Skip to main content
SpringerLink
Log in

On the Darboux Problem for a Hyperbolic System of Equations with Multiple Characteristics

  • Published:
Russian Mathematics Aims and scope Submit manuscript

Abstract

We prove the existence and uniqueness of a solution of the boundary value problem with conditions on one of the characteristics and on the free line for a system of hyperbolic equations with multiple characteristics. An analog of the Riemann–Hadamard method for this problem is developed, and a definition of the Riemann–Hadamard matrix is given. The solution of this problem is constructed in terms of the introduced Riemann–Hadamard matrix.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. A. V. Bitsadze, “On structural properties of solutions of hyperbolic systems of partial differential equations of the first order,” Mat. Model. 6 (6), 22–31 (1994).

    MathSciNet  MATH  Google Scholar 

  2. T. V. Chekmarev, “Formulas for solution of the Goursat problem for a linear system of partial differential equations,” Differ. Uravn. 18 (9), 1614–1622 (1982).

    MathSciNet  Google Scholar 

  3. T. V. Chekmarev, Systems of Equations of Mixed Type (Nizhegorod. Gos. Tekh. Univ., Nizhny Novgorod, 1995) [in Russian].

    Google Scholar 

  4. I. E. Pleshchinskaya, “The equivalence of some classes of first-order elliptic and hyperbolic systems and second-order partial differential equations,” Differ. Uravn. 23 (9), 1634–1637 (1987).

    MathSciNet  MATH  Google Scholar 

  5. R. K. Romanovskii, “On Riemann matrices of the first and second kind,” Math. USSR-Sb. 55 (2), 485–492 (1985). https://doi.org/10.1070/SM1986v055n02ABEH003016

    Article  Google Scholar 

  6. R. K. Romanovskii, “Exponentially splittable hyperbolic systems with two independent variables,” Math. USSR-Sb. 61 (2), 335–350 (1988). https://doi.org/10.1070/SM1988v061n02ABEH003211

    Article  MathSciNet  Google Scholar 

  7. E. V. Vorob’ëva and R. K. Romanovskii, “The method of characteristics for hyperbolic boundary value problems on the plane,” Sib. Math. J. 41 (3), 433–441 (2000). https://doi.org/10.1007/BF02674100

    Article  MathSciNet  Google Scholar 

  8. L. B. Mironova, “On the Riemann method in R n for a system with multiple characteristics,” Russ. Math. 50 (1), 32–37 (2006).

    MATH  Google Scholar 

  9. L. B. Mironova, “On characteristic problems for a system with double higher partial derivatives,” Vestn. Samar. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauki 43, 31–37 (2006). https://doi.org/10.14498/vsgtu450

    Article  Google Scholar 

  10. V. I. Zhegalov and L. B. Mironova, “One system of equations with double major partial derivatives,” Russ. Math. 51 (3), 9–18 (2007).

    Article  MATH  Google Scholar 

  11. R. K. Romanovskii and M. V. Mendziv, “Stability of solutions to the Cauchy problem for a plane hyperbolic system with time-periodic coefficients,” Sib. Math. J. 48 (5), 913–918 (2007). https://doi.org/10.1007/s11202-007-0093-2

    Article  Google Scholar 

  12. V. I. Zhegalov, “A problem with normal derivatives in boundary conditions for a system of differential equations,” Russ. Math. 52 (8), 58–60 (2008). https://doi.org/10.3103/S1066369X08080070

    Article  MathSciNet  MATH  Google Scholar 

  13. Yu. G. Voronova, “On Cauchy problem for linear hyperbolic systems of the equations with zero generalized Laplace invariants,” Ufim. Math. J. 2 (2), 20–26 (2010).

    MATH  Google Scholar 

  14. A. V. Zhiber and O. S. Kostrigina, “Goursat problem for nonlinear hyperbolic systems with integrals of the first and second order,” Ufa Math. J. 3 (3), 65–77 (2011).

    MathSciNet  MATH  Google Scholar 

  15. E. A. Sozontova, “Characteristic problems with normal derivatives for hyperbolic systems,” Russ. Math. 57 (10), 37–47 (2013). https://doi.org/10.3103/S1066369X13100046

    Article  MathSciNet  MATH  Google Scholar 

  16. R. K. Romanovskii and Yu. A. Medvedev, “Optimal two-sided boundary control of heat transmission in a rod. Hyperbolic model,” Russ. Math. 60 (6), 45–51 (2016). https://doi.org/10.3103/S1066369X16060062

    Article  MATH  Google Scholar 

  17. A. A. Andreev and Yu. O. Yakovleva, “The Cauchy problem for a system of the hyperbolic differential equations of the nth order with nonmultiple characteristics,” Vestn. Samar. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauki 21 (4), 752–759 (2017). https://doi.org/10.14498/vsgtu1577

    Article  Google Scholar 

  18. A. N. Mironov, L. B. Mironova, and Yu. O. Yakovleva, “The Riemann method for equations with a dominant partial derivative (a review),” Vestn. Samar. Gos. Tekh. Univ., Ser. Fiz.-Mat. Nauki 25 (2), 207–240 (2021). https://doi.org/10.14498/vsgtu1853

  19. L.B. Mironova, “Boundary-value problems with data on characteristics for hyperbolic systems of equations,” Lobachevskii J. Math. 41 (3), 400–406 (2020). https://doi.org/10.1134/S1995080220030130

    Article  MathSciNet  MATH  Google Scholar 

  20. A. N. Mironov and L. B. Mironova, “Riemann–Hadamard method for one system in three-dimensional space,” Differ. Equations 57 (8), 1034–1041 (2021). https://doi.org/10.1134/S0012266121080073

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Elabuga Institute, Kazan Federal University, 423600, Elabuga, Russia

    A. N. Mironov

  2. Samara State Technical University, 443100, Samara, Russia

    A. P. Volkov

Authors
  1. A. N. Mironov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. P. Volkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to A. N. Mironov or A. P. Volkov.

Ethics declarations

The authors declare that they have no conflicts of interest.

Additional information

Translated by L. Kartvelishvili

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mironov, A.N., Volkov, A.P. On the Darboux Problem for a Hyperbolic System of Equations with Multiple Characteristics. Russ Math. 66, 31–36 (2022). https://doi.org/10.3103/S1066369X22080060

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S1066369X22080060

Keywords:

Search

Navigation