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Статья
2017

A nonlocal problem for degenerate hyperbolic equation


O. A. RepinO. A. Repin, S. K. KumykovaS. K. Kumykova
Русская математика
https://doi.org/10.3103/S1066369X17070064
Abstract / Full Text

We consider a nonlocal problem for a degenerate equation in a domain bounded by characteristics of this equation. The boundary-value conditions of the problem include linear combination of operators of fractional integro-differentiation in the Riemann–Liouville sense. The uniqueness of solution of the problem under consideration is proved by means of the modified Tricomi method, and existence is reduced to solvability of either singular integral equation with the Cauchy kernel or Fredholm integral equation of second kind.

Author information
  • Samara State Economic University, ul. Sovetskoi Armii 141, Samara, 443090, RussiaO. A. Repin
  • Kabardino-Balkarian State University, ul. Chernyshevskogo 173, Nalchik, 360004, RussiaS. K. Kumykova
References
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  2. Nakhushev, A. M. Problems with Shifts for Partial Differential Equations (Nauka, Moscow, 2006) [in Russian].
  3. Repin, O. A., Kumykova, S. K. “Nonlocal Problem for an Equation of a Mixed Type in a Domain Whose Elliptic Part is a Half-Strip”, Differ. Equ. 50, No. 6, 805–814 (2014).
  4. Repin, O. A., Kumykova, S. K. “A ProblemWith Generalized Fractional Integro-Differentiation Operators of Arbitrary Order”, RussianMathematics 56, No. 12, 50–60 (2012).
  5. Repin, O. A., Kumykova, S. K. “A Nonlocal Problem for a Mixed-Type Equation Whose Order Degenerates Along the Line of Change of Type”, RussianMathematics 57, No. 8, 49–56 (2013).
  6. Smirnov, M. M. Degenerate Hyperbolic Equations (Vyseisaja Skola, Minsk, 1977) [in Russian].
  7. Kumykova, S. K. “Ein Randwertproblem mit Verschiebung fár eine innerhalb des Gebietes entartete hyperbolische Gleichung”, Differ.Uravn. 16, 93–104 (1980) [in Russian].