Vestnik KRAUNTS. Fiziko-matematicheskie nauki

19-26

Authors: Zh. A. Balkizov

In this paper, we study the boundary-value problem with displacement for a model inhomogeneous parabolic-hyperbolic equation of the third order. We prove the uniqueness and existence theorems for a regular solution of the problem under study. In the case when the coefficients of the problem are constant real numbers, the solution of the problem under study is written out in explicit form.

36-41

Authors: R. A. Kirzhinov

In this paper the A. A. Dezin problem analogue is considered for inhomogeneous parabolic–hyperbolic type equation of the second order. We proved the solution uniqueness of the solution to the problem under investigation. The solution representation is written out by the Green’s function method.

48-56

Authors: L. Kh. Gadzova

We consider an initial value problem for ordinary differential equation with discretely distributed fractional differentiation operator. We give the Lagrange formula, prove the existence and uniqueness theorem and construct an explicit form of solution.

67-75

Authors: R. Kh. Makaova

In this paper, we study the Tricomi problem for a third-order hyperbolic equation with degeneracy of order inside a mixed domain. The existence and uniqueness theorem for a regular solution is proved.

83-90

Authors: О. Kh. Masaeva

The existence and uniqueness of the solution of the Neumann problem is proved for the generalized Laplace equation with a fractional derivative in the upper half-plane.

98-105

Authors: V. A. Kim, R. I. Parovik

In the study of nonlinear systems, one of the important problems is the determination of the type of oscillations-periodic, quasi-periodic, random, chaotic. It is especially difficult to distinguish between quasi-periodic oscillations from chaotic and random oscillations, since quasi-periodic oscillations often have a very complex shape, visually weakly distinguishable from «random». A feature of chaotic oscillations is their high sensitivity to small changes in the initial conditions. Therefore, one of the most reliable ways of detecting chaos is to determine the rate of run-off of trajectories, which is estimated using the Lyapunov exponent spectrum. Using the construction of the spectrum of Max Lyapunov exponents, depending on the values of the control parameters, chaotic regimes of the Duffing fractal oscillator with variable power memory were found, and its phase trajectories.

116-123

Authors: O. D. Lipko

In this paper we study the conditions for the existence of chaotic and regular oscillatory regimes of the hereditary oscillator FitzHugh-Nagumo (EFN), a mathematical model for the propagation of a nerve impulse in a membrane. To achieve this goal, using the non-local explicit finite-difference scheme and Wolf’s algorithm with the Gram-Schmidt orthogonalization procedure, the Lyapunov maximum exponent spectra were constructed as a function of the values of the control parameters of the model of the EFN. The results of the calculations showed that there are spectra of maximum Lyapunov exponents both with positive values and with negative values. The results of the calculations were also confirmed with the help of oscillograms and phase trajectories, which indicates the possibility of the existence of both chaotic attractors and limit cycles.

131-139

Authors: I. Kh. Mashukov, V. A. Shapovalov, A. A. Adzhieva

With the help of the Vaisala EFM550 electric field meter, long-term studies (2010-2017) of the course of the electric field strength in the area of the city of Nalchik were carried out. The article considers the use of digital processing to improve the quality of data obtained with a network of sensors operating in the hardware and software complex in the monitoring center of the geophysical situation over the southern region of the Russian Federation at the FSBI «High-Mountain Geophysical Institute». The possibilities of applying Kalman filtering to refine the data of instrumental measurements of the electric field strength of the atmosphere are investigated. A comparative analysis of the obtained data on the course of the electric field strength before and after filtration is given.

148-157

Authors: D. A. Tvoyrdyj

The Cauchy problem for the Riccati equation with non-constant coefficients and taking into account variable power memory is proposed. Power memory is defined by the operator of a fractional derivative of a variable order generalizing the Gerasimov-Caputo derivative. In work with the help of numerical methods: the Newton method and the explicit finitedifference scheme, the solution of the proposed Cauchy problem is found, and also their calculation accuracy is determined using the Runge rule. It is shown that both methods can be used to solve the proposed Cauchy problem, but Newton’s method converges faster. Further in this work, the calculated curves and phase trajectories were constructed for a different choice of the fractional order function of the differentiation operator. It is assumed that the proposed model can be used in describing economic cyclical processes.

168-179

Authors: V. A. Shapovalov

A three-dimensional numerical model of a convective cloud is developed taking into account thermodynamic, microphysical and electrical processes. The model uses a detailed microphysics. The system of equations of the cloud model describing the time variation of the dynamic and microphysical characteristics of the cloud consists of 3 equations of motion, heat and moisture balance equations, 137 equations describing the spectrum of cloud droplets, crystals, and microbubble particles. In addition, in order for the solution to satisfy the continuity equation, it is necessary to solve the three-dimensional elliptic equation for the pressure perturbation at each time step. One of the methods widely used for solving such problems is the splitting method developed by G.I. Marchuk, an improved version of this method, the predictor scheme with a divergent corrector, was successfully used in the modeling of cumulus clouds by R. Pastushkov. The conducted studies showed that, despite the certain complexity in the implementation of this scheme, it provides the necessary stability of the count, an approximation of the second order of accuracy in space and time, and is conservative. Splitting methods for physical processes and componentwise splitting are used (locally — one-dimensional schemes). The equations of the cloud model in finite-difference form were approximated by central and directional differences for spatial variables, as well as directed time differences. The resulting algebraic system was solved by a sweep method.

184-189

Authors: M. A. Kazakov

The article suggests a constructive method for training neural networks in which neurons added just before the current epoch of training assume the main training load on the new class to ensure the stability of the network in relation to learning on new data classes. The results of computational experiments are presented