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Статья
2019

Application of Riemann Method to One System in Three-Dimensional Space


L. B. MironovaL. B. Mironova
Русская математика
https://doi.org/10.3103/S1066369X19060057
Abstract / Full Text

For a system of three equations of the second order we prove existence and uniqueness of solutions to the Cauchy problem and to problem with conditions on characteristics and a free surface. We construct solutions to these problems in terms of the Riemann matrix.

Author information
  • Elabuga Institute of Kazan Federal University, 89 Kazanskaya str., Elabuga, 423600, RussiaL. B. Mironova
References
  1. Bitsadze, A.V. “Structural Properties of Solutions of Hyperbolic Systems of First-order Partial Differential Equations”, Mat. Model. 6 (6), 22–31 (1994).
  2. Chekmarev, T.V. “Formulas for Solution of the Goursat Problem for a Linear System of Partial Differential Equations”, Differentsial’nye Uravneniya 18 (9), 1614–1622 (1982).
  3. Pleshchinskaya, I.E. “The Equivalence of Some Classes of First-order Elliptic and Hyperbolic Systems and Second-order Partial Differential Equations”, Differentsial’nye Uravneniya 23 (9), 1634–1637 (1987).
  4. Mironova, L.B. “On the Riemann Method in R n” for a System with Multiple Characteristics”, Russian Math. (Iz. VUZ) 50 (1), 32–37 (2006).
  5. Zhegalov, V.I., Mironova, L.B. “On a System of Equations with Double Higher Partial Derivatives”, Russian Math. (Iz. VUZ) 51 (3), 9–18 (2007).
  6. Zhegalov, V.I. “A Problem with Normal Derivatives in the Boundary Conditions for a System of Differential Equations”, Russian Math. (Iz. VUZ) 52 (8), 58–60 (2008).
  7. Voronova Yu. G. “About Problem of Cauchy for Linear Hyperbolic System of the Equations with Zero Generalized Laplace Invariants”, Ufimsk. Mat. Zh. 2 (2), 20–26 (2010).
  8. Zhiber, A.V., Kostrigina, O.S. “Goursat Problem for Nonlinear Hyperbolic Systems with Integrals of the First and Second Order”, Ufimsk. Mat. Zh. 3 (3), 67–79 (2011).
  9. Sozontova, E.A. “Characteristic Problems with Normal Derivatives for Hyperbolic Systems”, Russian Math. (Iz. VUZ) 57 (10), 37–47 (2013).
  10. Andreev, A.A., Yakovleva, Yu.O. “The Cauchy Problem for a System of Differential Equations of Hyperbolic Type of Order n with Non-multiple Characteristics”, Vestnik Sam. Gos. Tekhn. Univ., Ser. Fiz.-matem. Nauki 21 (4), 752–759 (2017).
  11. Zhegalov, V.I. “A Three-dimensional Analogue of the Goursat Problem” (in: Nonclassical Equations and Equations of Mixed Type, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 94–98 (1990)).
  12. Zhegalov, V.I., Sevast’yanov, V.A. “The Goursat Problem in Four-dimensional Space”, Differential Equations 32 (10), 1427–1428 (1996).
  13. Zhegalov, V.I. “On the Three-dimensional Riemann Function”, Siberian Math. J. 38 (5), 929–934 (1997).
  14. Mironov, A.N. “On the Construction of the Riemann Function for a Fourth-order Equation”, Differ. Equ. 37 (12), 1787–1791 (2001).
  15. Utkina, E.A. “On the General Case of the Goursat Problem”, Russian Math. (Iz. VUZ) 49 (8), 53–58 (2005).
  16. Mironov, A.N. “The Riemann Method for Equations with the Highest Partial Derivative in R n”, Siberian Math. J. 47 (3), 481–490 (2006).
  17. Romanovskiĭ, R.K. “Riemann Matrices of the First and Second Kinds”, Mat. Sb. (N.S.) 127 (169) (4), 494–501 (1985).
  18. Romanovskiĭ, R.K. “Exponentially Splittable Hyperbolic Systems with Two Independent Variables”, Math. USSR-Sb. 61 (2), 335–349 (1988).
  19. Vorob’eva, E.V., Romanovskii, R.K. “The Method of Characteristics for Hyperbolic Boundary Value Problems in the Plane”, Siberian Math. J. 41 (3), 433–441 (2000).
  20. Romanovskiĭ, R.K., Mendziv, M.V. “Stability of Solutions of the Cauchy Problem for a Hyperbolic System in the Plane with Time-periodic Coefficients”, Siberian Math. J. 48 (5), 913–918 (2007).
  21. Romanovskiĭ, R.K., Medvedev, Yu.A. “Optimal Two-sided Boundary Control of Heat Conduction in a Rod. A Hyperbolic Model”, Russian Math. (Iz. VUZ) 60 (6), 45–51 (2016).