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On solvability of non-linear semi-periodic boundary-value problem for system of hyperbolic equations

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Abstract

We study a non-linear semi-periodic boundary-value problem for a system of hyperbolic equations with mixed derivative. At that, the semi-periodic boundary-value problem for a system of hyperbolic equations is reduced to an equivalent problem, consisting of a family of periodic boundary-value problems for ordinary differential equations and functional relation. When solving a family of periodic boundary-value problems of ordinary differential equations we use the method of parameterization. This approach allowed to establish sufficient conditions for the existence of an isolated solution of non-linear semi-periodic boundary-value problem for a system of hyperbolic equations.

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References

  1. Cesari, L. “Periodic Solutions of Hyperbolic Partial Differential Equations” in Proc. Internat. Sympos. Nonlinear Vibrations, Kiev, 1961, 2, 440–457 (Izd. Akad. Nauk Ukrain. SSR, Kiev, 1963).

    MathSciNet  Google Scholar 

  2. Vejvoda, O. “Periodic Solutions of a Linear and Weakly Nonlinear Wave Equation in One Dimension. I”, Czech. Math. J. 14, No. 3, 341–382 (1964).

    MathSciNet  MATH  Google Scholar 

  3. Samoilenko, A. M. and Tkach, B. P. Numerical-Analytical Methods in Theory of Periodic Solutions to Equations with Partial Derivatives (Naukova Dumka, Kiev, 1992) [in Russian].

    Google Scholar 

  4. Ptashnik, B. I. Ill-posed Boundary Problems for Differential Equations with Partial Derivatives (Naukova Dumka, Kiev, 1984) [in Russian].

    Google Scholar 

  5. Kolesov, A. Yu., Mishchenko, E. F., and Rozov, N. Kh. Asymptotic Methods of Investigation of Periodic Solutions to Nonlinear Hyperbolic Equations (Nauka, Moscow, 1998) [in Russian].

    MATH  Google Scholar 

  6. Kiguradze, T. I. “Periodic Boundary Value Problems for Linear Hyperbolic Equations. I”, Differential Equations 29, No. 2, 231–245 (1993).

    MathSciNet  MATH  Google Scholar 

  7. Zhestkov, S. V. “Doubly Periodic Solutions of Nonlinear Hyperbolic Partial Differential Systems”, Ukrain. Mat. Zh. 39, No. 4, 521–523 (1987) [in Russian].

    MathSciNet  MATH  Google Scholar 

  8. Mitropol’skii, Yu. A., Khoma, G. P., and Gromyak, M. I. Asymptotic Methods of Investigation of Quasi-Wave Hyperbolic Type Equations (Naukova Dumka, Kiev, 1991) [in Russian].

    Google Scholar 

  9. Asanova, A. T. and Dzhumabaev, D. S. “Correct Solvability of Nonlocal Boundary Value Problems for Systems of Hyperbolic Equations”, Differ. Equ. 41, No. 3, 352–363 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  10. Orumbaeva, N. T. “On an Algorithm for Finding the Solution of a Periodic Boundary-Value Problem for a System of Quasilinear Hyperbolic Equations”, (Russian) Sib. Elektron. Mat. Izv. 10, 464–474 (2013) [in Russian].

    MathSciNet  MATH  Google Scholar 

  11. Dzhumabaev, D. S. “Conditions for the Unique Solvability of a Linear Boundary Value Problem for an Ordinary Differential Equation”, U.S.S.R. Comput. Math. and Math. Phys. 29, No. 1, 34–46 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  12. Orumbayeva, N. T. and Sabitbekova, G. “About the Solvability of a Family of Periodic Boundary Value Problems for Ordinary Differential Equations”, Vestnik Karagandinskogo Gos. Univ., Ser. Matem., No. 4(72), 89–96 (2013) [in Russian].

    Google Scholar 

  13. Trenogin, V. A. Functional Analysis (Nauka, Moscow, 1980) [in Russian].

    MATH  Google Scholar 

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Authors and Affiliations

  1. Buketov Karaganda State University, ul. Universitetskaya 28, Karaganda, 100028, Republic of Kazakhstan

    N. T. Orumbaeva

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  1. N. T. Orumbaeva
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Correspondence to N. T. Orumbaeva.

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Original Russian Text © N.T. Orumbaeva, 2016, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2016, No. 9, pp. 26–41.

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Orumbaeva, N.T. On solvability of non-linear semi-periodic boundary-value problem for system of hyperbolic equations. Russ Math. 60, 23–37 (2016). https://doi.org/10.3103/S1066369X16090036

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