The European Physical Journal Special Topics

, Volume 222, Issue 8, pp 1939–1950 | Cite as

Multi-time fractional diffusion equation

  • A. V. PskhuEmail author
Regular Article


We construct a fundamental solution of a multi-time diffusion equation with the Dzhrbashyan-Nersesyan fractional differentiation operator with respect to the time variables. We give a representation for a solution of the Cauchy problem and prove the uniqueness theorem in the class of functions of fast growth. The corresponding results for equations with Riemann-Liouville and Caputo derivatives are obtained as particular cases of the proved assertions.


Soliton Cauchy Problem Fundamental Solution Fractional Order European Physical Journal Special Topic 
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  1. 1.
    M.M. Dzhrbashyan, A.B. Nersesyan, Izv. Akad. Nauk Armenian SSR Matem. 3, 3 (1968) (Russian)zbMATHGoogle Scholar
  2. 2.
    M. Caputo, Elasticita e Dissipazione (Zanichelli, Bologna, 1969) (Italian)Google Scholar
  3. 3.
    R. Hilfer (ed.), Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000), p. 463Google Scholar
  4. 4.
    J. Sabatier, O.P. Agrawal, J.A. Tenreiro Machado (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering (Springer, Dordrecht, 2007), p. 552Google Scholar
  5. 5.
    V.V. Uchaikin, Method of Fractional Derivatives (Artishok, Ulyanovsk, 2008), p. 512 (Russian)Google Scholar
  6. 6.
    V.E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media (Higher Education Press, Beijing and Springer-Verlag, Berlin Heidelberg, 2010), p. 505Google Scholar
  7. 7.
    F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models (World Scientific, Singapore, 2010), p. 368Google Scholar
  8. 8.
    K. Oldham, J. Spanier, The Fractional Calculus (Academic Press, New York, 1974), p. 235Google Scholar
  9. 9.
    K. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equation (John Wiley, New York, 1993), p. 367Google Scholar
  10. 10.
    S. Samko, A. Kilbas, O. Marichev, Fractional Integral and Derivative. Theory and Applications (Gordon and Breach, Switzerland, 1993), p. 977Google Scholar
  11. 11.
    I. Podlubny, Fractional Differential Equations (Academic Press, San Diego New York London, 1999), p. 340Google Scholar
  12. 12.
    A.M. Nakhushev, Fractional calculus and its applications (Fizmatlit, Moscow, 2003), p. 271 (Russian)Google Scholar
  13. 13.
    A.V. Pskhu, Partial differential equations of fractional order (Nauka, Moscow, 2005), p. 199 (Russian)Google Scholar
  14. 14.
    A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations (North-Holland Math. Stud., vol. 204, Elsevier, Amsterdam, 2006), p. 523Google Scholar
  15. 15.
    W. Wyss, J. Math. Phys. 27, 2782 (1986)MathSciNetADSCrossRefzbMATHGoogle Scholar
  16. 16.
    W.R. Schneider, W. Wyss, J. Math. Phys. 30, 134 (1989)MathSciNetADSCrossRefzbMATHGoogle Scholar
  17. 17.
    A.N. Kochubei, Differential Equations 26, 485 (1990)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Y. Fujita, Osaka J. Math. 27, 309, 77 (1990)Google Scholar
  19. 19.
    F. Mainardi, Appl. Math. Lett. 9, 23 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    F. Mainardi, Chaos, Solitons Fractals 7, 1461 (1996)MathSciNetADSCrossRefzbMATHGoogle Scholar
  21. 21.
    H. Engler, Differential Integral Equations 10, 815 (1997)MathSciNetzbMATHGoogle Scholar
  22. 22.
    E. Buckwar, Yu. Luchko, J. Math. Anal. Appl. 227, 81 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Yu. Luchko, R. Gorenflo, Fract. Calc. Appl. Anal. 1, 63 (1998)MathSciNetzbMATHGoogle Scholar
  24. 24.
    F. Mainardi, R. Gorenflo, J. Comput. Appl. Math. 118, 283 (2000)MathSciNetADSCrossRefzbMATHGoogle Scholar
  25. 25.
    A.V. Pskhu, Differ. Equ. 39, 1359 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    A.V. Pskhu, Differ. Equ. 39, 1509 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    S.D. Eidelman, A.N. Kochubei, J. Differential Eq. 199, 211 (2004)MathSciNetCrossRefzbMATHADSGoogle Scholar
  28. 28.
    Yu. Luchko, J. Math. Anal. Appl. 351, 218 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    A.V. Pskhu, Izvestiya: Math. 73, 351 (2009)MathSciNetADSCrossRefzbMATHGoogle Scholar
  30. 30.
    E.M. Wright, J. London Math. Soc. 8, 71 (1933)CrossRefGoogle Scholar
  31. 31.
    E.M. Wright, Quart. J. Math., Oxford Ser. 11, 36 (1940)MathSciNetADSCrossRefGoogle Scholar

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© EDP Sciences and Springer 2013

Authors and Affiliations

  1. 1.Scientific Research Institute of Applied Mathematics and AutomationKabardino-Balkar Scientific Centre of the Russian Academy of SciencesNalchikRussian Federation

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