BSU Bulletin. Mathematics, Informatics
Bibliographic description:
, , , , DIRICHLET PROBLEM FOR HELMHOLTZ EQUATIONS IN GYROTROPIC ELLIPTICAL REGION WITH LONGITUDINAL MAGNETIZATION // BSU Bulletin. Mathematics, Informatics. - 2019. №3. . - С. 17-31.
Title:
DIRICHLET PROBLEM FOR HELMHOLTZ EQUATIONS IN GYROTROPIC ELLIPTICAL REGION WITH LONGITUDINAL MAGNETIZATION
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We have formulated and solved the Dirichlet problem for Helmholtz equations of electromagnetic waves, propagating in an elliptical cylinder filled with longitu- dinally magnetized ferrite which is described by a second-rank tensor. It is assumed that the cylinder has an infinitely conductive wall. To solve the boundary value problem of Helmholtz equations for longitudinal components of electromagnetic waves, we have used the method of shortening the initial differential equation and the method of variables separation. The solution of the above boundary value prob- lem in elliptic coordinates is associated with the use of even and odd ordinary and modified Mathieu functions of the first kind. Using the results obtained, we have determined all the components of electromagnetic waves for even and odd solu- tions. Applying the Dirichlet condition to the components of electromagnetic waves and solving the system of linear homogeneous algebraic equations, we have ob- tained dispersion equations. The found dispersion equations of electromagnetic waves are of great practical importance and allow studying the propagation of hy- brid waves in this region.
Keywords:
elliptical cylinder; ferrite; Dirichlet problem; Helmholtz equation; electromagnetic wave; longitudinal magnetization; gyrotropic region; transverse com- ponents of the electromagnetic field; Mathieu functions; dispersion equation.
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Lax B., Button K. J. Microwave Ferrites and Ferrimagnetics. New York: Mac-Graw-Hill, 1962. 752 p.
McLachlan N. W. Theory and Application of Mathieu Functions. London: Oxford University Press, 1947.
Itigilov G. B., Shirapov D. Sh. Metod invariantnykh preobrazovanii dlya opredeleniya poperechnykh komponent elektromagnitnogo polya v girotropnykh ogranichennykh oblastyakh [Method of Invariant Transformations for Determining the Transverse Components of Electromagnetic Field in Gyrotropic Bounded Areas]. Vestnik Buryatskogo gosudarstvennogo universiteta. Matematika, informatika. 2012. V. 9. Pp.162–166.
Shirapov D. Sh., Itigilov G. B. Obobshchennye uravneniya Gelmgoltsa giro- tropnykh volnovodov proizvolnoi formy poperechnogo secheniya [Generalized Helm- holtz Equations of Gyrotropic Waveguides with Arbitrary Cross-Section]. Sovremennye problemy distantsionnogo zondirovaniya, radiolokatsii, rasprostraneniya i di- fraktsii voln. Proc. 2nd All-Russ. conf. (June 26–28, 2018). Pp. 209–219.
Mikaelyan A. L. Teoriya i primenenie ferritov na sverkhvysokikh chastotakh [Theory and Application of Ferrites at Microwave Frequencies]. Leningrad: Gosener- goizdat Publ., 1963. 664 p.
Nazarov A. V., Raevskii S. B. Elektromagnitnye volny v strukturakh, soderz- hashchikh prodolno namagnichennye ferritovye sloi [Electromagnetic Waves in Struc- tures Containing Longitudinally Magnetized Ferrite Layers]. Fizika volnovykh protsessov i radiotekhnicheskie sistemy. 2007. V. 10. No. 1. Pp. 76–82.
Bitsadze A. V. Uravneniya matematicheskoi fiziki [Equations of Mathematical Physics]. 2nd rev. ed. Moscow: Nauka Publ., 1982. 336 p.
Ango A. Matematika dlya elektro- i radioinzhenerov [Mathematics for Elec- trical and Radio Engineers]. Moscow: Nauka Publ., 1967. 780 p.
Bateman H., Erdélyi A. Higher Transcendental Functions. New York: McGraw-Hill, 1953–1955.
Nazarov A. V. Novoselova N. A., Raevskii S. B. O polnote sistemy reshenii kraevykh zadach dlya ferritovykh volnovodov, poluchennykh metodom ukorocheniya differentsialnogo uravneniya [On Completeness of the System of Boundary Value Problems Solutions for Ferrite Waveguides Obtained by the Method of Differential Equation Shortening]. Telecommunications and Radio Engineering. 2016. V. 7 (227). Pp. 63–66.
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