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Lipschitz stability in inverse source and inverse coefficient problems for a first- and half-order time-fractional diffusion equation. (English) Zbl 1431.35229

Summary: We consider inverse problems for the first- and half-order time-fractional equation. We establish the stability estimates of Lipschitz type in inverse source and inverse coefficient problems by means of the Carleman estimates.

MSC:

35R11 Fractional partial differential equations
35R30 Inverse problems for PDEs
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[1] B. Amaziane, A. Bourgeat, M. Goncharenko, and L. Pankratov, Characterization of the flow for a single fluid in an excavation damaged zone, C. R. Mecanique, 332 (2004), pp. 79-84. · Zbl 1176.76128
[2] B. Amaziane, L. Pankratov, and A. Piatnitski, Homogenization of a single-phase flow through a porous medium in a thin layer, Math. Models Methods Appl. Sci., 17 (2007), pp. 1317-1349. · Zbl 1154.76052
[3] A. Ashyralyev, Well-posedness of the Basset problem in spaces of smooth functions, Appl. Math. Lett., 24 (2011), pp. 1176-1180. · Zbl 1217.34006
[4] A. B. Basset, On the descent of a sphere in a viscous liquid, Q. J. Math., 42 (1910), pp. 369-381. · JFM 41.0826.01
[5] E. Bazhlekova and I. Dimovski, Exact solution of two-term time-fractional Thornley’s problem by operational method, Integral Transforms Spec. Funct., 25 (2014), pp. 61-74. · Zbl 1297.35265
[6] C. E. Brennen, Fundamentals of Multiphase Flow, Cambridge University Press, Cambridge, UK, 2005. · Zbl 1127.76001
[7] A. L. Bukhgeim and M. V. Klibanov, Uniqueness in the large of a class of multidimensional inverse problems, Dokl. Akad. Nauk SSSR, 260 (1981), pp. 269-272 (in Russian).
[8] M. Caputo, Linear model of dissipation whose Q is almost frequency independent-II, Geophys. J. R. Astr. Soc., 13 (1967), pp. 529-539
[9] T. Carleman, Sur un problème d’unicité pour les systèmes d’équations aux dérivées partielles à deux variables indépendantes, Ark. Mat. Astr. Fys., 26B (1939), pp. 1-9. · Zbl 0022.34201
[10] A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series 34, Seoul National University, Seoul, Korea, 1996. · Zbl 0862.49004
[11] O. Y. Imanuvilov, Controllability of parabolic equations, Sb. Math., 186 (1995), pp. 879-900. · Zbl 0845.35040
[12] O. Y. Imanuvilov and M. Yamamoto, Lipschitz stability in inverse parabolic problems by the Carleman estimate, Inverse Problems, 14 (1998), pp. 1229-1245. · Zbl 0992.35110
[13] V. Isakov, Inverse Problems for Partial Differential Equations, 2nd ed., Springer-Verlag, Berlin, 2006. · Zbl 1092.35001
[14] D. Jiang, Z. Li, Y. Liu, and M. Yamamoto, Weak unique continuation property and a related inverse source problem for time-fractional diffusion-advection equations, Inverse Problems, 33 (2017), 055013. · Zbl 1372.35364
[15] A. Kawamoto, Lipschitz stability estimates in inverse source problems for a fractional diffusion equation of half order in time by Carleman estimates, J. Inverse Ill-Posed Probl., 26 (2018), pp. 647-672. · Zbl 1401.35345
[16] A. Kawamoto, Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation, Inverse Probl. Imaging, 12 (2018), pp. 315-330. · Zbl 1395.35196
[17] M. V. Klibanov, Inverse problems in the “large” and Carleman bounds, Differ. Equ., 20 (1984), pp. 755-60. · Zbl 0573.35083
[18] M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), pp. 575-596. · Zbl 0755.35151
[19] M. V. Klibanov, Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, J. Inverse Ill-Posed Probl., 21 (2013), pp. 477-560. · Zbl 1273.35005
[20] M. V. Klibanov and A. A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht, 2004. · Zbl 1069.65106
[21] G. P. Langlois, M. Farazmand, and G. Haller, Asymptotic dynamics of inertial particles with memory, J. Nonlinear. Sci., 25 (2015), pp. 1225-1255. · Zbl 1361.37070
[22] X. Huang, Z. Li, and M. Yamamoto, Carleman estimates for the time-fractional advection-diffusion equations and applications, Inverse Problems, 35 (2019), 045003. · Zbl 1418.35050
[23] Z. Li, Y. Liu, and M. Yamamoto, Initial-boundary value problems for multi-term time-fractioal diffusion equations with positive constant coefficients, Appl. Math. Comput., 257 (2015), pp. 381-397. · Zbl 1338.35471
[24] Z. Li, O. Y. Imanuvilov, and M. Yamamoto, Uniqueness in inverse boundary value problems for fractional diffusio equations, Inverse Problems, 32 (2016), 015004. · Zbl 1332.35396
[25] Z. Li and M. Yamamoto, Uniqueness for inverse problems of determining orders of multi-term time-fractional derivatives of diffusion equation, Appl. Anal., 94 (2015), pp. 570-579. · Zbl 1327.35408
[26] C.-L. Lin and G. Nakamura Unique continuation property for multi-terms time fractional diffusion equations, Math. Ann., pp. 1-24.
[27] Y. Liu, Strong maximum principle for multi-term time-fractional diffusion equations and its application to an inverse source problem, Comput. Math. Appl., 73 (2017), pp. 96-108. · Zbl 1368.35273
[28] Y. Luchko, Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl., 351 (2009), pp. 218-223. · Zbl 1172.35341
[29] Y. Luchko, Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation, J. Math. Anal. Appl., 374 (2011), pp. 538-548. · Zbl 1202.35339
[30] M. R. Maxey and J. J. Riley, Equation of motion for a small rigid sphere in a nonuniform flow, Phys. Fluids, 26 (1983), pp. 883-889. · Zbl 0538.76031
[31] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. · Zbl 0924.34008
[32] C. Ren and X. Xu, Local stability for an inverse coefficient problem of a fractional diffusion equation, Chin. Ann. Math. Ser. B, 35 (2014), pp. 429-446. · Zbl 1297.35287
[33] K. Sakamoto and M. Yamamoto, Inverse source problem with a final overdetermination for a fractional diffusion equation, Math. Control Relat. Fields, 1 (2011), pp. 509-518. · Zbl 1241.35220
[34] X. Xu, J. Cheng and M. Yamamoto, Carleman estimate for fractional diffusion equation with half order and application, Appl. Anal., 90 (2011), pp. 1355-1371. · Zbl 1236.35199
[35] M. Yamamoto, Carleman estimates for parabolic equations and applications, Inverse Problems, 25 (2009), 123013. · Zbl 1194.35512
[36] M. Yamamoto and Y. Zhang, Conditional stability in determining a zeroth-order coefficient in a half-order fractional diffusion equation by a Carleman estimate, Inverse Problems, 28 (2012), 105010. · Zbl 1256.35195
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