On conditions of the solvability of nonlocal multi-point boundary value problems for quasi-linear systems of hyperbolic equations

A. T. Assanova ; A. E. Imanchiev

Eurasian mathematical journal, Tome 6 (2015) no. 4, pp. 19-28.

Résumé

A nonlocal multi-point boundary value problem for a system of quasi-linear hyperbolic equations is investigated. Based on the results for linear problems coefficient conditions are established ensuring the existence of classical solutions to nonlocal multi-point boundary value problem for a system of quasi-linear hyperbolic equations, and algorithms of finding these solutions are suggested.

Bibliographie

[1] R. P. Agarwal, I. Kiguradze, “On multi-point boundary value problems for linear ordinary differential equations with singularities”, Journal of Mathematical Analysis and Applications, 297:2 (2004), 131–151 | DOI | MR | Zbl

[2] A. T. Asanova, D. S. Dzhumabaev, “Unique Solvability of the boundary value problem for systems of hyperbolic equations with data on the characteristics”, Computational Mathematics and Mathematical Physics, 42:11 (2002), 1609–1621 | MR | Zbl

[3] A. T. Asanova, D. S. Dzhumabaev, “Unique solvability of nonlocal boundary value problems for systems of hyperbolic equations”, Differential Equations, 39:10 (2003), 1414–1427 | DOI | MR | Zbl

[4] A. T. Asanova, D. S. Dzhumabaev, “Criteria of well-posed solvability of boundary value problem for system of hyperbolic equations”, Izvestia NAN Respubl. Kazakhstan. Ser. phyz.-mathem., 2002, no. 3, 20–26 (in Russian) | MR

[5] A. T. Asanova, D. S. Dzhumabaev, “Correct solvability of a nonlocal boundary value problem for systems of hyperbolic equations”, Doklady Mathematics, 68:1 (2003), 46–49 | MR

[6] A. T. Asanova, D. S. Dzhumabaev, “Well-posed solvability of nonlocal boundary value problems for systems of hyperbolic equations”, Differential Equations, 41:3 (2005), 352–363 | DOI | MR | Zbl

[7] A. T. Asanova, “A nonlocal boundary value problem for systems of quasilinear hyperbolic equations”, Doklady Mathematics, 74:3 (2006), 787–791 | DOI

[8] A. T. Asanova, “On the unique solvability of a family of two-point boundary-value problems for systems of ordinary differential equations”, Journal of Mathematical Sciences, 150:5 (2008), 2302–2316 | DOI | MR | Zbl

[9] A. T. Asanova, “On the solvability of a family of multi-point boundary value problems for system of differential equations and their applications to nonlocal boundary value problems”, Mathematical journal, 13:3 (2013), 38–42 (in Russian) | MR

[10] A. T. Asanova, “On multipoint problem for system of hyperbolic equations with mixed derivative”, Nonlinear Oscillations (Neliniini Kolyvannya), 17:3 (2014), 295–313 (in Russian) | MR

[11] L. Byszewski, “Existence and uniqueness of solutions of nonlocal problems for hyperbolic equation $u_{xt}=F(x,t,u,u_x)$”, Journal of Applied and Stochastic Analysis, 3:3 (1990), 163–168 | DOI | MR | Zbl

[12] L. Cesari, “Periodic solutions of hyperbolic partial differential equations”, Proc. Internat. Sympos. Non-linear Vibrations (Kiev, 1961), v. 2, Izd. Akad. Nauk Ukrain. SSR, Kiev, 1963, 440–457 | MR

[13] D. S. Dzhumabaev, A. T. Asanova, “Well-posed solvability of a linear nonlocal boundary value problem for systems of hyperbolic equations”, Dopovidi NAN Ukraine, 2010, no. 4, 7–11 (in Russian) | MR | Zbl

[14] D. S. Dzhumabaev, “Conditions the unique solvability of a linear boundary value problem for ordinary differential equations”, USSR Computational Mathematics and Mathematical Physics, 29:1 (1989), 34–46 | DOI | MR | Zbl

[15] D. S. Dzhumabaev, A. E. Imanchiev, “Well-posed solvability of a linear multi-point boundary value problem”, Mathematical journal, 5:1(15) (2005), 30–38 (in Russian) | MR | Zbl

[16] I. T. Kiguradze, “Boundary-value problems for system of ordinary differential equations”, Journal of Soviet Mathematics, 43:2 (1988), 2259–2339 | DOI | MR

[17] T. Kiguradze, “Some boundary value problems for systems of linear partial differential equations of hyperbolic type”, Mem. Differential Equations Math. Phys., 1 (1994), 1–144 | MR | Zbl

[18] Yu. A. Mitropol'skii, L. B. Urmancheva, “About two-point problem for systems of hyperbolic equations”, Ukranian mathematical Journal, 42:12 (1990), 1657–1663 (in Russian) | MR

[19] A. M. Nakhushev, Problems with replased for partial differential equations, Nauka, M., 2006 (in Russian)

[20] B. I. Ptashnyck, Ill-posed boundary value problems for partial differential equations, Naukova Dumka, Kiev, Ukraine, 1984 (in Russian) | MR

[21] A. M. Samoilenko, V. N. Laptinskii, K. K. Kenzhebayev, “Constructive methods of investigation of periodic and multi-point boundary value problems”, Proceedings of Institute of Mathematics of NAS of Ukraine, 29, Institute of Mathematics NAS Ukraine, Kiev, 1999, 1–220 (in Russian)

[22] A. M. Samoilenko, B. P. Tkach, Numerical-analytical methods in the theory periodical solutions of equations with partial derivatives, Naukova Dumka, Kiev, Ukraine, 1992 (in Russian)

[23] A. M. Samoilenko, N. I. Ronto, Numerical-analytical methods for investigation of a solutions of boundary value problems, Naukova Dumka, Kiev, Ukraine, 1985 (in Russian) | MR

[24] L. B. Urmancheva, Two-point and multi-point problems for systems of partial differential equations hyperbolic type, Preprint 93.2, Institute of Mathematics NAS Ukraine, Kiev, 1993, 40 pp. (in Russian)