Russian Mathematics

, Volume 61, Issue 4, pp 35–40 | Cite as

A problem with operators of fractional differentiation in boundary condition for mixed-type equation

  • O. A. Repin
  • S. K. Kumykova


We consider a question on unique solvability of a boundary-value problem with fractional derivatives for a mixed-type equation of second order. We prove first a uniqueness theorem. The existence theorem is proved by means of reduction to Fredholm equation of the second kind, and its unconditional solvability follows from the uniqueness of solution.


operator of fractional differentiation Gauss hypergeometric function Cauchy problem Fredholm integral equation 


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  1. 1.
    Samko, S. G., Kilbas, A. A., Marichev. O. I. Integrals and Derivatives of Fractional Orders with Applications (Nauka i tekhnika, Minsk, 1987) [in Russian].MATHGoogle Scholar
  2. 2.
    Nakhushev, A. I. Fractional Calculus with Applications (Fizmatgiz, Moscow, 2003) [in Russian].MATHGoogle Scholar
  3. 3.
    Nakhushev, A. I. Boundary-Value Problems with Shifts for Equations with Partial Derivatives (Nauka, Moskow, 2006) [in Russian].Google Scholar
  4. 4.
    Repin, O. A., Kumykova, S. K. “On a Boundary Value Problem with Shift for an Equation ofMixed Type in an Unbounded Domain”, Differ. Equ. 48, No. 8, 1127–1136 (2012).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Repin, O. A., Kumykova, S. K. “Nonlocal Problem for an Equation of Mixed type in a Domain Whose Elliptic Part is a Half-Strip”, Differ. Equ. 50, No. 6, 805–814 (2014).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Smirnov, M. M. Equations of Mixed Type (Vysshaya Shkola, Moscow, 1985) [in Russian].Google Scholar
  7. 7.
    Ivashkina, G. A. “On problems of Bitsadze-Samarskij type for the equation U xx + sgny|y|m U yy = 0(0 <m < 1)”, Differ. Uravn. 17, 1078–1089 (1981) [in Russian].MathSciNetMATHGoogle Scholar
  8. 8.
    Bitsadze, A. V. Equations of Mathematical Physics (Nauka, Moscow, 1976) [in Russian].MATHGoogle Scholar

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© Allerton Press, Inc. 2017

Authors and Affiliations

  1. 1.Samara State Economic UniversitySamaraRussia
  2. 2.Kabardino-Balkarian State UniversityNal’chikRussia

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