Нажмите "Enter", чтобы перейти к контенту

Доклады АМАН. Т. 19, 1. С. 31-41. ISSN 1726-9946

Доклады АМАН. Т. 19, №1. С. 31-41. ISSN 1726-9946

Содержание выпуска

МАТЕМАТИКА

УДК 517.956 Научная статья

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

1Институт математики и математического моделирования, Алматы
2Южно-Казахстанский государственный университет им. М. Ауезова, Шымкент
E-mail: sadybekov@math.kz; abdisalam@mail.ru

    В этой статье рассматривается одна обратная задача для одномерного вырождающегося уравнения дробной теплопроводности с инволюцией и с периодическими граничными условиями относительно пространственной переменной. Эта проблема имитирует процесс распространения тепла в тонкой замкнутой проволоке, обернутой вокруг слабо проницаемой изоляции. Обратная задача состоит в восстановлении (одновременно с решением) уравнения неизвестная правая часть уравнения, зависящая только от пространственная переменная. Условиями переопределения являются начальное и конечное состояния. Результаты существования и единственности для данной задачи получены методом разделения переменных.

Ключевые слова: обратная задача, уравнение теплопроводности, уравнение с инволюцией, субдиффузионный процесс, уравнение с вырождением, периодические граничные условия, метод разделения переменных.

© М.А. Садыбеков,
А.А. Сарсенби, 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.

Для цитирования. Sadybekov M.A., Sarsenbi A.A. On one inverse problem of reconstructing a subdiffusion process with degeneration from nonlocal data // Докл.  Адыгской (Черкесской) международной академии наук. 2019. Т. 19, № 1. C. 31-41.

Читать статью

ДОКЛАДЫ АМАН

©​ | 2022 | Адыгская (Черкесская) Международная академия наук