 115017 (2020) · Zbl 1206.65226
[15] Zhang, Y.; Xu, X., Inverse source problem for a fractional diffusion equation, Inverse Prob., 27, 035010 (2011) · Zbl 1211.35280
[16] Xiong, XT; Zhou, Q.; Hon, YC, An inverse problem for fractional diffusion equation in 2-dimensional case: stability analysis and regularization, J. Math. Anal. Appl., 393, 185-199 (2012) · Zbl 1245.35144
[17] Zhao, JJ; Liu, SS; Liu, T., An inverse problem for space-fractional backward diffusion problem, Math. Methods Appl. Sci., 37, 1147-1158 (2014) · Zbl 1476.35340
[18] Liu, SS; Feng, LX, A modified kernel method for a time-fractional inverse diffusion problem, Adv. Differ. Equ., 342, 1-11 (2015)
[19] Liu, SS; Feng, LX, A posteriori regularization parameter choice rule for a modified kernel method for a time-fractional inverse diffusion problem, J. Comput. Appl. Math., 353, 355-366 (2019) · Zbl 1432.65139
[20] Liu, SS; Feng, LX, Filter regularization method for a time-fractional inverse advection-dispersion problem, Adv. Differ. Equ., 222, 1-14 (2019)
[21] Liu, F.; Burrage, K., Novel techniques in parameter estimation for fractional dynamical models arising from biological systems, Comput. Math. Appl., 62, 822-833 (2011) · Zbl 1228.93114
[22] Yu, B.; Jiang, XY; Wang, C., Numerical algorithms to estimate relaxation parameters and Caputo fractional derivative for a fractional thermal wave model in spherical composite medium, Appl. Math. Comput., 274, 106-118 (2016) · Zbl 1410.80021
[23] Metzler, R.; Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep., 339, 1-77 (2000) · Zbl 0984.82032
[24] Fan, W.; Jiang, X.; Qi, H., Parameter estimation for the generalized fractional element network Zener model based on the Bayesian method, Phys. A, 427, 40-49 (2015)
[25] Yuste, SB; Acedo, L.; Lindenberg, K., Reaction front in an \(a+b\rightarrow c\) reaction-subdiffusion process, Phys. Rev. E, 69, 036126 (2004)
[26] Liu, F.; Anh, V.; Turner, I., Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math., 166, 209-219 (2004) · Zbl 1036.82019
[27] Trong, DD; Hai, DND; Minh, ND, Optimal regularization for an unknown source of space-fractional diffusion equation, Appl. Math. Comput., 349, 184-206 (2019) · Zbl 1429.65221
[28] Yang, F.; Fu, CL; Li, XX, Identifying an unknown source in space-fractional diffusion equation, Acta Math. Sci., 34B, 1012-1024 (2014) · Zbl 1324.35204
[29] Li, XX; Lei, JL; Yang, F., An a posteriori Fourier regularization method for identifying the unknown source of the space-fractional diffusion equation, J. Inequal. Appl., 434, 1-13 (2014)
[30] Mainardi, F.; Luchko, Y.; Pagnini, G., The fundamental solution of the space-time fractional diffusion equation, Fract. Calculus Appl. Anal., 4, 153-192 (2001) · Zbl 1054.35156
[31] Kirsch, A., An Introduction to the Mathematical Theory of Inverse Problem (1999), New York: Springer, New York
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