Skip to main content
SpringerLink
Log in

Functionally invariant solutions to Maxwell’s system

  • Published:
Journal of Applied and Industrial Mathematics Aims and scope Submit manuscript

Abstract

We consider the problem of the existence of functionally invariant solutions to Maxwell’s system. The solutions found contain functional arbitrariness, which is used for determining the parameters of Maxwell’s system (the dielectric and magnetic constants).

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. N. P. Erugin and M. M. Smirnov, “Functionally Invariant Solutions of Differential Equations,” Differentsial’nye Uravneniya 17 (5), 853–865 (1981) [Differential Equations 17, 563–573 (1981)].

    MathSciNet  MATH  Google Scholar 

  2. V. M. Babich and V. S. Buldyrev, Asymptotic Methods in Problems of Diffraction of Short Waves (Nauka, Moscow, 1972) [in Russian].

    MATH  Google Scholar 

  3. V. M. Babich, “The Multidimensional WKB Method or RayMethod. Its Analogs and Generalizations,” Itogi Nauki Tekh. Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 34, 93–134 (1988).

    MATH  Google Scholar 

  4. M. V. Neshchadim, “Classes of Generalized Functionally Invariant Solutions of the Wave Equation. I,” Siberian Electron. Math. Reports 10, 418–435 (2013) [see http://semr.math.nsc.ru/v10/p418-435. pdf].

    MathSciNet  MATH  Google Scholar 

  5. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. II: Partial Differential Equations (Interscience, New York, 1962;Mir, Moscow, 1964).

    MATH  Google Scholar 

  6. V. I. Smirnov and S. L. Sobolev, “A NewMethod for Solving the Planar Problemof Elastic Vibrations,” Trudy Seismolog. Inst. Akad. Nauk SSSR 20, 1–37 (1932).

    Google Scholar 

  7. V. I. Smirnov and S. L. Sobolev, “On the Application of a New Method to the Study of Elastic Vibrations in Space in the Presence of Axial Symmetry,” Trudy Seismolog. Inst. Akad. Nauk SSSR 29, 43–51 (1933).

    Google Scholar 

  8. S. L. Sobolev, “Functionally Invariant Solutions to the Wave Equation,” Trydy Fiz.-Mat. Inst. Steklov. 5, 259–264 (1934).

    Google Scholar 

  9. M. S. Shneerson, “Maxwell’s Equations and Functionally Invariant Solutions to the Wave Equation,” Differentsial’nyeUravneniya 4 (4), 743–758 (1968).

    MathSciNet  Google Scholar 

  10. N. T. Stel’mashuk, “Construction of Functionally Invariant Solutions to Maxwell’s System for an Electromagnetic Field in Vacuum,” Vestnik Akad. Nauk BSSR. Ser. Fiz.-Mat. Nauki No. 4, 35–39 (1974).

    MathSciNet  Google Scholar 

  11. M. V. Neshchadim, “Solutions toMaxwell’s System with Zero Invariants,” Vestnik Novosib. Gos. Univ. Ser. Mat., Mekh., Inform. 6 (3), 59–61 (2006).

    MathSciNet  MATH  Google Scholar 

  12. Yu. E. Anikonov and M. V. Neshchadim, “On Analytical Methods in the Theory of Inverse Problems for Hyperbolic Equations. I,” Sibirsk. Zh. Industr. Mat. 14 (1), 27–39 (2011) [J. Appl. Indust. Math. 5 (4), 506–518 (2012)].

    MathSciNet  MATH  Google Scholar 

  13. Yu. E. Anikonov and M. V. Neshchadim, “On Analytical Methods in the Theory of Inverse Problems for Hyperbolic Equations. II,” Sibirsk. Zh. Industr. Mat. 14 (2), 28–33 (2011) [J. Appl. Indust. Math. 6 (1), 6–11 (2012)].

    MATH  Google Scholar 

  14. L. V. Ovsyannikov, Group Analysis of Differential Equations (Nauka, Moscow, 1978; Academic Press, New York, 1982).

    MATH  Google Scholar 

  15. P. J. Olver, Applications of Lie Groups toDifferential Equations (Springer, New York, 1986;Mir, Moscow, 1989).

    Google Scholar 

  16. M. V. Neshchadim, “Equivalent Transformations and Some Exact Solutions to the System of Maxwell’s Equations,” Selcuk J. Appl. Math. 3 (2), 99–108 (2002).

    MathSciNet  MATH  Google Scholar 

  17. Yu. E. Anikonov and M. V. Neshchadim, “Representations for the Solutions and Coefficients of Second-Order Differential Equations,” Sibirsk. Zh. Industr. Mat. 15 (4), 17–23 (2012) [J. Appl. Indust. Math. 7 (1), 15–21 (2013)].

    MATH  Google Scholar 

  18. Yu. E. Anikonov and M. V. Neshchadim, “Representations for the Solutions and Coefficients of Evolution Equations,” Sibirsk. Zh. Industr. Mat. 16 (2), 40–49 (2013) [J. Appl. Indust. Math. 7 (3), 326–334 (2013)].

    MathSciNet  MATH  Google Scholar 

  19. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 2: The Classical Theory of Fields (Nauka, Moscow, 1988; Pergamon Press, Oxford, 1980).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090, Russia

    M. V. Neshchadim

  2. Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia

    M. V. Neshchadim

Authors
  1. M. V. Neshchadim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. V. Neshchadim.

Additional information

Original Russian Text © M.V. Neshchadim, 2017, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2017, Vol. XX, No. 4, pp. 66–74.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Neshchadim, M.V. Functionally invariant solutions to Maxwell’s system. J. Appl. Ind. Math. 11, 107–114 (2017). https://doi.org/10.1134/S1990478917010124

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1990478917010124

Keywords

Search

Navigation