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# Reports AIAS. Vol. 19, no 1. P. 31-41. ISSN 1726-9946

Contents of this issue

MATHEMATICS

Research Article

## On one inverse problem of reconstructing a subdiffusion process with degeneration from nonlocal data

### Sadybekov M.A.1 – corresponding member of NAS RK, academician of AIAS, Sarsenbi A.A.1,2

1Institute of Mathematics and Mathematical Modeling, Almaty
2M. Auezov South Kazakhstan State University, Shymkent
E-mail: sadybekov@math.kz; abdisalam@mail.ru

In this article, we consider an inverse problem for one-dimensional degenerate fractional heat equation with involution and with periodic boundary conditions with respect to a spatial variable. This problem simulates the process of heat propagation in a thin closed wire wrapped around a weakly permeable insulation. The inverse problem consists in the restoration (simultaneously with the solution) of an unknown right-hand side of the equation, which depends only on the spatial variable. The conditions for redefinition are initial and final states. Existence and uniqueness results for the given problem are obtained via the method of separation of variables.

Keywords: inverse problem, heat equation, equation with involution, subdiffusion process, equation with degeneration, periodic boundary conditions, method of separation of variables.

© M.A. Sadybekov,
A.A. Sarsenbi, 2019

Список литературы (ГОСТ)

1. Ahmad B., Alsaedi A., Kirane M., Tapdigoglu R.G. An in-verse problem for space and time fractional evolution equations with an involution perturbation // Quaestiones Mathematicae. 2017. V. 40, № 2. Pp. 151-160. DOI: 10.2989/16073606.2017.1283370.
2. Cabada A., Tojo A.F. Equations with involutions, Workshop on Differential Equations // Malla Moravka, Czech Republic. 2014. 240 p. Available from: http://users.math.cas.cz/sremr/wde2014/prezentace/cabada.pdf.
3. Sadybekov M., Dildabek G., Ivanova M. On an inverse problem of reconstructing a heat conduction process from nonlocal data // Advances in Mathematical Physics. (Article ќ 8301656). 2018. Pp. 1-8.
4. Dildabek G., Ivanova M. On a class of inverse problems on a source restoration in the heat conduction process from nonlocal data // Mathematical Journal. 2018. V. 18, № 2. Pp. 87-106.
5. Kirane M., Sadybekov M.A., Sarsenbi A.A. On an inverse problem of reconstructing a subdiffusion process from nonlocal data // Mathematical Methods in the Applied Sciences. 2018 (Accepted for publication). Pp. 1-10.
6. Ashyralyev A., Sarsenbi A. Well-posedness of a parabolic equation with nonlocal boundary condition // Boundary Value Problems. 2015. № 38. DOI: 10.1186/s13661-015-0297-5.
7. Kirane M., Al-Salti N. Inverse problems for a nonlocal wave equation with an involution perturbation // J. Nonlinear Sci. Appl. 2016. V. 9. Pp. 1243-1251.
8. Ashyralyev A., Sarsenbi A. Well-Posedness of a Parabolic Equation with Involution // Numerical Functional Analysis and Optimization. 2017. Pp. 1-10. DOI: 10.1080/01630563.2017.1316997
9. Orazov I., Sadybekov M.A. One nonlocal problem of determination of the temperature and density of heat sources // Russian Math. 2012. V. 56, ќ 2. Pp. 60_64. DOI: 10.3103/S1066369X12020089.
10. Orazov I., Sadybekov M.A. On a class of problems of determining the temperature and density of heat sources given initial and final temperature // Sib. Math. J. 2012. V. 53, № 1. Pp. 146-151. DOI: 10.1134/S0037446612010120.
11. Ivanchov M.I. Some inverse problems for the heat equation with nonlocal boundary conditions // Ukrainian Mathematical Journal. 1993. V. 45, № 8. Pp. 1186-1192.
12. Kaliev I.A., Sabitova M.M. Problems of determining the temperature and density of heat sources from the initial and final temperatures // Journal of Applied and Industrial Mathematics. 2010. V. 4, № 3. Pp. 332-339. DOI: 10.1134/S199047891003004X.
13. Kaliev I.A., Mugafarov M.F., Fattahova O.V. Inverse problem for forwardbackward parabolic equation with generalized conjugation conditions // Ufa Mathematical Journal. 2011.V. 3, №2. Pp. 33-41.
14. Kirane M., Malik A.S. Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time // Appl. Math. Comput. 2011. V. 218, № 1. Pp. 163-170. DOI: 10.1016/j.am.2011.05.084.
15. Ismailov M.I., Kanca F. The inverse problem of finding the time-dependent diffusion coefficient of the heat equation from integral overdetermination data // Inverse Problems in Science and Engineering. 2012. V. 20. Pp. 463-476. DOI: 10.1080/17415977.2011.629093.
16. Kirane M., Malik A.S., Al-Gwaiz M.A. An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions // Math. Methods Appl. Sci. 2013. V. 36, № 9. Pp. 1056-1069. DOI: 10.1002/mma.2661.
17. Ashyralyev A., Sharifov Y.A. Counterexamples in inverse problems for parabolic, elliptic, and hyperbolicequations // Advances in Difference Equations. 2013. V. 2013, № 173. Pp. 797-810. DOI: 10.1186/1687-1847-2013-173.
18. Kanca F. Inverse coefficient problem of the parabolic equation with periodic boundary and integral overdetermination conditions // Abstract and Applied Analysis. 2013. V. 2013. (Article ќ 659804). Pp. 1-7. DOI: 10.1155/2013/659804.
19. Lesnic D., Yousefi S.A., Ivanchov M. Determination of a time-dependent diffusivity form nonlocal conditions // Journal of Applied Mathematics and Computation. 2013. V. 41. Pp. 301-320. DOI: 10.1007/s12190-012-0606-4.
20. Miller L., Yamamoto M. Coefficient inverse problem for a fractional diffusion equation // Inverse Problems. 2013. V. 29, № 7. (Article № 075013). Pp. 1-8. DOI: 10.1088/0266-5611/29/7/075013.
21. Li G., Zhang D., Jia X., Yamamoto M. Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation // Inverse Problems. 2013. V. 29, № 6. (Article № 065014). Pp. 1-36. DOI: 10.1088/0266-5611/29/6/065014.
22. Kostin A.B. Counterexamples in inverse problems for parabolic, elliptic, and hyperbolic equations // Computational Mathematics and Mathematical Physics. 2014. V. 54, № 5. Pp. 797-810. DOI: 10.1134/S0965542514020092.
23. Ashyralyev A., Hanalyev A. Well-posedness of nonlocal parabolic differential problems with dependent operators // The Scientific World Journal. 2014. V. 2014. (Article № 519814). Pp. 1-11. DOI: 10.1155/2014/519814.
24. Ashyralyev A., Sarsenbi A. Well-posedness of a parabolic equation with nonlocal boundary condition // Boundary Value Problems. 2015. V. 2015, № 1. DOI: 10.1186/s13661-015-0297-5.
25. Orazov I., Sadybekov M.A. On an inverse problem of mathematical modeling of the extraction process of polydisperse porous materials // AIP Conference Proceedings. 2015. V. 1676. (Article № 020005). DOI: 10.1063/1.4930431.
26. Orazov I., Sadybekov M.A. One-dimensional diffusion problem with not strengthened regular boundary conditions // AIP Conference Proceedings. 2015. V.1690. (Article № 040007). DOI: 10.1063/1.4936714
27. Tuan N.H., Hai D.N.D., Long L.D., Thinh N.V., Kirane M. On a Riesz-Feller space fractional backward diffusion problem with a nonlinear source // J. Comput. Appl. Math. 2017. V. 312. Pp. 103-126. DOI: 10.1016/j.cam.2016.01.003.
28. Sadybekov M., Oralsyn G., Ismailov M. An inverse problem of finding the time-dependent heat transfer coefficient from an integral condition // International Journal of Pure and Applied Mathematics. 2017. V. 113, № 4. Pp. 139-149. DOI: 10.12732/ijpam.v113i4.13.
29. Tuan N.H., Kirane M., Hoan L.V.C., Long L.D. Identification and regularization for unknown source for a time-fractional diffusion equation // Computers & Mathematics with Applications. 2017. V. 73, № 6. Pp. 931-950. DOI: 10.1016/j.camwa.2016.10.002.
30. Torebek B.T., Tapdigoglu R. Some inverse problems for the nonlocal heat equation with Caputo fractional derivative // Math Meth Appl Sci. 2017. V. 40. Pp. 6468-6479. DOI: 10.1002/mma.4468.
31. Kirane M. Samet, B., Torebek B.T. Determination of an unknown source term temperature distribution for the sub-diffusion equation at the initial and final data // Electronic Journal of Differential Equations. 2017. V. 2017. (Article № 257). Pp. 1-13.
32. Sarsenbi A.M. Unconditional bases related to a nonclassical second-order differential operator // Differential Equations. 2010. V. 46, № 4. Pp. 506-511. DOI: 10.1134/S0012266110040051.
33. Kurdyumov V.P., Khromov A.P. The Riesz bases consisting of eigen and associated functions for a functional differential operator with variable structure // Russian Mathematics. 2010. V. 2. Pp. 39-52. DOI: 10.3103/S1066369X10020052.
34. Sarsenbi A., Tengaeva A.A. On the basis properties of root functions of two generalized eigenvalue problems // Differential Equations. 2012. V. 48, № 2. Pp. 306-308. DOI: 10.1134/S0012266112020152.
35. Sadybekov M.A., Sarsenbi A.M. Criterion for the basis property of the eigenfunction system of a multiple differentiation operator with an involution // Differential Equations. 2012. V. 48, № 8. Pp. 1112-1118. DOI: 10.1134/S001226611208006X.
36. Kopzhassarova A., Sarsenbi A. Basis Properties of Eigenfunctions of Second Order Differential Operators with Involution // Abstr. Appl. Anal. 2012. V. 2012. (Article № 576843). Pp. 1-6. DOI: 10.1155/2012/576843.
37. Kopzhassarova A.A., Lukashov A.L., Sarsenbi A.M. Spectral Properties of non-self-adjoint perturbations for a spectral problem with involution // Abstr. Appl. Anal. 2012. V. 2012. (Article № 590781). DOI: 10.1155/2012/590781.
38. Sarsenbi A., Sadybekov M. Eigenfunctions of a fourth order operator pencil // AIP Conference Proceedings. 2014. V. 1611. Pp. 241-245. DOI: 10.1063/1.4893840.
39. Kritskov L.V., Sarsenbi A.M. Spectral properties of a nonlocal problem for the differential equation with involution // Differential Equations. 2015. V. 51, № 8. Pp. 984-990. DOI: 10.1134/S0012266115080029.
40. Kritskov L.V., Sarsenbi A.M. Basicity in Lp of root functions for differential equations with involution // Electron. J. Differ. Equ. 2015. V. 2015, № 278.
41. Sadybekov M.A., Sarsenbi A., Tengayeva A. Description of spectral properties of a generalized spectral problem with involution for differentiation operator of the second order // AIP Conference Proceedings. 2016. V. 1759. (Article № 020154). DOI: 10.1063/1.4959768.
42. Baskakov A.G., Krishtal I.A., Romanova E.Y. Spectral analysis of a differential operator with an involution // Journal of Evolution Equations. 2017. V. 17, № 2. Pp. 669-684. DOI: 10.1007/s00028-016-0332-8.
43. Kritskov L.V., Sarsenbi A.M. Riesz basis property of system of root functions of second-order differential operator with involution // Differential Equations. 2017. V. 53, № 1. Pp. 33-46. DOI: 10.1134/S0012266117010049.
44. Kilbas A.A., Srivastava H.M., Trujillo J.J. Theory and applications of fractional differential equations. North-Holland Mathematics Studies 204, Elsevier. 2006.

For citation. Sadybekov M.A., Sarsenbi A.A. On one inverse problem of reconstructing a subdiffusion process with degeneration from nonlocal data. Reports Adyghe (Circassian) International Academy of Sciences. 2019, vol. 19, no. 1, pp. 31-41.

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