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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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).
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).
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].
Ptashnik, B. I. Ill-posed Boundary Problems for Differential Equations with Partial Derivatives (Naukova Dumka, Kiev, 1984) [in Russian].
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].
Kiguradze, T. I. “Periodic Boundary Value Problems for Linear Hyperbolic Equations. I”, Differential Equations 29, No. 2, 231–245 (1993).
Zhestkov, S. V. “Doubly Periodic Solutions of Nonlinear Hyperbolic Partial Differential Systems”, Ukrain. Mat. Zh. 39, No. 4, 521–523 (1987) [in Russian].
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].
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).
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].
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).
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].
Trenogin, V. A. Functional Analysis (Nauka, Moscow, 1980) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © N.T. Orumbaeva, 2016, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2016, No. 9, pp. 26–41.
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X16090036