Uniqueness of solution for the inverse problem of finding two minor coefficients in a semilinear time fractional telegraph equation

H. P. Lopushanska1, A. O. Lopushansky2
1) Ivan Franko National University of Lviv, Lviv, Ukraine; 2) Faculty of Mathematical and Natural Sciences, Rzeszow University Rzeszow, Poland
We find sufficient conditions of the uniqueness of a solution for the inverse problem of determining two continuous minor coefficients in a semilinear time fractional telegraph equation under two integral overdetermination conditions.
fractional derivative; inverse problem; integral overdetermination condition; Greens vectorfunction; integral equation
1. Y. Povstenko, Theories of thermal stresses based on space-time fractional telegraph equations, Comput. Math. Appl., 64 (2012), 33213328.

2. T.S. Aleroev, M. Kirane, S.A. Malik, Determination of a source term for a time fractional diffusion equation with an integral type overdetermination condition, Electron. J. Differ. Equ., 2013 (2013), 116.

3. Y. Hatano, J. Nakagawa, Sh.Wang, M. Yamamoto, Determination of order in fractional diffusion equation, J. Math-for-Ind., 5A (2013), 5157.

4. M.I. Ismailov, Inverse source problem for a time-fractional diffusion equation with nonlocal boundary conditions, Appl. Math. Model., 40 (2016), 7/8, 48914899.

5. J. Janno, K. Kasemets, Uniqueness for an inverse problem foe a semilinear time-fractional diffusion equation, // Inverse Probl. Imaging, 11 (2017), 1, 125149. doi: 10.3934/ipi.2017007

6. B. Jin, W. Rundell, A turorial on inverse problems for anomalous diffusion processes, Inverse Probl., 31 (2015), 3. doi:10.1088/0266-5611/31/3/035003.

7. H. Lopushanska, V. Rapita, Inverse coefficient problem for semi-linear fractional telegraph equation, Electron. J. Differ. Equ., 2015 (2015), 153, 113.

8. H. Lopushanska, V. Shumska, The inverse problem of finding minor coefficients in the telegraph equation with fractional derivatives in time, In: Nonclassical problems of the theory of differential equations: a collection of scientific works devoted to the 80th anniversary of B.Y. Ptashnyk. Under the Ed. Kushnir R.M., Pelykh V.O., Lviv: Ya.S. Pidstryhach IAPMM of NAS of Ukraine, 2017. 161172. (in Ukrainian)

9. H. Lopushanska, A problem with an integral boundary condition for a time fractional diffusion equation and an inverse problem, Fract. Differ. Calc., 6 (2016), 1, 133145. doi:10.7153/fdc-06-09

10. A. Lopushanskyj, Solvability of inverse boundary value problem for equation with fractional derivative, Visn. Lviv Univ. Ser. Mech. Math., 79 (2014), 97110. (in Ukrainian)

11. W. Rundell, X. Xu, L. Zuo, The determination of an unknown boundary condition in fractional diffusion equation, Appl. Anal., 1 (2012), 116.

12. K. Sakamoto, M. Yamamoto, Initial value/boundary-value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 1, 426447.

13. Y. Zhang, X. Xu, Inverse source problem for a fractional diffusion equation, Inv. probl., 27 (2011), 112.

14. M. Ivanchov, Inverse problem for semilinear parabolic equation, Mat. Stud., 29 (2008), 2, 181191.

15. N.P. Protsakh, Asymptotic behavior of solution of the inverse problem for weakly nonlinear ultraparabolic equation, Carpathian Math. Publ., 5 (2013), 2, 326335. doi: 10.15330/cmp.5.2.326-335 (in Ukrainian)

16. V.V. Anh, N.N. Leonenko, Spectral analysis of fractional kinetic equations with random datas, J. Statist. Phys., 104 (2001), 5/6, 13491387.

17. S.D. Eidelman, S.D. Ivasyshen, A.N. Kochubei, Analytic methods in the theory of differential and pseudodifferential equations of parabolic type, Basel-Boston-Berlin, Birkhauser Verlag, 2004.

18. A. Friedman, Partial differential equations of parabolic type, Englewood Cliffs, N. J., Prentice-Hall, 1964.

19. M.I. Matijchuk, The connection between fundamental solutions of parabolic equations and fractional equations, Bukovyn Math. J., 4 (2016), 34, 101114. (in Ukrainian)

20. A.A. Voroshylov, A.A. Kilbas, Conditions of the existence of classical solution of the Cauchy problem for diffusion-wave equation with Caputo partial derivative, Dokl. Ak. Nauk., 414 (2007), 4, 14.

21. P.P. Zabreiko, A.I. Koshelev, M.A. Krasnoselsky, S.G. Mikhlin, L.S. Rakovshchik, V.Ya. Stetsenko, Integral equations, Nauka, Moskow, 1968. (in Russian)

Matematychni Studii
Full text of paper
Table of content of issue