doi: 10.3934/mcrf.2022017
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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. ChengJ. NakagawaM. 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. GorenfloY. 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. HatanoJ. NakagawaS. 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. KrasnoschokS. PereverzyevS. 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. LiD. ZhangX. 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. MetzlerW. 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. TatarR. 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. YuX. 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.

show all references

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. ChengJ. NakagawaM. 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. GorenfloY. 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. HatanoJ. NakagawaS. 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. KrasnoschokS. PereverzyevS. 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. LiD. ZhangX. 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. MetzlerW. 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. TatarR. 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. YuX. 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.

[1]

Qing Tang. On an optimal control problem of time-fractional advection-diffusion equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 761-779. doi: 10.3934/dcdsb.2019266

[2]

S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3747-3761. doi: 10.3934/dcdss.2020435

[3]

Nguyen Huy Tuan, Donal O'Regan, Tran Bao Ngoc. Continuity with respect to fractional order of the time fractional diffusion-wave equation. Evolution Equations and Control Theory, 2020, 9 (3) : 773-793. doi: 10.3934/eect.2020033

[4]

Assyr Abdulle. Multiscale methods for advection-diffusion problems. Conference Publications, 2005, 2005 (Special) : 11-21. doi: 10.3934/proc.2005.2005.11

[5]

Jaan Janno, Kairi Kasemets. Uniqueness for an inverse problem for a semilinear time-fractional diffusion equation. Inverse Problems and Imaging, 2017, 11 (1) : 125-149. doi: 10.3934/ipi.2017007

[6]

Xiaohua Jing, Masahiro Yamamoto. Simultaneous uniqueness for multiple parameters identification in a fractional diffusion-wave equation. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022019

[7]

Lena-Susanne Hartmann, Ilya Pavlyukevich. Advection-diffusion equation on a half-line with boundary Lévy noise. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 637-655. doi: 10.3934/dcdsb.2018200

[8]

Abdollah Borhanifar, Maria Alessandra Ragusa, Sohrab Valizadeh. High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5495-5508. doi: 10.3934/dcdsb.2020355

[9]

Rui Qian, Rong Hu, Ya-Ping Fang. Local smooth representation of solution sets in parametric linear fractional programming problems. Numerical Algebra, Control and Optimization, 2019, 9 (1) : 45-52. doi: 10.3934/naco.2019004

[10]

Editorial Office. WITHDRAWN: Fractional diffusion equation described by the Atangana-Baleanu fractional derivative and its approximate solution. Discrete and Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2020173

[11]

Kaouther Bouchama, Yacine Arioua, Abdelkrim Merzougui. The numerical solution of the space-time fractional diffusion equation involving the Caputo-Katugampola fractional derivative. Numerical Algebra, Control and Optimization, 2022, 12 (3) : 621-636. doi: 10.3934/naco.2021026

[12]

Kim-Ngan Le, William McLean, Martin Stynes. Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2765-2787. doi: 10.3934/cpaa.2019124

[13]

Michael Taylor. Random walks, random flows, and enhanced diffusivity in advection-diffusion equations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1261-1287. doi: 10.3934/dcdsb.2012.17.1261

[14]

Liming Ling. The algebraic representation for high order solution of Sasa-Satsuma equation. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1975-2010. doi: 10.3934/dcdss.2016081

[15]

Claude Bardos, François Golse, Ivan Moyano. Linear Boltzmann equation and fractional diffusion. Kinetic and Related Models, 2018, 11 (4) : 1011-1036. doi: 10.3934/krm.2018039

[16]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4907-4926. doi: 10.3934/dcdsb.2020319

[17]

Tarek Saanouni. Energy scattering for the focusing fractional generalized Hartree equation. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3637-3654. doi: 10.3934/cpaa.2021124

[18]

Atsushi Kawamoto. Hölder stability estimate in an inverse source problem for a first and half order time fractional diffusion equation. Inverse Problems and Imaging, 2018, 12 (2) : 315-330. doi: 10.3934/ipi.2018014

[19]

Yones Esmaeelzade Aghdam, Hamid Safdari, Yaqub Azari, Hossein Jafari, Dumitru Baleanu. Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2025-2039. doi: 10.3934/dcdss.2020402

[20]

Haili Qiao, Aijie Cheng. A fast high order method for time fractional diffusion equation with non-smooth data. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 903-920. doi: 10.3934/dcdsb.2021073

2021 Impact Factor: 1.141

Metrics

  • PDF downloads (133)
  • HTML views (73)
  • Cited by (0)

Other articles
by authors

[Back to Top]