Skip to main content
SpringerLink
Log in

A General Integral of a Quasilinear Equation and Application to a Nonlinear Characteristic Problem

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

We describe a method for constructing general integrals for some nonstrictly hyperbolic quasilinear equations and prove a nonlinear analog of Asgeirsson’s mean value theorem. Using a general integral, we study the nonlinear version of the Goursat characteristic problem.

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. Monge G., “Mémoire sur le calcul intégral des équations aux différences partielles,” Mém. Acad Sci. Paris: Imprimerie Royale, 118–192 (1784).

    Google Scholar 

  2. Darboux G., Leçons sur la théorie générale des surfaces et les applications geometriques du calcul infinitésimal, Gauthier-Villars et Fils, Paris (1894).

    MATH  Google Scholar 

  3. Goursat E., A Course in Mathematical Analysis. Vol. 3 [Russian translation], Gostekhizdat, Moscow and Leningrad (1933).

    Google Scholar 

  4. Bitsadze A. V., Certain Classes of Partial Differential Equations, Gordon and Breach, New York (1958).

    Google Scholar 

  5. Baghaturia G., “Nonlinear versions of hyperbolic problems for one quasi-linear equation of mixed type,” J. Math. Sci., vol. 208, no. 6, 621–634 (2015).

    Article  MathSciNet  Google Scholar 

  6. Menteshashvili M. Z., “On the Cauchy problem with initial data on the unit circle,” Soobshch. Akad. Nauk GruzSSR, vol. 148, no. 2, 190–192 (1993).

    MathSciNet  MATH  Google Scholar 

  7. Gvazava D. K., Some Classes of Hyperbolic Equations and Equations of Mixed Type [Georgian], Metsniereba, Tbilisi (1992).

    MATH  Google Scholar 

  8. Gvazava J. K., “Nonlocal and initial problems for quasi-linear, non-strictly hyperbolic equations with general solutions represented by superposition of arbitrary functions,” Georgian Math. J., vol. 10, 687–707 (2003).

    MathSciNet  MATH  Google Scholar 

  9. Bitsadze R. G., “General representation of the solutions of a quasilinear equation of a problem of nonlinear oscillations,” Soobshch. Akad. Nauk GruzSSR, vol. 128, no. 3, 493–496 (1987).

    MathSciNet  MATH  Google Scholar 

  10. Gunter N. M., Integration of First-Order Partial Differential Equations [Russian], ONTI GTTI, Moscow and Leningrad (1934).

    Google Scholar 

  11. Goursat E., Lecons sur l’intégration des équations aux dérivées partielles du second ordre à deux variables indépendantes. Vol. 2, Hermann, Paris (1898).

    MATH  Google Scholar 

  12. Hörmander L., Lectures on Nonlinear Hyperbolic Differential Equations, Springer-Verlag, Berlin (1997).

    MATH  Google Scholar 

  13. Ásgeirsson L., “Über eine Mittelwertseigenschaft von Losungen homogener linearer partieller Differentialgleichungen 2. Ordnung mit konstanten Koeffizienten,” Math. Ann., vol. 113, no. 1, 321–346 (1937).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Niko Muskhelishvili Institute of Computational Mathematics, Georgian Technical University, Tbilisi, Georgia

    G. G. Baghaturia & M. Z. Menteshashvili

Authors
  1. G. G. Baghaturia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Z. Menteshashvili
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to G. G. Baghaturia or M. Z. Menteshashvili.

Additional information

Russian Text © The Author(s), 2019, published in Sibirskii Matematicheskii Zhurnal, 2019, Vol. 60, No. 6, pp. 1209–1222.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baghaturia, G.G., Menteshashvili, M.Z. A General Integral of a Quasilinear Equation and Application to a Nonlinear Characteristic Problem. Sib Math J 60, 940–951 (2019). https://doi.org/10.1134/S0037446619060028

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0037446619060028

Keywords

Search

Navigation