Russian Mathematics

, Volume 63, Issue 6, pp 42–50 | Cite as

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

  • L. B. MironovaEmail author


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.


hyperbolic system Riemann method Riemann matrix Cauchy problem characteristics 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bitsadze, A.V. “Structural Properties of Solutions of Hyperbolic Systems of First-order Partial Differential Equations”, Mat. Model. 6 (6), 22–31 (1994).MathSciNetzbMATHGoogle Scholar
  2. 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).MathSciNetGoogle Scholar
  3. 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).MathSciNetzbMATHGoogle Scholar
  4. 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).MathSciNetzbMATHGoogle Scholar
  5. 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).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 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).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 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).zbMATHGoogle Scholar
  8. 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).MathSciNetzbMATHGoogle Scholar
  9. 9.
    Sozontova, E.A. “Characteristic Problems with Normal Derivatives for Hyperbolic Systems”, Russian Math. (Iz. VUZ) 57 (10), 37–47 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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).CrossRefzbMATHGoogle Scholar
  11. 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)).Google Scholar
  12. 12.
    Zhegalov, V.I., Sevast’yanov, V.A. “The Goursat Problem in Four-dimensional Space”, Differential Equations 32 (10), 1427–1428 (1996).MathSciNetzbMATHGoogle Scholar
  13. 13.
    Zhegalov, V.I. “On the Three-dimensional Riemann Function”, Siberian Math. J. 38 (5), 929–934 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mironov, A.N. “On the Construction of the Riemann Function for a Fourth-order Equation”, Differ. Equ. 37 (12), 1787–1791 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Utkina, E.A. “On the General Case of the Goursat Problem”, Russian Math. (Iz. VUZ) 49 (8), 53–58 (2005).MathSciNetzbMATHGoogle Scholar
  16. 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).MathSciNetCrossRefGoogle Scholar
  17. 17.
    Romanovskiĭ, R.K. “Riemann Matrices of the First and Second Kinds”, Mat. Sb. (N.S.) 127 (169) (4), 494–501 (1985).MathSciNetGoogle Scholar
  18. 18.
    Romanovskiĭ, R.K. “Exponentially Splittable Hyperbolic Systems with Two Independent Variables”, Math. USSR-Sb. 61 (2), 335–349 (1988).MathSciNetCrossRefGoogle Scholar
  19. 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).MathSciNetCrossRefGoogle Scholar
  20. 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).MathSciNetCrossRefGoogle Scholar
  21. 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).MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Allerton Press, Inc. 2019

Authors and Affiliations

  1. 1.Elabuga Institute of Kazan Federal UniversityElabugaRussia

Personalised recommendations