A. N. Bogolyubov

An approach to introducing fractional integro-differentiation in classical electrodynamics

Analogs for Maxwell’s equations with fractional derivatives are obtained using the concepts of an effective current and the velocity of a charged particle in a medium. The calibration invariance is considered and a diffusion-wave equation is found and analyzed for scalar and vector potentials. It is shown that the stochastic nature of charged particle motion in a medium influences the dynamics of an electromagnetic field.

Calculation of a parallel-plate waveguide with a chiral insert by the mixed finite element method

A parallel-plate waveguide with metallic boundaries, containing an insert of a chiral substance, is considered. The field distribution inside the insert is studied when the system is excited by the guide’s normal wave incident on the insert. The problem is considered in the full vector formulation. The waveguide is calculated by the mixed finite element method, which makes it possible to avoid spurious modes (so-called spirits).

Mathematical simulation of an irregular waveguide with reentering edges

A mathematical model is proposed to study an irregular waveguide with reentering edges. Theoretical estimates for the behavior of the solution near the reentrant corners are used to estimate the rate of convergence of the numerical solution to the exact one. The mode structure of the waveguide field is analyzed.

Mathematical modeling of waveguides with fractal insets

The paper is devoted to mathematical modeling of rectangular dielectric waveguides with local periodic and fractal insets. A three-dimensional scalar boundary-value problem for the Helmholtz equation is considered. The solution is obtained numerically by means of incomplete Galerkin method and finite-difference techniques. The transmission spectra for one-, two- and three-dimensional periodic and fractal insets are computed and compared with each other. Practical applications are suggested.

Excitation of electromagnetic oscillations in a domain with chiral filling

The problem of excitation of electromagnetic oscillations by a given distribution of charges and currents in a domain with inhomogeneous chiral filling is examined. The domain in which the problem is considered may either be finite with a perfectly conducting boundary surface or be the complement of a perfectly conducting bounded body. A special functional space is defined on which a generalized initial-boundary value problem is formulated. The Galerkin method is used to prove the existence and uniqueness of a weak solution of this problem.

Behavior of solutions to elliptic boundary value problems in a neighborhood of corner points of discontinuity lines of the coefficients

A representation of the solution to an elliptic boundary value problem in the vicinity of a corner point on the discontinuity line of the coefficient of the higher order derivative is constructed. The study is based on the method of additive separation of singularities proposed by Kondrat’ev.

Mathematical modeling of waveguide transitions

Results of simulation of waveguide transitions performed at the Department of Mathematics of the Faculty of Physics, Moscow State University, are presented. Synthesis of waveguide transitions is based on the incomplete Galerkin method and the finite-difference technique formulated in the direct and projection forms (the finite-element method) and involves the Tikhonov regularization method.

The electromagnetic initial boundary value problem in a domain with a chiral filling

The problem of the excitation of electromagnetic oscillations by given charge and current distributions in a domain with a nonhomogeneous chiral filling is investigated. The domain where the problem is considered may be both a finite one bounded by an ideally conducting surface and an infinite supplement to an ideally conducting bounded object. The initial boundary value problem is shown to arise, for which the generalized formulation in a special functional space is given. The existence of a unique weak solution to the problem is proven using the Galerkin method.

A study of the propagation of electromagnetic pulses through photonic crystal structures

A mathematical model for the study of two- and three-dimensional photonic crystals and of waveguide systems based on them was designed and applied. The numerical algorithm for the construction of the solution was built using such numerical methods as the method of finite differences in the time domain (FDTD), the TS/SF method, and the method of a perfectly matched layer (PML), which makes it possible to apply the algorithm under consideration to the study of other waveguide systems as well. The results of the simulation of particular waveguide systems are presented.

Optical diffraction on fractal lattices

The basic properties of fractals and fractal objects are described. Papers devoted to diffraction on fractal lattices are surveyed. The optical Fourier transform is considered as a method for determining the fractal dimension of a diffraction lattice.