[go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

New exact solutions to the nonautonomous Liouville equation

  • Published:
Siberian Mathematical Journal Aims and scope Submit manuscript

Abstract

We obtain new formulas for the exact analytic solutions to the nonautonomous elliptic Liouville equation in the two-dimensional coordinate space with the free function dependent specially on an arbitrary harmonic function. We present new exact solutions to the wave Liouville equation with two arbitrary functions, providing original formulas for the general solution for the classical (autonomous) and wave Liouville equations. Some equivalence transformations are presented for the elliptic Liouville equation depending on conjugate harmonic functions. In particular, we indicate a transformation that reduces the equation under study to an autonomous form.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. Sabitov I. Kh., “Solutions to the equation Δu = f(x, y)e cu in some special cases,” Sb.: Math., 192, No. 6, 879–894 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  2. Semenov È. I., “Properties of the fast diffusion equation and its multidimensional exact solutions,” Siberian Math. J., 44, No. 4, 680–685 (2003).

    Article  MathSciNet  Google Scholar 

  3. Semenov È. I., “Multidimensional exact solutions to a quasilinear parabolic equation with anisotropic heat conductivity,” Siberian Math. J., 47, No. 2, 376–382 (2006).

    Article  MathSciNet  Google Scholar 

  4. Svirshchevskii S. R., “Group classification and invariant solutions of nonlinear polyharmonic equations,” Differentsial’nye Uravneniya, 29, No. 10, 1772–1781 (1993).

    MathSciNet  Google Scholar 

  5. Polyanin A. D. and Zaizev V. F., Handbook on Nonlinear Equations of Mathematical Physics: Exact Solutions [in Russian], Fizmatlit, Moscow (2002).

    Google Scholar 

  6. Ibragimov N. Kh., Transformation Groups in Mathematical Physics [in Russian], Nauka, Moscow (1983).

    MATH  Google Scholar 

  7. Polyanin A. D. and Zaizev V. F., Handbook of Exact Solutions for Ordinary Differential Equations, CRC Press, Boca Raton, FL (2001).

    Google Scholar 

  8. Pukhnachov V. V., “Equivalence transformations and hidden symmetry of evolution equations,” Dokl. Akad. Nauk SSSR, 294, No. 3, 535–538 (1989).

    Google Scholar 

  9. Pukhnachov V. V., “Reciprocal transformations of radial equations of nonlinear heat conduction,” Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 213, 151–163 (1994).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of System Dynamics and Control Theory, Irkutsk, Russia

    È. I. Semenov

Authors
  1. È. I. Semenov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to È. I. Semenov.

Additional information

Original Russian Text Copyright © 2008 Semenov È. I.

__________

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 49, No. 1, pp. 207–217, January–February, 2008.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Semenov, È.I. New exact solutions to the nonautonomous Liouville equation. Sib Math J 49, 166–174 (2008). https://doi.org/10.1007/s11202-008-0017-9

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11202-008-0017-9

Keywords