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  • 1.
    Pankratova, Iryna
    et al.
    Department of Technology, Narvik University College, Narvik, Norway.
    Pettersson, Klas
    Department of Technology, Narvik University College, Narvik, Norway.
    Spectral asymptotics for an elliptic operator in a locally periodic perforated domain2015Ingår i: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 94, nr 6, s. 1207-1234Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider the homogenization of an elliptic spectral problem with a large potential stated in a thin cylinder with a locally periodic perforation. The size of the perforation gradually varies from point to point. We impose homogeneous Neumann boundary conditions on the boundary of perforation and on the lateral boundary of the cylinder. The presence of a large parameter 1/ε in front of the potential and the dependence of the perforation on the slow variable give rise to the effect of localization of the eigenfunctions. We show that the jth eigenfunction can be approximated by a scaled exponentially decaying function that is constructed in terms of the jth eigenfunction of a one-dimensional harmonic oscillator operator.

  • 2.
    Shestopalov, Yury
    Högskolan i Gävle, Akademin för teknik och miljö, Avdelningen för elektroteknik, matematik och naturvetenskap, Matematik.
    Resonance frequencies of arbitrarily shaped dielectric cylinders2023Ingår i: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 102, nr 6, s. 1618-1632Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider eigenvalue problems for dielectric cylindrical scatterers of arbitrary cross section with generalized conditions at infinity that enable one to take into account complex eigenvalues. The existence of resonance (scattering) frequencies associated with these eigenvalues is proved. The technique involves the determination of characteristic numbers (CNs) of the Fredholm operator-valued functions of the problems constructed using Green's potentials. Separating principal parts in the form of meromorphic operator pencils, we apply the operator generalization of Rouché's theorem to verify the occurrence of CNs in close proximities of the pencil poles. The results are illustrated in detail using the case of a dielectric cylinder of circular cross section.

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  • 3.
    Shestopalov, Yury
    Högskolan i Gävle, Akademin för teknik och miljö, Avdelningen för elektroteknik, matematik och naturvetenskap, Matematik.
    Trigonometric and cylindrical polynomials and their applications in electromagnetics2020Ingår i: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 99, nr 16, s. 2807-2822Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    Development of the theory of trigonometric polynomials (TPs) is considered that involves generalization of the notion of TP and extension of the methods and results to cylindrical polynomials (CPs). On the basis of the proposed augmentation of TPs and CPs, a general approach is presented to the analysis of guided waves and resonances in electromagnetics and beyond. The method employs the known explicit forms of dispersion equations (DEs) describing eigenoscillations and normal waves in layered structures and is based on the development of the theory of generalized TPs and CPs performed in the study. The approach enables one to complete rigorous proofs of existence and determine domains of localization of the TP and CP zeros and the DE roots and validate iterative numerical solution techniques.

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  • 4.
    Shestopalov, Yury
    et al.
    Karlstad University, Karlstad, Sweden.
    Smirnov, Yury
    Penza State University, Penza, Ryssland.
    Eigenwaves in Waveguides with Dielectric Inclusions : Spectrum2014Ingår i: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 93, nr 2, s. 408-427Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We consider fundamental issues of the mathematical theory of the wave propagation in waveguides with inclusions. Analysis is performed in terms of a boundary eigenvalue problem for the Maxwell equations which is reduced to an eigenvalue problem for an operator pencil. We formulate the definition of eigenwaves and associated waves using the system of eigenvectors and associated vectors of the pencil and prove that the spectrum of normal waves forms a non-empty set of isolated points localized in a strip with at most finitely many real points.

  • 5.
    Shestopalov, Yury
    et al.
    Karlstads universitet.
    Smirnov, Yury
    Penza State University, Penza, Ryssland.
    Eigenwaves in waveguides with dielectric inclusions: completeness2014Ingår i: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 93, nr 9, s. 1824-1845Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    We formulate the definition of eigenwaves and associated waves in a nonhomogeneously filled waveguide using the system of eigenvectors and associated vectors of a pencil and prove its double completeness with a finite defect or without a defect. Then, we prove the completeness of the system of transversal components of eigenwaves and associated waves as well as the ‘mnimality’ of this system and show that this system is generally not a Schauder basis. This work is a continuation of the paper Eigenwaves in waveguides with dielectric inclusions: spectrum. Appl. Anal. 2013. doi:10.1080/00036811.2013.778980 by Y. Smirnov and Y. Shestopalov. Therefore, we omit the problem statements and all necessary basic definitions given in the previous paper.

  • 6.
    Smirnov, Yury
    et al.
    Penza State University, Penza, Ryssland.
    Smolkin, Eugen
    Penza State University, Russia.
    Shestopalov, Yury
    Högskolan i Gävle, Akademin för teknik och miljö, Avdelningen för elektroteknik, matematik och naturvetenskap, Matematik.
    Surface waves in a Goubau line filled with nonlinear anisotropic inhomogeneous medium2022Ingår i: Applicable Analysis, ISSN 0003-6811, E-ISSN 1563-504X, Vol. 10, nr 17, s. 6172-6190Artikel i tidskrift (Refereegranskat)
    Abstract [en]

    The propagation of monochromatic nonlinear symmetric hybrid surface waves in a cylindrical nonlinear anisotropic inhomogeneous metal-dielectric waveguide is considered. The physical problem is reduced to solving a transmission eigenvalue problem for a system of ordinary differential equations. Spectral parameters of the problem are propagation constants of the waveguide. The setting is reduced to a new type of nonlinear eigenvalue problem and an analytical method of its solution is developed. The propagation modes are found that have not been previously reported in the literature. For the numerical solution, a method is proposed based on solving an auxiliary Cauchy problem (a version of the shooting method). As a result of comprehensive numerical modeling, new propagation regimes are discovered.

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