On the Fredholm property of integral equations system in the problem of electromagnetic waves propagation in a graphene-coated rod 

Yuriy G. Smirnov, Doctor of physical and mathematical sciences, professor, head of the sub-department of mathematics and supercomputer modeling, Penza State University (40 Krasnaya street, Penza, Russia), E-mail: mmm@pnzgu.ru 

Background. The problem of electromagnetic waves propagation in a dielectric rod of arbitrary cross-section covered with a layer of graphene, which is considered infinitely thin, is considered. The main problem in describing the process of wave propagation in the waveguiding structure is to obtain and analyze the system of integral equations to determine propagation constants. Materials and methods. Maxwell’s equations are solved in the frequency domain. The coupling conditions contain the conductivity of graphene. In this article, we neglect the nonlinearity of graphene. The method of Green’s functions is applied. Results and conclusions. The system of integral equations for determining the propagation constants is obtained. The Fredholm property of the system and the discreteness of the spectrum of the problem are proved. 

graphene, integral equation, discreteness of the spectrum 

Smirnov Yu.G. О фредгольмовости системы интегральных уравнений в задаче о распространении электромагнитных волн в стержне, покрытом графеном. Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fiziko-matematicheskie nauki = University proceedings. Volga region. Physical and mathematical sciences. 2023;(3):74–86. (In Russ.). doi: 10.21685/2072-3040-2023-3-6