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The paper analyzes the methods of studying the process of interaction of soil environments with the tillage tools of soil processing machines. The mathematical methods of numerical modeling are considered in detail, which make it possible to overcome the disadvantages of analytical and empirical approaches. A classification and overview of the possibilities the continuous (FEM — finite element method, CFD — computational fluid dynamics) and discrete (DEM — discrete element method, SPH — hydrodynamics of smoothed particles) numerical methods is presented. Based on the discrete element method, a mathematical model has been developed that represents the soil in the form of a set of interacting small spherical elements. The working surfaces of the tillage tool are presented in the framework of the finite element approximation in the form of a combination of many elementary triangles. The model calculates the movement of soil elements under the action of contact forces of soil elements with each other and with the working surfaces of the tillage tool (elastic forces, dry and viscous friction forces). This makes it possible to assess the influence of the geometric parameters of the tillage tools, technological parameters of the process and soil parameters on the geometric indicators of soil displacement, indicators of the self-installation of tools, power loads, quality indicators of loosening and spatial distribution of indicators. A total of 22 indicators were investigated (or the distribution of the indicator in space). This makes it possible to reproduce changes in the state of the system of elements of the soil (soil cultivation process) and determine the total mechanical effect of the elements on the moving tillage tools of the implement. A demonstration of the capabilities of the mathematical model is given by the example of a study of soil cultivation with a disk cultivator battery. In the computer experiment, a virtual soil channel of 5×1.4 m in size and a 3D model of a disk cultivator battery were used. The radius of the soil particles was taken to be 18 mm, the speed of the tillage tool was 1 m/s, the total simulation time was 5 s. The processing depth was 10 cm at angles of attack of 10, 15, 20, 25 and 30°. The verification of the reliability of the simulation results was carried out on a laboratory stand for volumetric dynamometry by examining a full-scale sample, made in full accordance with the investigated 3D-model. The control was carried out according to three components of the traction resistance vector: $F_x$, $F_y$ and $F_z$. Comparison of the data obtained experimentally with the simulation data showed that the discrepancy is not more than 22.2%, while in all cases the maximum discrepancy was observed at angles of attack of the disk battery of 30°. Good consistency of data on three key power parameters confirms the reliability of the whole complex of studied indicators.

The paper reviews possibilities to predict buckling of thin cylindrical shells with non-destructive techniques during operation. It studies shallow shells made of high strength materials. Such structures are known for surface displacements exceeding the thickness of the elements. In the explored shells relaxation oscillations of significant amplitude can be generated even under relatively low internal stresses. The problem of the cylindrical shell oscillation is mechanically and mathematically modeled in a simplified form by conversion into an ordinary differential equation. To create the model, the researches of many authors were used who studied the geometry of the surface formed after buckling (postbuckling behavior). The nonlinear ordinary differential equation for the oscillating shell matches the well-known Duffing equation. It is important that there is a small parameter before the second time derivative in the Duffing equation. The latter circumstance enables making a detailed analysis of the obtained equation and describing the physical phenomena — relaxation oscillations — that are unique to thin high-strength shells.

It is shown that harmonic oscillations of the shell around the equilibrium position and stable relaxation oscillations are defined by the bifurcation point of the solutions to the Duffing equation. This is the first point in the Feigenbaum sequence to convert the stable periodic motions into dynamic chaos. The amplitude and the period of relaxation oscillations are calculated based on the physical properties and the level of internal stresses within the shell. Two cases of loading are reviewed: compression along generating elements and external pressure.

It is highlighted that if external forces vary in time according to the harmonic law, the periodic oscillation of the shell (nonlinear resonance) is a combination of slow and stick-slip movements. Since the amplitude and the frequency of the oscillations are known, this fact enables proposing an experimental facility for prediction of the shell buckling with non-destructive techniques. The following requirement is set as a safety factor: maximum load combinations must not cause displacements exceeding specified limits. Based on the results of the experimental measurements a formula is obtained to estimate safety against buckling (safety factor) of the structure.

In this work, we consider a special approximation of the general perturbation formula for the electromagnetic field by a set of electrically small inhomogeneities located in the domain of interest. The problem considered in this paper arises in many applications of technical electrodynamics, radar technologies and subsurface remote sensing. In the general case, it is formulated as follows: at some point in the perturbed domain, it is necessary to determine the amplitude of the electromagnetic field. The perturbation of electromagnetic waves is caused by a set of electrically small scatterers distributed in space. The source of electromagnetic waves is also located in perturbed domain. The problem is solved by introducing the far field approximation and through the formulation for the scatterer radar cross section value. This, in turn, allows one to significantly speed up the calculation process of the perturbed electromagnetic field by a set of a spherical inhomogeneities identical to each other with arbitrary electrophysical parameters. In this paper, we consider only the direct scattering problem; therefore, all parameters of the scatterers are known. In this context, it may be argued that the formulation corresponds to the well-posed problem and does not imply the solution of the integral equation in the generalized formula. One of the features of the proposed algorithm is the allocation of a characteristic plane at the domain boundary. All points of observation of the state of the system belong to this plane. Set of the scatterers is located inside the observation region, which is formed by this surface. The approximation is tested by comparing the results obtained with the solution of the general formula method for the perturbation of the electromagnetic field. This approach, among other things, allows one to remove a number of restrictions on the general perturbation formula for E-filed analysis.

Corrosion damage to metals and alloys is one of the main problems of strength and durability of metal structures and products operated in contact with chemically aggressive environments. Recently, there has been a growing interest in computer modeling of the evolution of corrosion damage, especially pitting corrosion, for a deeper understanding of the corrosion process, its impact on the morphology, physical and chemical properties of the surface and mechanical strength of the material. This is mainly due to the complexity of analytical and high cost of experimental in situ studies of real corrosion processes. However, the computing power of modern computers allows you to calculate corrosion with high accuracy only on relatively small areas of the surface. Therefore, the development of new mathematical models that allow calculating large areas for predicting the evolution of corrosion damage to metals is currently an urgent problem.

In this paper, the evolution of the corrosion front in the interaction of a polycrystalline metal surface with a liquid aggressive medium was studied using a computer model based on a cellular automat. A distinctive feature of the model is the specification of the solid body structure in the form of Voronoi polygons used for modeling polycrystalline alloys. Corrosion destruction was performed by setting the probability function of the transition between cells of the cellular automaton. It was taken into account that the corrosion strength of the grains varies due to crystallographic anisotropy. It is shown that this leads to the formation of a rough phase boundary during the corrosion process. Reducing the concentration of active particles in a solution of an aggressive medium during a chemical reaction leads to corrosion attenuation in a finite number of calculation iterations. It is established that the final morphology of the phase boundary has a fractal structure with a dimension of 1.323 ± 0.002 close to the dimension of the gradient percolation front, which is in good agreement with the fractal dimension of the etching front of a polycrystalline aluminum-magnesium alloy AlMg6 with a concentrated solution of hydrochloric acid. It is shown that corrosion of a polycrystalline metal in a liquid aggressive medium is a new example of a topochemical process, the kinetics of which is described by the Kolmogorov–Johnson– Meil–Avrami theory.

There are conditions of uneven combustion for tubular powder elements of large elongation used in artillery propelling charges. Here it is necessary to consider in parallel the processes of combustion and movement of powder gases inside and outside the channels of the powder tubes. Without this, it is impossible to adequately formulate and solve the problems of ignition, erosive combustion and stress-strain state of tubular powder elements in the shot process. The paper presents a physical and mathematical formulation of the main problem of the internal ballistics of an artillery shot for a charge consisting of a set of powder tubes. Combustion and movement of a bundle of powder tubes along the barrel channel is modeled by an equivalent tubular charge of all-round combustion. The end and cross-sectional areas of the channel of such a charge (equivalent tube) are equal to the sum of the areas of the ends and cross-sections of the channels of the powder tubes, respectively. The combustion surface of the channel is equal to the sum of the inner surfaces of the tubes in the bundle. The outer combustion surface of the equivalent tube is equal to the sum of the outer surfaces of the tubes in the bundle. It is assumed that the equivalent tube moves along the axis of the bore. The speed of motion of an equivalent tubular charge and its current position are determined from Newton’s second law. To calculate the flow parameters, we used two-dimensional axisymmetric equations of gas dynamics, for the solution of which an axisymmetric orthogonalized difference mesh is constructed, which adapts to the flow conditions. When the tube moves and burns, the difference grid is rearranged taking into account the changing regions of integration. The control volume method is used for the numerical solution of the system of gas-dynamic equations. The gas parameters at the boundaries of the control volumes are determined using a self-similar solution to the Godunov problem of decay for an arbitrary discontinuity. The developed technique was used to calculate the internal ballistics parameters of an artillery shot. This approach is considered for the first time and allows a new approach to the design of tubular artillery charges, since it allows obtaining the necessary information in the form of fields of velocity and pressure of powder gases for calculating the process of gradual ignition, unsteady erosive combustion, stress-strain state and strength of powder elements during the shot. The time dependences of the parameters of the internal ballistics process and the distribution of the main parameters of the flow of combustion products at different times are presented.

Migration has a significant impact on the shaping of the demographic structure of the territories population, the state of regional and local labour markets. As a rule, rapid change in the working-age population of any territory due to migration processes results in an imbalance in supply and demand on labour markets and a change in the demographic structure of the population. Migration is also to a large extent a reflection of socio-economic processes taking place in the society. Hence, the issues related to the study of migration factors, the direction, intensity and structure of migration flows, and the prediction of their magnitude are becoming topical issues these days.

Mathematical tools are often used to analyze, predict migration processes and assess their consequences, allowing for essentially accurate modelling of migration processes for different territories on the basis of the available statistical data. In recent years, quite a number of scientific papers on modelling internal and external migration flows using mathematical methods have appeared both in Russia and in foreign countries in recent years. Consequently, there has been a need to systematize the currently most commonly used methods and tools applied in migration modelling to form a coherent picture of the main trends and research directions in this field.

The presented review considers the main approaches to migration modelling and the main components of migration modelling methodology, i. e. stages, methods, models and model classification. Their comparative analysis was also conducted and general recommendations on the choice of mathematical tools for modelling were developed. The review contains two sections: migration modelling methods and migration models. The first section describes the main methods used in the model development process — econometric, cellular automata, system-dynamic, probabilistic, balance, optimization and cluster analysis. Based on the analysis of modern domestic and foreign publications on migration, the most common classes of models — regression, agent-based, simulation, optimization, probabilistic, balance, dynamic and combined — were identified and described. The features, advantages and disadvantages of different types of migration process models were considered.

The paper is devoted to the secondary use of spectrum in telecommunication networks. It is emphasized that one of the solutions to this problem is the use of cognitive radio technologies and dynamic spectrum access for the successful functioning of which a large amount of information is required, including the parameters of base stations and network subscribers. Storage and processing of information should be carried out using a radio environment map, which is a spatio-temporal database of all activity in the network and allows you to determine the frequencies available for use at a given time. The paper presents a two-level model for forming a map of the radio environment of a cellular communication system LTE, in which the local and global levels are highlighted, which is described by the following parameters: a set of frequencies, signal attenuation, signal propagation map, grid step, current time count. The key objects of the model are the base station and the subscriber unit. The main parameters of the base station include: name, identifier, cell coordinates, range number, radiation power, numbers of connected subscriber devices, dedicated resource blocks. For subscriber devices, the following parameters are used: name, identifier, location, current coordinates of the device cell, base station identifier, frequency range, numbers of resource blocks for communication with the station, radiation power, data transmission status, list of numbers of the nearest stations, schedules movement and communication sessions of devices. An algorithm for the implementation of the model is presented, taking into account the scenarios of movement and communication sessions of subscriber devices. A method for calculating a map of the radio environment at a point on a coordinate grid, taking into account losses during the propagation of radio signals from emitting devices, is presented. The software implementation of the model is performed using the MatLab package. The approaches are described that allow to increase the speed of its work. In the simulation, the choice of parameters was carried out taking into account the data of the existing communication systems and the economy of computing resources. The experimental results of the algorithm for the formation of a radio environment map are demonstrated, confirming the correctness of the developed model.

Since 1950s the field of city transport modelling has progressed rapidly. The first equilibrium distribution models of traffic flow appeared. The most popular model (which is still being widely used) was the Beckmann model, based on the two Wardrop principles. The core of the model could be briefly described as the search for the Nash equilibrium in a population demand game, in which losses of agents (drivers) are calculated based on the chosen path and demands of this path with correspondences being fixed. The demands (costs) of a path are calculated as the sum of the demands of different path segments (graph edges), that are included in the path. The costs of an edge (edge travel time) are determined by the amount of traffic on this edge (more traffic means larger travel time). The flow on a graph edge is determined by the sum of flows over all paths passing through the given edge. Thus, the cost of traveling along a path is determined not only by the choice of the path, but also by the paths other drivers have chosen. Thus, it is a standard game theory task. The way cost functions are constructed allows us to narrow the search for equilibrium to solving an optimization problem (game is potential in this case). If the cost functions are monotone and non-decreasing, the optimization problem is convex. Actually, different assumptions about the cost functions form different models. The most popular model is based on the BPR cost function. Such functions are massively used in calculations of real cities. However, in the beginning of the XXI century, Yu. E. Nesterov and A. de Palma showed that Beckmann-type models have serious weak points. Those could be fixed using the stable dynamics model, as it was called by the authors. The search for equilibrium here could be also reduced to an optimization problem, moreover, the problem of linear programming. In 2013, A.V.Gasnikov discovered that the stable dynamics model can be obtained by a passage to the limit in the Beckmann model. However, it was made only for several practically important, but still special cases. Generally, the question if this passage to the limit is possible remains open. In this paper, we provide the justification of the possibility of the above-mentioned passage to the limit in the general case, when the cost function for traveling along the edge as a function of the flow along the edge degenerates into a function equal to fixed costs until the capacity is reached and it is equal to plus infinity when the capacity is exceeded.

Rugose spiraling whitefly (RSW) is one of the major pests which affects the coconut trees. It feeds on the tree by sucking up the water content as well as the essential nutrients from leaves. It also forms sooty mold in leaves due to which the process of photosynthesis is inhibited. Biocontrol of pest is harmless for trees and crops. The experimental results in literature reveal that Pseudomallada astur is a potential predator for this pest. We investigate the dynamics of predator, Pseudomallada astur’s interaction with rugose spiralling whitefly, Aleurodicus rugioperculatus in coconut trees using a mathematical model. In this system of ordinary differential equation, the pest-predator interaction is modeled using Holling type III functional response. The parametric values are calculated from the experimental results and are tabulated. An approximate analytical solution for the system has been derived. The homotopy analysis method proves to be a suitable method for creating solutions that are valid even for moderate to large parameter values, hence we employ the same to solve this nonlinear model. The $\hbar$-curves, which give the admissible region of $\hbar$, are provided to validate the region of convergence. We have derived the approximate solution at fifth order and stopped at this order since we obtain a more approximate solution in this iteration. Numerical simulation is obtained through MATLAB. The analytical results are compared with numerical simulation and are found to be in good agreement. The biological interpretation of figures implies that the use of a predator reduces the whitefly’s growth to a greater extent.

Technical carbon (soot) is a product obtained by thermal decomposition (pyrolysis) of hydrocarbons (usually oil) in a stream of heat carrier gas. Technical carbon is widely used as a reinforcing component in the production of rubber and plastic masses. Tire production uses 70% of all carbon produced. In furnace carbon production, the liquid hydrocarbon feedstock is injected into the natural gas combustion product stream through nozzles. The raw material is atomized and vaporized with further pyrolysis. It is important for the raw material to be completely evaporated before the pyrolysis process starts, otherwise coke, that contaminates the product, will be produced. It is impossible to operate without mathematical modeling of the process itself in order to improve the carbon production technology, in particular, to provide the complete evaporation of the raw material prior to the pyrolysis process. Mathematical modelling is the most important way to obtain the most complete and detailed information about the peculiarities of reactor operation.

A three-dimensional mathematical model and calculation method for raw material atomization and evaporation in the thermal gas flow are being developed in the FlowVision software package PC. Water is selected as a raw material to work out the modeling technique. The working substances in the reactor chamber are the combustion products of natural gas. The motion of raw material droplets and evaporation in the gas stream are modeled in the framework of the Eulerian approach of interaction between dispersed and continuous media. The simulation results of raw materials atomization and evaporation in a real reactor for technical carbon production are presented. Numerical method allows to determine an important atomization characteristic: average Sauter diameter. That parameter could be defined from distribution of droplets of raw material at each time of spray forming.