Skip to main content
SpringerLink
Log in

An analytic method for the initial value problem of the electric field system in vertical inhomogeneous anisotropic media

  • Published:
Applications of Mathematics Aims and scope Submit manuscript

Abstract

The time-dependent system of partial differential equations of the second order describing the electric wave propagation in vertically inhomogeneous electrically and magnetically biaxial anisotropic media is considered. A new analytical method for solving an initial value problem for this system is the main object of the paper. This method consists in the following: the initial value problem is written in terms of Fourier images with respect to lateral space variables, then the resulting problem is reduced to an operator integral equation. After that the operator integral equation is solved by the method of successive approximations. Finally, a solution of the original initial value problem is found by the inverse Fourier transform.

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.D. Andersen: Real Paley-Wiener theorems for the inverse Fourier transform on a Riemannian symmetric space. Pac. J. Math. 213 (2004), 1–13.

    Article  MATH  Google Scholar 

  2. T.M. Apostol: Calculus I. Blaisdell Publishing Company, Waltham, Massachusetts-To-ronto-London, 1967.

    Google Scholar 

  3. H.H. Bang: Functions with bounded spectrum. Trans. Am. Math. Soc. 347 (1995), 1067–1080.

    Article  MathSciNet  MATH  Google Scholar 

  4. R. Burridge, J. Qian: The fundamental solution of the time-dependent system of crystal optics. Eur. J. Appl. Math. 17 (2006), 63–94.

    Article  MathSciNet  MATH  Google Scholar 

  5. G.C. Cohen: Higher-Order Numerical Methods for TransientWave Equations. Springer, Berlin, 2002.

    Google Scholar 

  6. R. Courant, D. Hilbert: Methods of Mathematical Physics, Vol. 2. Interscience Publishers, New York-London, 1962.

    Google Scholar 

  7. H. J. Eom: Electromagnetic Wave Theory for Boundary-Value Problems. Springer, Berlin, 2004.

    MATH  Google Scholar 

  8. L.C. Evans: Partial Differential Equations. AMS, Providence, 1998.

    MATH  Google Scholar 

  9. T.H. Gronwall: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math. 20 (1919), 292–296.

    Article  MathSciNet  Google Scholar 

  10. C.M. Krowne: Spectral-domain determination of propagation constant in biaxial planar media. Int. J. Electron. 3 (1985), 315–332.

    Article  Google Scholar 

  11. D. Ludwig: Conical refraction in crystal optics and hydromagnetics. Commun. Pure Appl. Math. 14 (1961), 113–124.

    Article  MathSciNet  MATH  Google Scholar 

  12. R.B. Melrose, G.A. Uhlmann: Microlocal structure of involutive conical refraction. Duke Math. J. 46 (1979), 571–582.

    Article  MathSciNet  MATH  Google Scholar 

  13. N. Ortner, P. Wagner: Fundamental matrices of homogeneous hyperbolic systems. Applications to cyrystal optics, elastodynamics, and piezoelectromagetism. ZAMM, Z. Agew. Math. Mech. 84 (2004), 314–346.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Sheen: Time harmonic electromagnetic fields in an biaxial anisotropic medium. J. Electromagn. Waves Appl. 19 (2005), 754–767.

    Article  MathSciNet  Google Scholar 

  15. V.K. Tuan, A. I. Zayed: Paley-Wiener-type theorems for a class of integral transforms. J. Math. Anal. Appl. 266 (2002), 200–226.

    Article  MathSciNet  MATH  Google Scholar 

  16. G.A. Uhlmann: Light intensity distribution in conical refraction. Commun. Pured Appl. Math. 35 (1982), 69–80.

    Article  MathSciNet  MATH  Google Scholar 

  17. V. S. Vladimirov: Equations of Mathematical Physics. Marcel Dekker, New York, 1971.

    Google Scholar 

  18. V.G. Yakhno: Constructing Green’s function for time-depending Maxwell system in anisotropic dielectrics. J. Phys. A: Math. Gen. 38 (2005), 2277–2287.

    Article  MathSciNet  MATH  Google Scholar 

  19. V.G. Yakhno, T.M. Yakhno, M. Kasap: A novel approach for modelling and simulation of electromagnetic waves in anisotropic dielectrics. Int. J. Solids Struct. 43 (2006), 6261–6276.

    Article  MathSciNet  MATH  Google Scholar 

  20. V.G. Yakhno: Computing and simulation of time-dependent electromagnetic fields in homogeneous anisotropic materials. Int. J. Eng. Sci. 46 (2008), 411–426.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Izmir, Turkey

    Valery Yakhno & Ali Sevimlican

  2. Electrical and Electronics Engineering Department, Muhandislik Fak., Dokuz Eylul University, Tinazpete kampus, Buca, Izmir, 35160, Turkey

    Valery Yakhno

  3. Department of Mathematics, Fen-Edebiyat Fak., Dokuz Eylul University, Tinazpete kampus, Buca, Izmir, 35160, Turkey

    Ali Sevimlican

Authors
  1. Valery Yakhno
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ali Sevimlican
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Valery Yakhno.

Additional information

This work was supported by Dokuz Eylul University of Turkey under the research grant 2006.KB.FEN.024.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yakhno, V., Sevimlican, A. An analytic method for the initial value problem of the electric field system in vertical inhomogeneous anisotropic media. Appl Math 56, 315–339 (2011). https://doi.org/10.1007/s10492-011-0019-y

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10492-011-0019-y

Keywords

MSC 2010

Search

Navigation