Differential Equations

, Volume 55, Issue 1, pp 138–141 | Cite as

Some Cases of the Cauchy Problem for First-Order Differential Equations with Discontinuous Coefficients

  • D. S. AnikonovEmail author
  • D. S. Konovalova
Short Communications


We consider the Cauchy problem in three-dimensional space for a first-order almost linear differential equation with discontinuous coefficients of the derivatives. Two special cases related to the behavior of characteristics are singled out and studied.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bibikov, Yu.N., Kurs obyknovennykh differentsial’nykh uravnenii (Course of Ordinary Differential Equations), Moscow: Vysshaya Shkola, 1991.Google Scholar
  2. 2.
    Petrovskii, I.G., Lektsii po teorii obyknovennykh differentsial’nykh uravnenii (Lectures in the Theory of Ordinary Differential Equations), Moscow: Nauka, 1970.Google Scholar
  3. 3.
    Petrova, G. and Popov, B., Linear transport equations with discontinuous coefficients, Commun. Partial Differ. Equations, 1999, vol. 24, no. 9–10, pp. 1849–1873.Google Scholar
  4. 4.
    Bouchut, F. and Jame, F., One-dimensional transport equations with discontinuous coefficients, Nonlinear Anal. Theory Methods Appl., 1998, vol. 32, no. 7, pp. 891–933.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kulikovskii, A.G., Sveshnikova, E.I., and Chugainova, A.P., Matematicheskie methody izucheniya razryvnykh reshenii nelineinykh giperbolicheskikh sistem uravnenii (Mathematical Methods for Studying Discontinuous Solutions of Nonlinear Hyperbolic Systems of Equations), Lecture Courses of Scientific-Educational Center, Moscow: MIAN, 2010, no. 16.Google Scholar
  6. 6.
    Tadmor, E., Local error estimates for discontinuous solutions of nonlinear hyperbolic equations, SIAM J. Numer. Anal., 1991, vol. 28, pp. 891–906.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kalitkin, N.N., Chislennye metody (Numerical Methods), Moscow: Nauka, 1978.Google Scholar
  8. 8.
    Gel’fand, I.M., Several problems of the theory of quasilinear equations, Uspekhi Mat. Nauk, 1959, vol. 14, no. 2 (86), pp. 87–158.Google Scholar
  9. 9.
    Filippov, A.F., Differentsial’nye uravneniya s razryvnoi pravoi chast’yu (Differential Equations with Discontinuous Right-Hand Side), Moscow: Nauka, 1985.zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Novosibirsk State UniversityNovosibirskRussia
  2. 2.Sobolev Institute of MathematicsSiberian Branch of the Russian Academy of SciencesNovosibirskRussia

Personalised recommendations