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Cybernetics and Systems Analysis

, Volume 55, Issue 5, pp 796–804 | Cite as

Some Nonlocal Boundary-Value Problems for the Biparabolic Evolution Equation and Its Fractional-Differential Analog

  • V. M. BulavatskyEmail author
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Abstract

For the biparabolic partial differential evolution equation and its fractional differential generalization, statements are made and closed-form solutions of some boundary-value problems with nonlocal boundary conditions are obtained. Variants of direct and inverse problem statements are considered. The mathematical formulation of the inverse problem involves the search, together with the solution of the original integro-differential equation of fractional order, of its unknown right-hand side as well, which functionally depends only on the geometric variable.

Keywords

biparabolic evolution equation fractional-differential analog of biparabolic equation nonlocal boundary-value problem inverse problem biorthogonal systems of functions 

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References

  1. 1.
    G. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Oxford Science Publications), Oxford University Press (1986).zbMATHGoogle Scholar
  2. 2.
    E. I. Kartashov, Analytical Methods in the Theory of Heat Conduction in Solids [in Russian], Vysshaya Shkola, Moscow (1979).Google Scholar
  3. 3.
    A. V. Lykov, Heat and Mass Exchange [in Russian], Energiya, Moscow (1978).Google Scholar
  4. 4.
    C. Cattaneo, “Sur une forme de l’équation de la chaleur éliminant le paradoxe d’une propagation instantanée,” Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, Vol. 247, No. 4, 431–433 (1958).MathSciNetzbMATHGoogle Scholar
  5. 5.
    V. I. Fushchich, A. S. Galitsyn, and A. S. Polubinskii, “A new mathematical model of heat conduction processes,” Ukr. Math. J., Vol. 42, No. 2, 210–216 (1990).MathSciNetCrossRefGoogle Scholar
  6. 6.
    V. I. Fushchich, “Symmetry and partial solutions to some multidimensional equations of mathematical physics,” Theoretical-Algebraic Methods in Problems of Mathematical Physics [in Russian], Inst. Math. AS UkrSSR (1983), pp. 4–22.Google Scholar
  7. 7.
    V. M. Bulavatsky, “A biparabolic mathematical model of the filtration consolidation process,” Dopov. Nac. Akad. Nauk. Ukr., No. 8, 13–17 (1997).MathSciNetGoogle Scholar
  8. 8.
    V. M. Bulavatsky, “Mathematical modeling of filtrational consolidation of soil under motion of saline solutions on the basis of biparabolic model,” J. Autom. Inform. Sci., Vol. 35, No. 8, 13–22 (2003).Google Scholar
  9. 9.
    V. M. Bulavatsky and V. V. Skopetsky, “Generalized mathematical model of the dynamics of consolidation processes with relaxation,” Cybern. Syst. Analysis, Vol. 44, No. 5, 646–654 (2008).MathSciNetCrossRefGoogle Scholar
  10. 10.
    V. V. Uchaikin, The Method of Fractional Derivatives [in Russian], Artishok, Ulyanovsk (2008).Google Scholar
  11. 11.
    M. M. Djrbashian, Harmonic Analysis and Boundary-Value Problems in the Complex Domain, Springer Basel AG, Basel (1993).CrossRefGoogle Scholar
  12. 12.
    S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Sci. Publ., Philadelphia (1993).zbMATHGoogle Scholar
  13. 13.
    A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam (2006).zbMATHGoogle Scholar
  14. 14.
    I. Podlubny, Fractional Differential Equations, Academic Press, New York (1999).zbMATHGoogle Scholar
  15. 15.
    F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London (2010).CrossRefGoogle Scholar
  16. 16.
    M. Caputo, “Models of flux in porous media with memory,” Water Resources Research, Vol. 36, 693–705 (2000).CrossRefGoogle Scholar
  17. 17.
    A. M. Nakhushev, Fractional Calculus and its Application [in Russian], Fizmatlit, Moscow (2003).Google Scholar
  18. 18.
    R. P. Meilanov, V. D. Beibalaev, and M. R. Shibanova, Applied Aspects of Fractional Calculus, Palmarium Acad. Publ., Saarbrucken (2012).Google Scholar
  19. 19.
    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).MathSciNetCrossRefGoogle Scholar
  20. 20.
    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).MathSciNetCrossRefGoogle Scholar
  21. 21.
    K. M. Furati, O. S. Iyiola, and M. Kirane, “An inverse problem for a generalized fractional diffusion,” Applied Mathematics and Computation, Vol. 249, 24–31 (2014).MathSciNetCrossRefGoogle Scholar
  22. 22.
    V. M. Bulavatsky and V. A. Bogaenko, “Mathematical modelling of the fractional differential dynamics of the relaxation process of convective diffusion under conditions of planed filtration,” Cybern. Syst. Analysis, Vol. 51, No. 6, 886–895 (2015).CrossRefGoogle Scholar
  23. 23.
    V. M. Bulavatsky, “Fractional differential analog of biparabolic evolution equation and some its applications,” Cybern. Syst. Analysis, Vol. 52, No. 5, 737–747 (2016).CrossRefGoogle Scholar
  24. 24.
    N. I. Ionkin, “Solution of one boundary-value problem of the theory of thermal conductivity with the nonclassical boundary condition,” Diff. Uravneniya, Vol. 13, No. 2, 294–304 (1977).MathSciNetGoogle Scholar
  25. 25.
    E. I. Moiseyev, “Solving one nonlocal boundary-value problem by the spectral method,” Diff. Uravneniya, Vol. 35, No. 8, 1094–1100 (1999).Google Scholar
  26. 26.
    V. M. Bulavatsky, Iu. G. Kryvonos, and V. V. Skopetsky, Nonclassical Mathematical Models of Heat and Mass Transfer Processes [in Ukrainian], Naukova Dumka, Kyiv (2005).Google Scholar
  27. 27.
    I. A. Kaliev and M. M. Sabitova, “Problems of determining the temperature and density of heat sources from the initial and terminal temperatures,” Sibirskii Zhurnal Industr. Matem., Vol. 12, No. 1 (37), 89–97 (2009).Google Scholar
  28. 28.
    R. Gorenflo, A. A. Kilbas, F. Mainardi, and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer-Verlag, Berlin (2014).zbMATHGoogle Scholar
  29. 29.
    A. A. Kilbas, M. Saigo, and R. K. Saxena, “Generalized Mittag-Leffler function and generalized fractional calculus operators,” Integral Transforms and Special Functions, Vol. 15, No. 1, 31–49 (2004).MathSciNetCrossRefGoogle 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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