Том 47, № 2 (2024)

Previously, boundary control problems for a heat conduction equation with Dirichlet boundary condition were studied in a bounded domain. In this paper, we consider the boundary control problem for the heat conduction equation with Neumann boundary condition in a bounded one-dimensional domain. On the part of the border of the considered domain, the value of the solution with control parameter is given. Restrictions on the control are given in such a way that the average value of the solution in some part of the considered domain gets a given value. The studied initial boundary value problem is reduced to the Volterra integral equation of the first type using the method of separation of variables. It is known that the solution of Volterra’s integral equation of the first kind cannot always be shown to exist. In our work, the existence of a solution to the Volterra integral equation of the first kind is shown using the method of Laplace transform. For this, the necessary estimates for the kernel of the integral equation were found. Finally, the admissibility of the control function is proved.

The paper proposes a mathematical model of the nonlinear Van der Pol-Airy oscillator taking into account heredity. The nonlinearity of the oscillator is due to the dependence of the friction coefficient on the square of the displacement function, which is typical for the Van der Pol oscillator. Also, the natural frequency of oscillations is a function of time, which increases linearly as it increases. The latter is typical for the Airy oscillator. Heredity effects are introduced into the model equation through fractional derivatives in the Gerasimov-Caputo sense. They indicate that the oscillatory system may have memory effects that manifest themselves depending on its current state from previous ones. For the proposed mathematical model, a numerical algorithm was developed based on an explicit first-order finite-difference scheme. The numerical algorithm was implemented in a computer program in the Maple language, with the help of which the simulation results were visualized. Oscillograms and phase trajectories were constructed for various values of the model parameters. It is shown that a fractional mathematical model can have various oscillatory modes: from self-oscillatory, damped and chaotic. An interpretation of the simulation results is given.

The lithosphere physical state, including its surface layers, in a certain way characterizes the likelihood of the risk of catastrophic seismic events, indicates the degree of threat to human life and the risk of economic damage. One of the directions for assessing the lithosphere state is the analyzing result the characteristics variability of the geoacoustic emission signal at the observation point. The studies results carried out at the Institute of Cosmophysical Research and Radio Wave Propagation Far Eastern Branch of the Russian Academy of Sciences are presented. This make it possible to identify the dynamic characteristics of the geoacoustic emission signal associated with changes in the near-surface of the lithosphere. Recognition and assessment of the characteristics variability of geoacoustic emission signals on a given time scale of observations was carried out using a neural network approach. A methodology has been developed for classifying observed anomalies in the dynamics of geoacoustic emission signal characteristics.

The work carried out processing and analysis of long-term series of vertical temperature distribution in the middle atmosphere (altitude interval 15-60 km) over Tomsk, obtained on the basis of regular measurements of the lidar station of the Institute of Atmospheric Optics SB RAS for the period 2010-2023. During this period, about 1,000 individual night measurements were carried out and the corresponding total backscatter signals at a wavelength of 532 nm were accumulated and processed.. Based on the experimental data obtained, regional features of intra- and interannual variability of the thermal state of the middle atmosphere over Western Siberia were identified. For the warm period of each year (from May to September), a stable temperature distribution has been established with differences of up to several K in individual years. In spring and autumn, the difference between the average monthly temperature profiles increases to 5-10 K and in January reaches a maximum of 15 K. A fundamental difference between the vertical temperature distribution constructed from lidar data and the proposed CIRA-86 model has been revealed. From October to April, in the altitude range from 15 to 25 km, the lidar temperature profile is shifted from the model one in the negative direction, and above from 25 to 50 km, in the positive direction. The maximum negative shift in profiles is observed in December and is 16 K, and negative up to 15 K in January. A description is given of methods for analyzing a number of experimental data for the presence of values that differ significantly from the rest of the data and can lead to distortion of the results obtained.

Capillary-gravity waves significantly change the general circulation of the water surface of the World Ocean: attenuation and collapse of longer waves, gas exchange and mixing in the upper layer, kinematics of surface suspensions, and, therefore, require additional study. In the framework of a two-dimensional problem, the paper considers surface capillary-gravity waves without taking into account wind effects, but taking into account the isobaric approximation. The approach proposed by Keldysh is chosen as a quantitative basis. The object of perturbation located at a finite depth is a vortex dipole with constant curl. Two asymptotic solutions were obtained on the basis of analytical calculations: the first solution describes the profile of capillary-gravity waves located behind the vortex dipole; the second solution is the profile of capillary-gravity waves in front of the perturbation source. It is shown that the capillary component of the wave dominates in the formation of waves in front of the perturbation source, and the gravitational component dominates behind the obstacle. The relations between isobaric and barotropic effects on the free surface are qualitatively analyzed. The case for the gravitational component is considered; for this purpose, the technique of representing the wave profile using a Taylor polynomial was used.

Determining the possibilities of using the ground penetrating radar method, which is a method of shallow geophysics, in seismically active areas will increase the information content and reliability of the results, especially in engineering and geological surveys to ensure seismic-resistant construction in geodynamically active areas, which include Kamchatka. To optimize the ground penetrating radar method, it is necessary to compare and analyze the results of observations in various geological conditions. Since the ground penetrating radar method has been intensively developing in recent years, theoretical research in the field of data processing must be illustrated by numerous examples of the practical use of ground penetrating radars at various research sites. The purpose of the research is to create a scientific and methodological basis for the georadar method in relation to Kamchatka, including methods for processing and interpreting data, taking into account practical experience of application at various sites. The object of the research is the geological features and soil conditions of the territory of the Petropavlovsk-Kamchatsky city, requiring clarification and detailed study. The article describes the experience and features of working with the «Triton-M» georadar device in the Petropavlovsk-Kamchatsky city. Some results of ground penetrating radar profiling are characterized. Recommended values of measurement parameters during work are given. From a practical point of view, the most important parameters of the ground penetrating radar are the probing depth and the depth resolution. The «Triton-M» ground penetrating radar device is applicable for solving geological problems in the territory of the Petropavlovsk-Kamchatsky city, while optimally selected values of measurement parameters minimize the recording of interference and provide more accurate useful information about the probed environment. For accurate interpretation of radargrams, it is necessary to use information about the geological structure of the studied territory.