The spectral resolution of the finite-difference theta-method for the heat conduction equation is investigated. By analogy between the frequency response of the heat conduction equation and a low pass filter, we have found an equivalent representation of the finite-difference scheme in the form of two first-order IIR filters with zero group delay. On the basis of the spectral consistency, the error estimate of the discrete filtering model is obtained. The optimal parameters of the IIR filters providing a minimum error of the frequency response within a given spectral range are found. It is remarkable that the optimal ratio of spatial and temporal steps for the theta-method coincides with the ratio provided by the filtering model with co efficients corresponding to a minimum root-mean-square error of the frequency response for the given spectral range. It is shown that the optimized scheme provides a manifold (by a factor of 5–7) reduction in the root-mean-square error of the frequency response in comparison with the 6th order accuracy theta-method. The optimal time step is a little bit larger in comparison with its value in the 6th order accuracy scheme and tends to the last one when the spectral resolution range tends to zero. The obtained results can be used to optimize discretization parameters using the finite-difference methods for the heat conduction equation.

The equations can be written as L_0u= − u∆+a(ε) δu =f which appear in different applications and are studied intensively. In this equation, δu is not determined in the classical theory of generalized functions, so one of the main objectives is to give meaning to the expression on the left-hand side of the equation, that is, it is an actual construction of the operator that corresponds to a given formal expression. This is achieved by special approximations of multiplication of the operator by the δ-function. To study equations with δ-shaped coefficients we have applied the approach, the main steps of which are: constructing the approximations of the considered expressions with operators of finite rank; finding the explicit form approximating a resolvent family; determining a resolvent limit and allocating resonance cases; describing the spectrum of the constructed limit of operators; studying the behavior of the eigenvalues of approximating operators. The purpose of this work is to find the asymptotic behavior of vector-functions for approximations, built in [2]. Thus, the main result of this work is the construction of the asymptotic behavior of the vector-functions in different cases of resonance.

Rational Fourier series were constructed by M. M. Dzhrbashian in 1956. A compact representation of their Dirichlet kernel was also found. Later V. N. Rusak introduced rational operators of Fejér, de la Vallée Poussin and Jackson type. Partial sums of rational Fourier series, operators of de la Vallée Poussin and Jackson type are widely used for finding classes of functions, for which rational approximation is better than polynomial approximation. But in our opinion, rational operators of Fejér type are still unexpolred, so it’s interesting to investigate their approximation characteristics for elementary functions. The periodic function | sin | x plays almost the same role in approximation theory as the function |x |. In this article, we have obtained some exact and asymptotic ratios for approximation of | sin x | by Fejér-type rational operators.

The problem under our consideration is to construct systems with a perturbed linear center of special form that have no more than one limit cycle in the entire phase plane for all real values of the perturbation parameter μ. To solve this problem, we have proposed a method for constructing a Dulac – Cherkas function as a second-degree polynomial with respect to a phase variable y, whose coefficients smoothly depend on the second-phase variable x and continuously depend on the parameter μ. The construction of the Dulac – Cherkas function is based on reducing the auxiliary polynomial Φ(x,y,μ) to the function Φ₀(x,μ) depending only on the variable x and the parameter μ. A regular method for such reduction is proposed. Examples of the constructed systems having a unique limit cycle in the entire phase plane are presented.

The problems of the motion of free particles in the three-dimensional Lobachevsky space are interpreted as scattering by space. The classical and quantum-mechanical cases are considered. A mechanical interpretation of parallel straight lines of the Lobachevsky space is given as the trajectories of non-interacting material points emitted from a point at infinity. Due to the properties of parallel lines in the Lobachevsky space, they can be considered as trajectories of particles scattered at an infinitely distant point. The concept of differential scattering cross sections in the horosphere element for the classical and quantum-mechanical problems is introduced. An analytical expression for the differential cross section in the quantum-mechanical problem is obtained. To derive this expression, we used the solutions of the Schrödinger equation in horospherical coordinates. It is noted that some part of a horosphere is a secant beam of parallel trajectories, can be considered as a model of a two-dimensional flat universe in the three-dimensional space with curvature – Lobachevsky space.

The generalized Schrődinger equation for a scalar Cox particle is studied in the presence of a magnetic field on the background of Lobachevsky space. Separation of variables is performed. An equation describing the particle motion along the z axis appears to be much more complex than that when describing the Cox particle in Minkowski space. The form of the effective potential curve says that we have a quantum-mechanical problem of tunneling type. The derived equation has 6 regular singular points. Singular points 0 and 1 of the derived equation correspond to the physical domains z = ±∞. The solutions of the equation are constructed with the help of power series. Convergence of the series is examined by the Poincare – Perrone method. These series are convergent within the whole physical domain z ∈ (-∞,+∞). When considering an ordinary particle in Lobachevsky space, a simpler problem of tunneling type arises, which is exactly solved in terms of hypergeometric functions.

In the article, the Euler – Lagrange equations, which describe first-order phase transition dynamics in a con-figuration Finsler space of a Langmuir monolayer, have been obtained. An approximate method for analysis of the equations has been developed. The method is based on a combination of analytical and numerical calculations using the zero-order approximation with a fixed relaxation time and the more exact approximation with a model distribution of relaxation times. Heterogeneous dynamics of the system has been demonstrated. Such dynamics corresponds to the monolayer metastable state with different relaxation times of phase nuclei. The relaxation time distribution has a maximum and a maximum height depends on a monolayer compression rate. The increase of the maximum height at enhancement of a compression rate is accompanied by an explicit plateau of the isotherm that displays the characteristic behavior of the monolayer isotherm in the region of phase transition. The dynamics of a two-dimensional phase transition has been numerically studied at the compression rate as being sufficiently low, and a comparative analysis of the system behavior at two approximations (the approximation of fixed relaxation time and the approximation of model distribution of relaxation times) has been made. It has been found that the presence of phase nuclei with different relaxation times causes an effective centrifugal force, the magnitude of which depends on the gradient of electrocapillary forces.

The features of acousto-optic diffraction of a Bessel light beam (BLB) on a Bessel acoustic beam (BAB) in transversely anisotropic crystals are investigated. The scheme of acousto-optic (AO) interaction is considered when the incident TH-polarized BLB propagating along the optical c axis of an optically uniaxial crystal is excited in the crystal of the TE- po-la rized BLB due to the process of anisotropic TH→TE diffraction. The diffraction problem is solved for transversely isotropic crystals, whose cylindrical symmetry is fully consistent with the symmetry of both BLB and BAB propagating along the optical axis and the AO interaction occurs without distortion of the spatial structure of beams. It is shown that in the process of backward acousto-optic scattering, the use of Bessel light beams with a large cone angle makes it possible to significantly reduce the frequency of the acoustic wave necessary for satisfying the longitudinal synchronism condition (values of less than 1 GHz). It is established that the efficiency of the AO interaction of the BLB and the BAB is determined not only by the intensity of the acoustic field, longitudinal wave detuning and the interaction length, but also by the period of transverse oscillations of Bessel beams. These oscillations determine the value of the overlap integrals and, consequently, the effective AO parameters. When the conditions of longitudinal and transverse synchronisms are realized, it is possible to achieve the high diffraction efficiency close to unity. The angular width of transverse synchronism is equal to about 0.5 mrad and increases with increasing acoustic power. It is shown that when the BLB is diffracted, the order of its phase dislocation changes by a value that is equal to the order of BAB phase dislocation. Because of the narrow angular spectrum of a scattered field, the back ward AO diffraction of Bessel light beams is promising for the development of low-frequency AO filters and spectrum analyzers. The property of self-reconstruction of the transverse profile of an optical field is promising for applications of Bessel light beams in defectroscopy.

Numerical modeling of the electronic structure of a quantum dot, induced by an electric field of a nanosized disc-shaped gate, is carried out in the presence of external magnetic field. The dependences of an electronic energy spectrum on electric and magnetic fields are calculated using the finite element method. It has been found that a series of anti-crossing points for electronic levels takes place at relatively small magnetic fields. The existence of groups of close-energy levels (electronic shells) has been found. It has been shown that despite the essential distinction of the gate potential from the parabolic one, a model of a near-surface anisotropic harmonic oscillator can be effectively used for a qualitative description of the electronic structure of the electrically induced quantum dot. With the use of this model, the evolution of energy spectrum and wave function structure with magnetic and electric fields is described. In particular, the anisotropic oscillator model allows to predict anti-crossing points of electronic levels in external fields, as well as quasi-degeneracy of states having different values of the angular momentum projection.

Graphene has semiconductor properties in several layers structures (heterostructure). Zinc oxide/graphene (ZnO/graphene) and zinc sulfide/graphene (ZnS/graphene) have been studied by quantum-mechanical simulation using the VASP software. The structural properties of a typical layered material (black phosphorus) have been simulated by different electron density functionals. Thus, the DFT electron density functional implemented in the VASP software was chosen to take into account the Van der Waals forces. Interlayer distances have been determined to study systems by a suitable electron density functional (DFT-D2). The distance is 3.1 Å for black phosphorus, 3.16 Å (ZnO/graphene) and 3.45 Å (ZnS/graphene) for heterostructures. Energy band structures have been calculated. Thus, the influence of a zinc-containing material on the graphene energy band structure has been registered. A band gap has been observed in ZnS/graphene (0.35 eV), but it is absent in ZnO/graphene. Taking into account that the DFT method underestimates the band gap width, this value may be larger in experimental works.

Using the same n⁺–p diode structures, the effect of injection of minority charge carriers on the annealing of various interstitial defects has been studied in silicon irradiated with α-particles. It has been found that the self-interstitial silicon atoms (Sii) possess the highest sensitivity to forward current injection. At a liquid nitrogen temperature and a forward current density of 10–20 mA/cm², the time constant for Sii annealing is about a few seconds. To activate the interstitial boron atoms at Т ≤ 140 K, substantially higher direct current densities are required (≥ 100 mA/cm²). However, it has been found that the forward current injection not only enhances, but even causes the retardation of interstitial carbon annealing. It is suggested that only the reactions of interstitial atoms, characterized by a strong electron-phonon coupling, can be enhanced by recombination processes.

The parameters for a temperature dependence of the band gap in the temperature range of 10–300 K and for a temperature dependence of spin-orbit splitting energy were obtained using the experimental emission spectra for LEDs based on Ga_1–xIn_xAs_ySb_1–y/AlGaAsSb heterostructures. For temperatures of 10–80 K, the rise of the emission intensity is limited by the Auger recombination process, for which the recombination energy of an electron-hole pair is transferred to a hole with its transition to the spin-orbital band. With an increase in a temperature of more than 100 K, there is a rise of the coefficient of the Auger recombination process, for which the energy released by the recombination of an electron-hole pair excites another electron in the conduction band. The sum of these processes results in quenching the LED emission with increasing temperature over 150 K.