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Cybernetics and Systems Analysis

, Volume 55, Issue 2, pp 232–239 | Cite as

An Inverse Problem for Anomalous Diffusion Equation with Bi-Ordinal Hilfer’s Derivative

  • V. M. BulavatskyEmail author
Article
  • 5 Downloads

Abstract

The author formulates and solves the inverse problem of finding the field function and the source function dependent on the geometric variable for the anomalous diffusion equation with bi-ordinal Hilfer’s fractional derivative and variable direction of time. The existence and uniqueness of the solution of the problem are established.

Keywords

anomalous diffusion fractional differential diffusion equation bi-ordinal Hilfer’s derivative equations with variable direction of time inverse problem 

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References

  1. 1.
    S. L. Sobolev, “Locally inequilibrium models of transport processes,” Uspekhi Fiz. Nauk, Vol. 167, No. 10, 1095–1106 (1997).Google Scholar
  2. 2.
    M. M. Khasanov and G. T. Bulgakova, Nonlinear and Nonequilibrium Effects in Rheologically Complex Media [in Russian], Inst. Komp’yut. Issled., Moscow–Izhevsk (2003).Google Scholar
  3. 3.
    R. P. Meilanov, V.D. Beibalaev, and M.R.Shibanova, Applied Aspects of Fractional Calculus, Palmarium Acad. Publ., Saarbrucken (2012).Google Scholar
  4. 4.
    S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Sci. Publ., Yverdon, Switzerland (1993).Google Scholar
  5. 5.
    A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006).zbMATHGoogle Scholar
  6. 6.
    I. Podlubny, Fractional Differential Equations, Academic Press, New York (1999).zbMATHGoogle Scholar
  7. 7.
    F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London (2010).CrossRefzbMATHGoogle Scholar
  8. 8.
    A. S. Berdyshev, B. Zh. Kadirkulov, and B.Kh. Turmetov, “Some inverse problems for a heat conduction equation of fractional order,” Vestnik Kazakh. NU, Ser. Matem., Mekhanika, Informatika, No. 2 (65), 36–41 (2010).Google Scholar
  9. 9.
    M. Kirane and S. A. Malik, “Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time,” Applied Mathematics and Computation, Vol. 218, 163–170 (2011).Google Scholar
  10. 10.
    T. S. Aleroev, M. Kirane, and S. A. Malik, “Determination of a source term for a time fractional diffusion equation with an integral type over-determining condition,” Electronic J. of Differential Equations, Vol. 270, 1–16 (2013).Google Scholar
  11. 11.
    F. Al-Musalhi, N. Al-Salti, and S. Kerbal, “Inverse problems of a fractional differential equations with Bessel operator,” arXiv: 1609.04587 v1 (2016).Google Scholar
  12. 12.
    K. M. Furati, O. S. Iyiola, M. Kirane, “An inverse problem for a generalized fractional diffusion,” Applied Mathematics and Computation, Vol. 249, 24–31 (2014).Google Scholar
  13. 13.
    K. M. Furati, O. S. Iyiola, and M. Kassem, “An inverse source problem for a two-parameter anomalous diffusion with local time datum,” arXiv: 1604.06886 v1 (2016).Google Scholar
  14. 14.
    B. Kh. Turmetov and K. M. Shinaliev, “Resolvability of some initial–boundary-value problems for the generalized heat conduction equation,” Vestnik Evraziiskogo Nats. Universiteta im. L. N. Gumileva, Ser. Estestvenno-Tekhnich. Nauki, Issue 6 (85), 8–14 (2011).Google Scholar
  15. 15.
    I. A. Kaliev and M.M. Pervushina, “Inverse problems for the heat conduction equation,” in: Modern Problems in Physics and Mathematics [in Russian], Vol. 1, Gilem, Ufa (2004), pp. 50– 55.Google Scholar
  16. 16.
    I. A. Kaliev and M. M. Sabitova, “Problems of determining temperature and density of heat sources based on initial and finite temperatures,” Sibirskii Zhurnal Industrial’noi Matematiki, Vol. 12, No. 1 (37), 89–97 (2009).Google Scholar
  17. 17.
    I. Orazov and M. A. Sadybekov, “One nonlocal problem of finding temperature and density of heat sources,” Izvestiya VUZ. Matematika, Vol. 2, 70–75 (2012).Google Scholar
  18. 18.
    I. A. Kaliev, M. F. Mugafarov, and O. V. Fattakhova, “Inverse problem for the parabolic equation with variable direction of time with generalized interface conditions,” Ufimskii Matematich. Zhurnal, Vol. 3, No. 2, 34–42 (2011).Google Scholar
  19. 19.
    N. N. Yanenko and V. A. Novikov, “One model of liquid with alternating coefficient of viscosity,” Chislennye Metody Mekhaniki Sploshnoi Sredy, Novosibirsk VTs SO AN SSSR, Vol. 4, No. 2, 142–147 (1973).Google Scholar
  20. 20.
    V. M. Bulavatsky, “Closed form of the solutions of some boundary-value problems for anomalous diffusion equation with Hilfer’s generalized derivative,” Cybern. Syst. Analysis., Vol. 50, No. 4, 570–577 (2014).Google Scholar
  21. 21.
    V. M. Bulavatsky and A. V. Gladky, “Mathematical modeling of the dynamics of a nonequilibrium diffusion process on the basis of integro-differentiation of fractional order,” Cybern. Syst. Analysis, Vol. 51, No. 1, 134–141 (2015).Google Scholar
  22. 22.
    T. Sandev, R. Metzler, and Z. Tomovski, “Fractional diffusion equation with a generalized Riemann–Liouville time fractional derivative,” J. of Physics A, Vol. 44, 5–52 (2011).Google Scholar
  23. 23.
    Z. Tomovski, T. Sandev, R. Metzler, and J. Dubbeldam, “Generalized space-time fractional diffusion equation with composite fractional time derivative,” Physica A, Vol. 391, 2527–2542 (2012).Google Scholar
  24. 24.
    M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York (1965).Google Scholar
  25. 25.
    R. Hilfer, “Fractional time evolution,” in: R. Hilfer (ed.), Applications of Fractional Calculus in Physics, World Scientific, Singapore (2000), pp. 87–130.Google Scholar
  26. 26.
    R. Gorenflo, A. A. Kilbas, F. Mainardi, and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer-Verlag, Berlin (2014).Google Scholar
  27. 27.
    M. S. Tulasynov, “The first boundary-value problem for one parabolic equation with variable direction of time with the complete matrix of conjugation conditions,” Vestnik Novosibirsk. Gos. Univer., Ser. Matem., Mekhanika, Informatika, Vol. 9, Issue 1, 57–68 (2009).Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.V. M. Glushkov Institute of CyberneticsNational Academy of Sciences of UkraineKyivUkraine

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