System of singular integral equations in the problem of electromagnetic oscillations of a graphene-coated dielectric ball 

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
Oleg V. Kondyrev, Postgraduate student, Penza State University (40 Krasnaya street, Penza, Russia) 

Background. Boundary value transmission problems for a system of Maxwell's equations are fundamental in electrodynamics. Recently, there has been interest in problems involving the presence of a thin layer of graphene on the surface, which changes the coupling conditions. The purpose of the study is to obtain a system of integral equations for the vector boundary value problem of electromagnetic oscillations of a dielectric ball coated with graphene in a spherical coordinate system. Materials and methods. Using the Stratton-Chu formula, the components of the field are expressed in terms of surface currents and transmission conditions are used to obtain singular integral equations on the surface of the ball. Results. A system of singular integral equations with 4 unknown scalar functions on the surface of the sphere is obtained. The properties of the system of integral equations are studied. Conclusions. The resulting system of singular integral equations can be solved numerically by known methods, for example, the Galerkin method or the collocation method. 

singular integral equations, Kerr nonlinearity, Maxwell equations, dielectric body, graphene 

Smirnov Yu.G., Kondyrev O.V. System of singular integral equations in the problem of electromagnetic oscillations of a graphene-coated dielectric ball. Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fiziko-matematicheskie nauki = University proceedings. Volga region. Physical and mathematical sciences. 2024;(3):3–17. (In Russ.). doi: 10.21685/2072-3040-2024-3-1