Russian Physics Journal

, Volume 59, Issue 9, pp 1482–1490 | Cite as

Method for Solving Physical Problems Described by Linear Differential Equations

  • B. A. Belyaev
  • V. V. Tyurnev

A method for solving physical problems is suggested in which the general solution of a differential equation in partial derivatives is written in the form of decomposition in spherical harmonics with indefinite coefficients. Values of these coefficients are determined from a comparison of the decomposition with a solution obtained for any simplest particular case of the examined problem. The efficiency of the method is demonstrated on an example of calculation of electromagnetic fields generated by a current-carrying circular wire. The formulas obtained can be used to analyze paths in the near-field magnetic (magnetically inductive) communication systems working in moderately conductive media, for example, in sea water.


current-carrying coil electromagnetic field linear equations in partial derivatives spherical harmonics near-field magnetic communication 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. N. Tikhonov and A. A. Samarskii, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1977).Google Scholar
  2. 2.
    N. S. Koshlyakov, È. B. Gliner, and M. M. Smirnov, Equations of Mathematical Physics in Partial Derivatives [in Russian], Vysshaya Shkola, Moscov (1970).Google Scholar
  3. 3.
    V. V. Nikolskii and T. I. Nikolskaya, Electrodynamics and Propagation of Radio Waves [in Russian], LIBROKOM, Moscow (2010).Google Scholar
  4. 4.
    L. A. Wainshtein, Electromagnetic Waves [in Russian], Radio Svyaz’, Moscow (1988).Google Scholar
  5. 5.
    G. I. Veselov, E. N. Egorov, Yu. N. Alekhin, et al., Microelectronic Microwave Devices, G. I. Veselov, ed. [in Russian], Vysshaya Shkola, Moscow (1988).Google Scholar
  6. 6.
    R. Mitra and S. W. Lee, Analytical Techniques in the Theory of Guided Waves, The Macmillan Company, New York (1971).Google Scholar
  7. 7.
    A. V. Kukushkin, Phys. Usp., 36, No. 2, 81 (1993).ADSCrossRefGoogle Scholar
  8. 8.
    R. Mittra et al., Computer Techniques for Electromagnetics, Pergamon Press, Oxford (1973).Google Scholar
  9. 9.
    V. V. Batygin and I. N. Toptygin, Collection of Tasks in Electrodynamics [in Russian], Pleiades Publishing Group, Moscow (2002).Google Scholar
  10. 10.
    W. R. Smythe, Static and Dynamic Electricity, McGraw-Hill Book Company, New York (1950).MATHGoogle Scholar
  11. 11.
    I. F. Akyildiz, P. Wang, and Z. Sun, IEEE Commun. Mag., November, 42 (2015).Google Scholar
  12. 12.
    B. Gulbahar and O. B. Akan, IEEE Trans. Wireless Commun., 11, No. 9, 3326 (2012).CrossRefGoogle Scholar
  13. 13.
    G. Korn and T. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill Book Company Inc., New York (1961).MATHGoogle Scholar
  14. 14.
    B. M. Yavorskii and A. A. Detlaf, Handbook on Physics for Engineers and Students of High Schools. [in Russian], Nauka, Moscow (1968).Google Scholar
  15. 15.
    M. Abramowitz and I. Stigan, eds., Handbook of Mathematical Functions, National Bureau of Standards, Washington, (1964).Google Scholar
  16. 16.
    D. J. Griffiths, Introduction to Electrodynamics. Instructor’s Solutions Manual, Prentice Hall, New Jersey (2004).Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.L. V. Kirensky Institute of Physics of the Federal Research Center of Krasnoyarsk Scientific Center of the Siberian Branch of the Russian Academy of SciencesKrasnoyarskRussia
  2. 2.Siberian Federal UniversityKrasnoyarskRussia
  3. 3.Siberian State Aerospace UniversityKrasnoyarskRussia

Personalised recommendations