# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2022017
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## Uniqueness for inverse problem of determining fractional orders for time-fractional advection-diffusion equations

 Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Meguro, Tokyo 153-8914, Japan, Honorary Member of Academy of Romanian Scientists, Ilfov, nr. 3, Bucuresti, Romania, Correspondence Member of Accademia Peloritana dei Pericolanti, Palazzo Università, Piazza S. Pugliatti 1 98122 Messina, Italy

* Corresponding author: Masahiro Yamamoto

Received  April 2021 Revised  November 2021 Early access April 2022

Fund Project: The author was supported by JSPS grant 20H00117 and NSFC grants 11771270, 91730303

We consider initial boundary value problems of time-fractional advection-diffusion equations with the zero Dirichlet boundary value $\partial_t^{\alpha} u(x, t) = -Au(x, t)$, where $-A = \sum_{i, j = 1}^d \partial_i(a_{ij}(x) \partial_j) + \sum_{j = 1}^d b_j(x) \partial_j + c(x)$. We establish the uniqueness for an inverse problem of determining an order $\alpha$ of fractional derivatives by data $u(x_0, t)$ for $0<t<T$ at one point $x_0$ in a spatial domain $\Omega$. The uniqueness holds even under assumption that $\Omega$ and $A$ are unknown, provided that the initial value does not change signs and is not identically zero. The proof is based on the eigenfunction expansions of finitely dimensional approximating solutions, a decay estimate and the asymptotic expansions of the Mittag-Leffler functions for large time.

Citation: Masahiro Yamamoto. Uniqueness for inverse problem of determining fractional orders for time-fractional advection-diffusion equations. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022017
##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] S. Agmon, Lectures on Elliptic Boundary Value Problems, D. van Nostrand, Princeton, 1965. [3] S. Alimov and R. Ashurov, Inverse problem of determining an order of the Caputo time-fractional derivative for a subdiffusion equation, J. Inverse and Ill-Posed Probl., 28 (2020), 651-658.  doi: 10.1515/jiip-2020-0072. [4] R. Ashurov and S. Umarov, Determination of the order of fractional derivative for subdiffusion equations, Fract. Calc. Appl. Anal., 23 (2020), 1647-1662.  doi: 10.1515/fca-2020-0081. [5] J. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki, Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Problems, 25 (2009), 115002.  doi: 10.1088/0266-5611/25/11/115002. [6] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. [7] R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer, Heidelberg, 2014. doi: 10.1007/978-3-662-43930-2. [8] R. Gorenflo, Y. Luchko and M. Yamamoto, Time-fractional diffusion equation in the fractional Sobolev spaces, Fract. Calc. Appl. Anal., 18 (2015), 799-820.  doi: 10.1515/fca-2015-0048. [9] Y. Hatano, J. Nakagawa, S. Wang and M. Yamamoto, Determination of order in fractional diffusion equation, J. Math-for-Ind., 5(A) (2013), 51-57. [10] J. Janno, Determination of the order of fractional derivative and a kernel in an inverse problem for a generalized time fractional diffusion equation, Electron. J. Differential Equations, 2016 (2016), Paper No. 199, 28 pp. [11] J. Janno and N. Kinash, Reconstruction of an order of derivative and a source term in a fractional diffusion equation from final measurements, Inverse Problems, 34 (2018), 025007.  doi: 10.1088/1361-6420/aaa0f0. [12] B. Jin and Y. Kian, Recovery of the order of derivation for fractional diffusion equations in an unknown medium, to appear, SIAM J. Appl. Math., arXiv: 2101.09165. [13] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995. [14] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. [15] M. Krasnoschok, S. Pereverzyev, S. V. Siryk and N. Vasylyeva, Determination of the fractional order in semilinear subdiffusion equations, Fract. Calc. Appl. Anal., 23 (2020), 694-722.  doi: 10.1515/fca-2020-0035. [16] A. Kubica, K. Ryszewska and M. Yamamoto, Time-Fractional Differential Equations: A Theoretical Introduction, Springer Japan, Tokyo, 2020. doi: 10.1007/978-981-15-9066-5. [17] G. Li, D. Zhang, X. Jia and M. Yamamoto, Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation, Inverse Problems, 29 (2013), 065014.  doi: 10.1088/0266-5611/29/6/065014. [18] Z. Li, O. Y. Imanuvilov and M. Yamamoto, Uniqueness in inverse boundary value problems for fractional diffusion equations, Inverse Problems, 32 (2016), 015004, 16 pp. doi: 10.1088/0266-5611/32/1/015004. [19] Z. Li, Y. Liu and M. Yamamoto, Inverse problems of determining parameters of the fractional partial differential equations, in Handbook of Fractional Calculus with Applications(ed: J. A. Tenreiro Machado, A. N. Kochubei and Y. Luchko) Vol. 2, De Gruyter, Berlin, 2019,431–442. [20] Z. Li and M. Yamamoto, Uniqueness for inverse problems of determining orders of multi-term time-fractional derivatives of diffusion equations, Appl. Anal., 94 (2015), 570-579.  doi: 10.1080/00036811.2014.926335. [21] Y. Luchko and M. Yamamoto, On the maximum principle for a time-fractional diffusion equation, Fract. Calc. Appl. Anal., 20 (2017), 1131-1145.  doi: 10.1515/fca-2017-0060. [22] F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos, Solitons and Fractals, 7 (1996), 1461-1477.  doi: 10.1016/0960-0779(95)00125-5. [23] R. Metzler, W. G. Glöckle and T. F. Nonnenmacher, Fractional model equations for anomalous diffusion, Physica A, 211 (1994), 13-24. [24] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phyics Reports, 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3. [25] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [26] A. Yu. Popov and A. M. Sedletskii, Distribution of roots of Mittag-Leffler functions, J. Math. Sci., 190 (2013), 209-409.  doi: 10.1007/s10958-013-1255-3. [27] H. E. Roman and P. A. Alemany, Continuous-time random walks and the fractional diffusion equation, J. Phys. A: Math. Gen., 27 (1994), 3407-3410.  doi: 10.1088/0305-4470/27/10/017. [28] 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), 426-447.  doi: 10.1016/j.jmaa.2011.04.058. [29] S. Tatar, R. Tinaztepe and S. Ulusoy, Simultaneous inversion for the exponents of the fractional time and space derivatives in the space-time fractional diffusion equation, Appl. Anal., 95 (2016), 1-23.  doi: 10.1080/00036811.2014.984291. [30] S. Tatar and S. Ulusoy, A uniqueness result for an inverse problem in a space-time fractional diffusion equation, Electron. J. Differential Equations, 2013 (2013), Paper No. 258, 9 pp. [31] V. Vergara and R. Zacher, Optimal decay estimates for time-fractional and other nonlocal subdiffusion equation via energy methods, SIAM J. Math. Anal., 47 (2015), 210-239.  doi: 10.1137/130941900. [32] M. Yamamoto, Uniqueness in determining the orders of time and spatial fractional derivatives, preprint, 2020, arXiv: 2006.15046. [33] M. Yamamoto, Uniqueness in determining fractional orders of derivatives and initial values, Inverse Problems, 37 (2021), 095006.  doi: 10.1088/1361-6420/abf9e9. [34] B. Yu, X. Jiang and H. Qi, An inverse problem to estimate an unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid, Acta Mech. Sin., 31 (2015), 153-161.  doi: 10.1007/s10409-015-0408-7.

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##### References:
 [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. [2] S. Agmon, Lectures on Elliptic Boundary Value Problems, D. van Nostrand, Princeton, 1965. [3] S. Alimov and R. Ashurov, Inverse problem of determining an order of the Caputo time-fractional derivative for a subdiffusion equation, J. Inverse and Ill-Posed Probl., 28 (2020), 651-658.  doi: 10.1515/jiip-2020-0072. [4] R. Ashurov and S. Umarov, Determination of the order of fractional derivative for subdiffusion equations, Fract. Calc. Appl. Anal., 23 (2020), 1647-1662.  doi: 10.1515/fca-2020-0081. [5] J. Cheng, J. Nakagawa, M. Yamamoto and T. Yamazaki, Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation, Inverse Problems, 25 (2009), 115002.  doi: 10.1088/0266-5611/25/11/115002. [6] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 2001. [7] R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer, Heidelberg, 2014. doi: 10.1007/978-3-662-43930-2. [8] R. Gorenflo, Y. Luchko and M. Yamamoto, Time-fractional diffusion equation in the fractional Sobolev spaces, Fract. Calc. Appl. Anal., 18 (2015), 799-820.  doi: 10.1515/fca-2015-0048. [9] Y. Hatano, J. Nakagawa, S. Wang and M. Yamamoto, Determination of order in fractional diffusion equation, J. Math-for-Ind., 5(A) (2013), 51-57. [10] J. Janno, Determination of the order of fractional derivative and a kernel in an inverse problem for a generalized time fractional diffusion equation, Electron. J. Differential Equations, 2016 (2016), Paper No. 199, 28 pp. [11] J. Janno and N. Kinash, Reconstruction of an order of derivative and a source term in a fractional diffusion equation from final measurements, Inverse Problems, 34 (2018), 025007.  doi: 10.1088/1361-6420/aaa0f0. [12] B. Jin and Y. Kian, Recovery of the order of derivation for fractional diffusion equations in an unknown medium, to appear, SIAM J. Appl. Math., arXiv: 2101.09165. [13] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1995. [14] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. [15] M. Krasnoschok, S. Pereverzyev, S. V. Siryk and N. Vasylyeva, Determination of the fractional order in semilinear subdiffusion equations, Fract. Calc. Appl. Anal., 23 (2020), 694-722.  doi: 10.1515/fca-2020-0035. [16] A. Kubica, K. Ryszewska and M. Yamamoto, Time-Fractional Differential Equations: A Theoretical Introduction, Springer Japan, Tokyo, 2020. doi: 10.1007/978-981-15-9066-5. [17] G. Li, D. Zhang, X. Jia and M. Yamamoto, Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation, Inverse Problems, 29 (2013), 065014.  doi: 10.1088/0266-5611/29/6/065014. [18] Z. Li, O. Y. Imanuvilov and M. Yamamoto, Uniqueness in inverse boundary value problems for fractional diffusion equations, Inverse Problems, 32 (2016), 015004, 16 pp. doi: 10.1088/0266-5611/32/1/015004. [19] Z. Li, Y. Liu and M. Yamamoto, Inverse problems of determining parameters of the fractional partial differential equations, in Handbook of Fractional Calculus with Applications(ed: J. A. Tenreiro Machado, A. N. Kochubei and Y. Luchko) Vol. 2, De Gruyter, Berlin, 2019,431–442. [20] Z. Li and M. Yamamoto, Uniqueness for inverse problems of determining orders of multi-term time-fractional derivatives of diffusion equations, Appl. Anal., 94 (2015), 570-579.  doi: 10.1080/00036811.2014.926335. [21] Y. Luchko and M. Yamamoto, On the maximum principle for a time-fractional diffusion equation, Fract. Calc. Appl. Anal., 20 (2017), 1131-1145.  doi: 10.1515/fca-2017-0060. [22] F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos, Solitons and Fractals, 7 (1996), 1461-1477.  doi: 10.1016/0960-0779(95)00125-5. [23] R. Metzler, W. G. Glöckle and T. F. Nonnenmacher, Fractional model equations for anomalous diffusion, Physica A, 211 (1994), 13-24. [24] R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phyics Reports, 339 (2000), 1-77.  doi: 10.1016/S0370-1573(00)00070-3. [25] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. [26] A. Yu. Popov and A. M. Sedletskii, Distribution of roots of Mittag-Leffler functions, J. Math. Sci., 190 (2013), 209-409.  doi: 10.1007/s10958-013-1255-3. [27] H. E. Roman and P. A. Alemany, Continuous-time random walks and the fractional diffusion equation, J. Phys. A: Math. Gen., 27 (1994), 3407-3410.  doi: 10.1088/0305-4470/27/10/017. [28] 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), 426-447.  doi: 10.1016/j.jmaa.2011.04.058. [29] S. Tatar, R. Tinaztepe and S. Ulusoy, Simultaneous inversion for the exponents of the fractional time and space derivatives in the space-time fractional diffusion equation, Appl. Anal., 95 (2016), 1-23.  doi: 10.1080/00036811.2014.984291. [30] S. Tatar and S. Ulusoy, A uniqueness result for an inverse problem in a space-time fractional diffusion equation, Electron. J. Differential Equations, 2013 (2013), Paper No. 258, 9 pp. [31] V. Vergara and R. Zacher, Optimal decay estimates for time-fractional and other nonlocal subdiffusion equation via energy methods, SIAM J. Math. Anal., 47 (2015), 210-239.  doi: 10.1137/130941900. [32] M. Yamamoto, Uniqueness in determining the orders of time and spatial fractional derivatives, preprint, 2020, arXiv: 2006.15046. [33] M. Yamamoto, Uniqueness in determining fractional orders of derivatives and initial values, Inverse Problems, 37 (2021), 095006.  doi: 10.1088/1361-6420/abf9e9. [34] B. Yu, X. Jiang and H. Qi, An inverse problem to estimate an unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid, Acta Mech. Sin., 31 (2015), 153-161.  doi: 10.1007/s10409-015-0408-7.
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