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.


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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