The European Physical Journal Special Topics

, Volume 222, Issue 8, pp 1767–1777 | Cite as

Axisymmetric solutions to fractional diffusion-wave equation in a cylinder under Robin boundary condition

  • Y. PovstenkoEmail author
Regular Article


The axisymmetric time-fractional diffusion-wave equation with the Caputo derivative of the order 0 < α ≤ 2 is considered in a cylinder under the prescribed linear combination of the values of the sought function and the values of its normal derivative at the boundary. The fundamental solutions to the Cauchy, source, and boundary problems are investigated. The Laplace transform with respect to time and finite Hankel transform with respect to the radial coordinate are used. The solutions are obtained in terms of Mittag-Leffler functions. The numerical results are illustrated graphically.


Cauchy Problem Fundamental Solution European Physical Journal Special Topic Fractional Derivative Fractional Calculus 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    F. Mainardi, in Fractals and Fractional Calculus in Continuum Mechanics, edited by A. Carpinteri, F. Mainardi (Springer Verlag, Wien, 1997), p. 291Google Scholar
  2. 2.
    Yu.A. Rossikhin, M.V. Shitikova, Appl. Mech. Rev. 50, 15 (1997)ADSCrossRefGoogle Scholar
  3. 3.
    R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)MathSciNetADSCrossRefzbMATHGoogle Scholar
  4. 4.
    R. Metzler, J. Klafter, J. Phys. A: Math. Gen. 37, R161 (2004)MathSciNetADSCrossRefzbMATHGoogle Scholar
  5. 5.
    G.M. Zaslavsky, Phys. Rep. 371, 461 (2002)MathSciNetADSCrossRefzbMATHGoogle Scholar
  6. 6.
    G.M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics (Oxford University Press, New York, 2005)Google Scholar
  7. 7.
    Y.Z. Povstenko, J. Thermal Stresses 28, 83 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    R.L. Magin, Fractional Calculus in Bioengineering (Begel House Publishers, Inc., Connecticut, 2006)Google Scholar
  9. 9.
    V.V. Uchaikin, Method of Fractional Derivatives (Arteshock, Ulyanovsk, 2008) (in Russian)Google Scholar
  10. 10.
    F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity (Imperial College Press, London, 2010)Google Scholar
  11. 11.
    M. Edelman, Commun. Nonlinear Sci. Numer. Simulat. 16, 4573 (2011)MathSciNetADSCrossRefzbMATHGoogle Scholar
  12. 12.
    W. Wyss, J. Math. Phys. 27, 2782 (1986)MathSciNetADSCrossRefzbMATHGoogle Scholar
  13. 13.
    W.R. Schneider, W. Wyss, J. Math. Phys. 30, 134 (1989)MathSciNetADSCrossRefzbMATHGoogle Scholar
  14. 14.
    Y. Fujita, Osaka J. Math. 27, 309 (1990)MathSciNetzbMATHGoogle Scholar
  15. 15.
    F. Mainardi, Appl. Math. Lett. 9, 23 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    B.N. Narahari Achar, J.W. Hanneken, J. Mol. Liq. 114, 147 (2004)CrossRefGoogle Scholar
  17. 17.
    Y.Z. Povstenko, J. Mol. Liq. 137, 46 (2008)CrossRefGoogle Scholar
  18. 18.
    Y.Z. Povstenko, Fract. Calc. Appl. Anal. 14, 418 (2011)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Y. Povstenko, Arch. Appl. Mech. 82, 345 (2012a)ADSCrossRefGoogle Scholar
  20. 20.
    N. Özdemir, D. Karadeniz, Phys. Lett. A 372, 5968 (2008)MathSciNetADSCrossRefzbMATHGoogle Scholar
  21. 21.
    N. Özdemir, D. Karadeniz, B.B. Iskender, Phys. Lett. A 373, 221 (2009)MathSciNetADSCrossRefzbMATHGoogle Scholar
  22. 22.
    N. Özdemir, O.P. Agrawal, D. Karadeniz, B.B. Iskender, Phys. Scr. T 136, 014024 (2009)ADSCrossRefGoogle Scholar
  23. 23.
    H. Qi, J. Liu, Meccanica 45, 577 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    A.K. Bazzaev, M.Kh. Shkhanukov-Lafishev, Comp. Math. Math. Phys. 50, 1141 (2010)MathSciNetCrossRefGoogle Scholar
  25. 25.
    J. Kemppainen, Abstr. Appl. Anal. 2011, 321903 (2011)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Y.Z. Povstenko, Int. J. Diff. Equat. 2012, 154085 (2012)MathSciNetGoogle Scholar
  27. 27.
    Y.Z. Povstenko, in Proceedings of the 13th International Carpathian Control Conference, edited by I. Petráš, I. Podlubny, K. Kostúr, A. Mojžišová, J. Kačur (High Tatras, Slovak Republic, 2012), p. 588Google Scholar
  28. 28.
    R.K. Saxena, A.M. Mathai, H.J. Haubold, Astrophys. Space Sci. 305, 289 (2006)ADSCrossRefzbMATHGoogle Scholar
  29. 29.
    V. Gafiychuk, B. Datsko, V. Meleshko, J. Comput. Appl. Math., 220, 215 (2008)Google Scholar
  30. 30.
    V. Gafiychuk, B. Datsko, V. Meleshko, D. Blackmore, Chaos, Solitons Fractals 41, 1095 (2009)MathSciNetADSCrossRefzbMATHGoogle Scholar
  31. 31.
    A.M. Mathai, R.K. Saxena, H.J. Haubold, The H-Function, Theory and Applications (Springer, New York, 2010)Google Scholar
  32. 32.
    V. Méndez, S. Fedotov, W. Horsthemke, Reaction-Transport Systems: Mesoscopic Foundations, Fronts, and Spatial Instabilities (Springer, Berlin, 2010)Google Scholar
  33. 33.
    R. Gorenflo, F. Mainardi, in Fractals and Fractional Calculus in Continuum Mechanics, edited by A. Carpinteri, F. Mainardi (Springer Verlag, Wien, 1997), p. 223Google Scholar
  34. 34.
    A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations (Amsterdam, Elsevier, 2006)Google Scholar
  35. 35.
    I. Podlubny, Fractional Differential Equations (San Diego, Academic Press, 1999)Google Scholar
  36. 36.
    A.S. Galitsyn, A.N. Zhukovsky, Integral Transforms and Special Functions in Heat Conduction Problems (Naukova Dumka, Kiev, 1976) (in Russian)Google Scholar

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© EDP Sciences and Springer 2013

Authors and Affiliations

  1. 1.Institute of Mathematics and Computer ScienceJan Długosz University in CzȩstochowaCzȩstochowaPoland

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