Geodesic
Inverse Problem for Finding the Order of the Fractional Derivative in the Wave Equation
Matematičeskie zametki, Tome 110 (2021) no. 6, pp. 824-836 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The paper investigates an inverse problem for finding the order of the fractional derivative in the sense of Gerasimov–Caputo in the wave equation with an arbitrary positive self-adjoint operator $A$ having a discrete spectrum. By means of the classical Fourier method, it is proved that the value of the projection of the solution onto some eigenfunction at a fixed time uniquely restores the order of the derivative. Several examples of the operator $A$ are discussed, including a linear system of fractional differential equations, fractional Sturm–Liouville operators, and many others.
Keywords: wave equation, fractional derivative in the sense of Gerasimov–Caputo, inverse problems for determining the order of the derivative. 
@article{MZM_2021_110_6_a1,
     author = {R. R. Ashurov and Yu. \`E. Fayziev},
     title = {Inverse {Problem} for {Finding} the {Order} of the {Fractional} {Derivative} in the {Wave} {Equation}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {824--836},
     year = {2021},
     volume = {110},
     number = {6},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2021_110_6_a1/}
}
TY  - JOUR
AU  - R. R. Ashurov
AU  - Yu. È. Fayziev
TI  - Inverse Problem for Finding the Order of the Fractional Derivative in the Wave Equation
JO  - Matematičeskie zametki
PY  - 2021
SP  - 824
EP  - 836
VL  - 110
IS  - 6
UR  - http://geodesic.mathdoc.fr/item/MZM_2021_110_6_a1/
LA  - ru
ID  - MZM_2021_110_6_a1
ER  - 
%0 Journal Article
%A R. R. Ashurov
%A Yu. È. Fayziev
%T Inverse Problem for Finding the Order of the Fractional Derivative in the Wave Equation
%J Matematičeskie zametki
%D 2021
%P 824-836
%V 110
%N 6
%U http://geodesic.mathdoc.fr/item/MZM_2021_110_6_a1/
%G ru
%F MZM_2021_110_6_a1
R. R. Ashurov; Yu. È. Fayziev. Inverse Problem for Finding the Order of the Fractional Derivative in the Wave Equation. Matematičeskie zametki, Tome 110 (2021) no. 6, pp. 824-836. http://geodesic.mathdoc.fr/item/MZM_2021_110_6_a1/

[1] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud., 204, Elsevier, Amsterdam, 2006 | MR

[2] R. Gorenflo, A. A. Kilbas, F. Mainardi, S. V. Rogozin, Mittag-Leffler Functions, Related Topics and Applications, Springer, Heidelberg, 2014 | MR

[3] Handbook of Fractional Calculus with Applications. Vol. 2. Fractional Differential Equations, eds. A. Kochubei, Yu. Luchko, De Gruyter, Berlin, 2019 | MR

[4] Z. Li, Y. Liu, M. Yamamoto, “Inverse problems of determining parameters of the fractional partial differential equations”, Handbook of Fractional Calculus with Applications. Vol. 2. Fractional Differential Equations, De Gruyter, Berlin, 2019, 431–442 | MR

[5] M. M. Dzhrbashyan, Integralnye preobrazovaniya i predstavleniya funktsii v kompleksnoi oblasti, Nauka, M., 1966 | MR | Zbl

[6] A. V. Pskhu, Uravneniya v chastnykh proizvodnykh drobnogo poryadka, Nauka, M., 2005 | MR | Zbl

[7] A. Alsaedi, B. Ahmad, M. Kirane, B. T. Torebek, “Blowing-up solutions of the time-fractional dispersive equations”, Adv. Nonlinear Anal., 10:1 (2020), 952–971 | DOI | MR

[8] B. Ahmad, A. Alsaedi, M. Kirane, “Blowing-up solutions of distributed fractional differential systems”, Chaos Solitons Fractals, 145 (2021), Paper No. 110747 | DOI | MR

[9] M. Kirane, B. T. Torebek, Maximum Principle for Space and Time-Space Fractional Partial Differential Equations, 2020, arXiv: 2002.09314v1

[10] J. Cheng, J. Nakagawa, M. Yamamoto, T. Yamazaki, “Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation”, Inverse Problems, 25:11 (2009), Paper No. 115002 | MR

[11] Z. Li, M. Yamamoto, “Uniqueness for inverse problems of determining orders of multi-term time-fractional derivatives of diffusion equation”, Appl. Anal., 94:3 (2015), 570–579 | DOI | MR

[12] M. Yamamoto, Uniqueness in Determining the Order of Time and Spatial Fractional Derivatives, 2020, arXiv: 2006.15046v1

[13] Z. Li, Y. Luchko, M. Yamamoto, “Analyticity of solutions to a distributed order time-fractional diffusion equation and its application to an inverse problem”, Comput. Math. Appl., 73:6 (2017), 1041–1052 | DOI | MR

[14] 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 | MR

[15] R. Ashurov, S. Umarov, “Determination of the order of fractional derivative for subdiffusion equations”, Fract. Calc. Appl. Anal., 23:6 (2020), 1647–1662 | DOI | MR

[16] Sh. Alimov, R. Ashurov, “Inverse problem of determining an order of the Caputo time-fractional derivative for a subdiffusion equation”, J. Inverse Ill-Posed Probl., 28:5 (2020), 651–658 | DOI | MR

[17] R. Ashurov, Yu. Fayziev, Determination of Fractional Order and Source Term in a Fractional Subdifusion Equation, , 2021 https://www.researchgate.net/publication/354997348

[18] R. R. Ashurov, Yu. E. Fayziev, “Uniqueness and existence for inverse problem of determining an order of time-fractional derivative of subdiffusion equation”, Lobachevskii J. Math, 42:3 (2021), 508–516 | DOI | MR

[19] R. Ashurov, R. Zunnunov, “Initial-boundary value and inverse problems for subdiffusion equation in $\mathbb R^N$”, Fract. Differ. Calc., 10:2 (2020), 291–306 | DOI | MR

[20] R. Ashurov, R. Zunnunov, Obratnaya zadacha po opredeleniyu poryadka drobnoi proizvodnoi v uravneniyakh smeshannogo tipa, 2021, arXiv: 2103.05287v1

[21] R. R. Ashurov, Inverse Problems of Determining an Order of Time-Fractional Derivative in a Wave Equation, 2021 | DOI

[22] C. Lizama, “Abstract linear fractional evolution equations”, Handbook of Fractional Calculus with Applications. Vol. 2. Fractional Differential Equations, De Gruyter, Berlin, 2019, 465–479 | MR

[23] O. Novozhenova, Biografiya i nauchnye trudy Alekseya Nikiforovicha Gerasimova. O lineinykh operatorov, uprugo-vyazkosti, elefteroze i drobnykh proizvodnykh, M., Izd-vo «Pero», 2018

[24] O. P. Agrawal, “Solution for a fractional diffusion-wave equation defined in a bounded domain”, Nonlinear Dynam., 29:1-4 (2002), 145–155 | DOI | MR

[25] A. V. Pskhu, “Nachalnaya zadacha dlya lineinogo obyknovennogo differentsialnogo uravneniya drobnogo poryadka”, Matem. sb., 202:4 (2011), 111–122 | DOI | MR | Zbl

[26] A. V. Pskhu, “Funktsiya Grina pervoi kraevoi zadachi dlya drobnogo diffuzionno-volnovogo uravneniya v mnogomernoi pryamougolnoi oblasti”, Materialy IV Mezhdunarodnoi nauchnoi konferentsii “Aktualnye problemy prikladnoi matematiki”, Ch. III, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 167, VINITI RAN, M., 2019, 52–61 | DOI

[27] S. Umarov, Introduction to Fractional and Pseudo-Differential Equations with Singular Symbols, Dev. Math., 41, Springer, Cham, 2015 | MR

[28] R. Ashurov, O. Muhiddinova, “Initial-boundary value problem for a time-fractional subdiffusion equation with an arbitrary elliptic differential operator”, Lobachevskii J. Math., 42:3 (2021), 517–525 | DOI | MR

[29] M. Ruzhansky, N. Tokmagambetov, B. T. Torebek, “On a non-local problem for a multi-term fractional diffusion-wave equation”, Fract. Calc. Appl. Anal., 23:2 (2020), 324–355 | DOI | MR

[30] A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher Transcendental Functions, Based, in part, on notes left by Harry Bateman, McGraw-Hill, New York, 1953 | MR

[31] V. A. Ilin, “O razreshimosti smeshannykh zadach dlya giperbolicheskogo i parabolicheskogo uravnenii”, UMN, 15:2 (92) (1960), 97–154 | MR | Zbl