The problem on conjugation on solutions of a hyperbolic system along unknown contact boundary in sector (in Ukrainian)

Author R. V. Andrusyak, N. O. Burdeyna, V. M. Kyrylych
Ivan Franko Natonal University of Lviv

Abstract The conditions of solvability in the generalized sense of a problem with unknown breaking line of initial data for a hyperbolic quasilinear system with two independent variables in the sector were established. Applying the method of characteristics a solution of the problem is reduced to finding the fixed point of the operator whose existence and uniqueness are proved using the Banach theorem.
Keywords hyperbolic problem; quasi-linear equations; free discontinuity line; method of characteristic
Reference 1. Rozhdestvenskyy B.L., Yanenko N.N. The systems of the guasi-linear equations and their application to gas dynamics. – M.: Nauka, 1978. – 592 p. (in Russian)

2. Li Ta-tsien. Global classical solutions for quasilinear hyperbolic systems. – New York: Masson, 1994. – 315 p.

3. Kulikovskiy A.G., Pogorelov N.V., Semenov A.Yu. Mathematical questions of the calculation solution of the hyperbolic systems. – M.: Fizmatlit, 2001. – 608 p. (in Russian)

4. Arguchintsev A.V. Optimization of the hyperbolic systems. – M.:FIZMATLIT, 2007. – 168 p. (in Russian)

5. Lax P.D. Hyperbolic differentiation partial equations. – M.-Izhevsk: NIZ “Regular and chaotic dynamics”, 2010. – 296 p. (in Russian)

6. Kuznetsov N.N. About hyperbolic systems of linear equations with disconected coefficients// ZHVM & MF. – 1963. – V.3, Ή2. – P. 299–313. (in Russian)

7. Mel’nyk Z.O., Myshkis A.D. The mixed problem for a two-dimensional first-order hyperbolic system with disconected coefficients// Mat. Sborn. – 1965. – V.68, Ή4. – P. 632–638. (in Russian)

8. Mel’nyk Z.O. An example of the nonclassical boundary problem for the fluctuation string equation// Ukr. Mat. Zhurn. – 1980. – V.32, Ή5. – P. 671–674. (in Russian)

9. Mel’nyk Z.O. Problems with integral limitations for general two-dimensional hyperbolic systems and equations// Diff. uravn. – 1985. – V.21, Ή2. – P. 246–253. (in Russian)

10. Sydorenko A.D. The problem with contact discontinuity for a system of three quasi-linear equations// Diff. uravn. – 1978. – V.14, Ή4. – P. 774–777. (in Russian)

11. Andrusyak R.V., Kyrylych V.M. The problem for a quasi-linear system of the hyperbolic type in the curvilinear sector with free boundaries// Nauk. Visn. Cherniv. Univ. Mathematics. – 2008. – V.421. – P. 5–12. (in Ukrainian)

12. Mel’nyk T.E. The conjugation of solutions of the second-order hyperbolic equation along unknown boundary// Doklady AN USSR. – 1980. – Ser. A, Ή12. – P. 10–12. (in Russian)

13. Kazakov K.Yu., Morozov S.F. About the detection of the unknown line of the solution discontinuity of mixed problem for a quasi-linear hyperbolic system// Ukr. Mat. Zhurn. – 1985. – V.37, Ή4. – P. 443–450. (in Russian)

14. Bassanini P., Turo J. Generalized solutions to free boundary problems for hyperbolic systems of functional partial differential equations// Ann. Math. Pura Appl. – 1990. – V.156, Ή4. – P. 211–230.

15. Kyrylych V.M. Some nonlinear problems with free boundaries for hyperbolic systems of quasi-linear equations// Visn. Lviv. Univ. Ser. Mech.-Mat. – 2009. – V.71. – P. 125–134. (in Ukrainian)

16. Andrusyak R.V., Burdeyna N.O., Kyrylych V.M. Classical solvability of the moving boundary problem for hyperbolic systems of quasi-linear equations// Ukr. Mat. Zhurn. – 2009. – V.61, Ή7. – P. 867–891. (in Ukrainian)

17. Andrusyak R.V., Burdeyna N.O., Kyrylych V.M. Quasi-linear hyperbolic Stefan problem with nonlocal boundary conditions// Ukr. Mat. Zhurn. – 2010. – V.62, Ή9. – P. 1173–1199. (in Ukrainian)

Pages 74-83
Volume 39
Issue 1
Year 2013
Journal Matematychni Studii
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