Kyiv Slavonic University

We establish necessary and sufficient conditions for the existence of solutions of an autonomous Noetherian boundary-value problem for a system of second-order ordinary differential equations in the critical case. For the construction of solutions of a nonlinear Noetherian boundary-value problem in the critical case, we propose an iterative scheme that combines the Newton–Kantorovich method and the leastsquares technique. The efficiency of the proposed method is demonstrated in the analysis of a periodic problem for a Lienard-type equation.

Means and families of operators generated by Bochner-Riesz kernels are studied. Some sharp results on their convergence are achieved. The equivalence of the approximation errors of these methods to smoothness quantities related to the Laplacian is proved.

The study is devoted to closed and transferred electromagnetic field in one-dimensional both transparent photonic crystals and crystals containing layers which absorb irradiation alternated by transparent layers. General theory of irrdiation closed inside the crystal is developed both in case transparent and adsorbing metallic photonic crystals in wide frequency range. True or absolute photonic gaps existing in the total reflection angle range are analyzed on the base of internal problem's analytical solution for 1D layered superlattices. We have investigated the frequency-angle diagrams for TM and TE photonic spectra trapped inside the whole reflection range for mesoscopic and macroscopic hierarchical 1D photonic crystals of varied topologies, geometry and materials. The effects of double refraction in binary multi-layered system and photon localization in finite periodic layered structures are investigated analytically and numerically. Energy storage and energy decay time in photonic crystals are considered with account for field amplitudes distribution inside the structure and angular band kinetics. The effect of sharp focusing of the butt-end emission for closed electromagnetic waves in 1D photonic crystals is discussed.

The system of one-dimensional photonic crystals combined with fiber waveguides, splitters and amplifiers is proposed as an optical binary signal's adder. We consider two or more angle distributed many-positional external binary signals hitting into the photonic crystal through the fiber entrances. Such kind of entrance construction due to spherical end of fiber allows external illumination to penetrate directly inside the range of total intrinsic reflection and populate photonic eigenmodes. Each component of complex signal hits into the 1D crystal at a fixed angle corresponding to mode angle at given frequency. The output sum signal is distributed onto the same fibers' angles. Physical principles and parameters of sub-picosecond infrared fiber-photonic processor are discussed.