THEORETICAL AND EXPERIMENTAL PHYSICS
An exact solution for the monomolecular adsorption isotherm on the surface with linear energetic heterogeneity has been obtained. It is shown that the interpretation of numerous experimental data that were earlier associated with a surface energy heterogeneity (the Temkin adsorption model) may require an extensive revision because of the existence of lengthy linear segments of an isotherm in semilogarithmic coordinates. The new properties of the isotherm are studied and the asymptotic and physical limiting cases have been considered. It has been shown that the surface energetic heterogeneity in the case of the polylogarithmic adsorption model is represented by two parameters: the energy dispersion, which is identical to the Temkin model heterogeneity parameter, and energy gradient. Depending on the sign of the latter, predatory surfaces and are distinguished. For predatory surfaces, the greater the adsorption heat of the surface area, the larger the number of such surface domains. For temperate surfaces, the greater adsorption heat is inherent in fewer surface domains. Physically, the greater adsorption gradient “pushes” the system in the same direction as the greater energy dispersion but less pronouncedly. Similar to the Temkin adsorption model, a simplified form of the polylogarithmic isotherm is obtained for the case of “medium coverage” (the term by Temkin).
TECHNICAL PHYSICS
DIFFERENTIAL EQUATIONS AND DYNAMIC SYSTEMS
A popular mathematical susceptible–infectious–recovered (SIR) model is used to describe the spread of the coronavirus. It is shown that using the first two integrals of the nonlinear system of three differential equations transforms it to one autonomous first-order differential equation with separable variables and two algebraic equations for calculating infected and recovered people with coronavirus. A feature of the reduced differential equation is that it is one-parameter model and its behavior is determined by a dimensionless parameter δ = β/(αN) depending on the transmission coefficient α, on the coefficient for the characterization of recovery β, and on the amount of contacting community N. It is demonstrated that the general solution of the SIR model can be represented in the form of quadrature. The influence of the dimensionless parameter δ and the influence of the infected patients on the characteristics describing the spread of coronavirus are investigated. Asymptotic dependences are given for the number of people that can become ill after contacts with infected people S(t), the number of ill peoples I(t), and the number of recovered an dead peoples R(t) depending on the initial number of infected people and the dimensionless parameter of the mathematical model. The results can be useful for considering the spread of COVID-19.
The nonlinear fourth-order partial differential equation is considered with power and nonlocal nonlinearities. This equation is used to describe the propagation of pulses in an optical fiber. Since the Cauchy problem for this equation cannot be solved by the inverse scattering transform method, the reduction of this partial differential equation to an ordinary differential equation (ODE) is considered. To construct the reduction, the traveling wave variables are used. Using a traveling wave solution, a system of ODEs is obtained composed of the imaginary and real parts of the equation. The Painlevé test is performed to check the integrability of this reduction. It is established that the constructed system of ODEs does not have the Painlevé property. Using the Fuchs indices obtained during the second step of the Painlevé test, the values of traveling wave velocity is found at which the model is simplified. In this case, only one four-order equation remains in the system of ODEs, and the simplest equation method is used to find exact solutions of it. As a result, solutions expressed in terms of the Jacobi elliptic function and the exponential function are constructed. The found exact solutions have two arbitrary constants and have the form of periodic and solitary waves.
A number of simple, but quite efficient, methods for constructing exact solutions of nonlinear partial differential equations that do not require special training and require a small amount of intermediate calculations are described. These methods are based on the following two main ideas: (i) simple exact solutions can serve as the basis for constructing more complex solutions of the equations under consideration; (ii) exact solutions of some equations can serve as the basis for constructing solutions of more complex equations. In particular, a method for constructing complex solutions based on simple solutions using translation and scaling transformations is proposed. It is shown that quite complex solutions can be obtain in some cases by adding terms to simpler solutions. Situations where a more complex composite solution can be constructed using similar simple solutions (nonlinear superposition of solutions) are considered. A method for constructing complex exact solutions of linear equations by introducing a complex parameter into more simple solutions is described. The efficiency of the proposed methods is illustrated by a large number of particular examples. Nonlinear heat conduction equations, reaction–diffusion equations, nonlinear wave equations, equations of motion in porous media, hydrodynamic boundary layer equations, equations of motion of a liquid film, gas dynamics equations, Navier–Stokes equations, etc. are considered. In addition to exact solutions of ordinary partial differential equations, some exact solutions of nonlinear functional-differential equations of the pantograph type with partial derivatives that, in addition to the required function, also contain functions with stretching or shrinking independent variables are described. The principle of analogy is formulated, which makes it possible to efficiently construct exact solutions of such functional-differential equations.
The mathematical model for describing the propagation of pulses in a nonlinear optical fiber with Bragg gratings is considered. The analytical properties of the model of wave propagation in the forward and backward directions in fiber Bragg gratings, described by coupled generalized nonlinear Schrödinger equations with qubic–quintic–septic nonlinearities, are studied. Using the traveling wave variables, the transition to the system of four ordinary differential equations obtained for the real and imaginary parts of the original system of equations is carried out. The compatibility condition for two linear differential equations of the system under study is presented. For two nonlinear differential equations of the system under study, first integrals are obtained, as well as constraints on the parameters for which the system does not contain fractional powers, and compatibility conditions under which the system has a general solution. Under the found constraints on the parameters of the model, the solution of the studied system of four ordinary differential equations is presented. The solution is given in the form of an optical soliton for coupled nonlinear partial differential equations of the Schrödinger type with nonlinearities of the third, fifth, and seventh degrees. The found solution is illustrated for different values of the parameters.
The generalized nonlinear Duffing equation, which is obtained from the equation for description of the pulse propagation in optical fibers using traveling wave variables, is considered. An ordinary second-order differential equation is written in the form of a dynamical system. For the generalized Duffing equation without external force, stationary points are found and their stability is investigated. The results of the study are presented in the table, where the type of stability is indicated for each of the three equilibrium points depending on the equation parameter. Using the presented Hamiltonian of the considered system of equations, phase portraits of the generalized Duffing equation are constructed excluding perturbation. The considered dynamical system in the presence of a periodic external force is numerically analyzed. For different amplitudes of the driving force, Poincaré sections of the dynamical system are constructed for two different values of the parameter. It is shown that with an increase in the amplitude of the perturbing force, the periodic trajectories of the solution of the system are destroyed and the area of the region of chaotic dynamics of the equations increases. According to Benettin’s algorithm, the senior Lyapunov exponent of the dynamical system is calculated as a function of the amplitude of the driving force. It has been found that with an increase in the amplitude, the senior Lyapunov exponent increases, and as a consequence, the exponential divergence of the trajectories increases, which agrees with the previously obtained Poincaré mappings.
A system of nonlinear partial differential equations of the fourth order with nonlocal nonlinearity is investigated. The system of equations describes the propagation of two waves in an optical fiber. The Cauchy problem for the considered system is unsolvable by the inverse scattering method. Traveling wave variables are used to transform the system of partial differential equations to a system of ordinary differential equations of the fourth order. The simplest equation method is applied to seed exact solutions in the form of solitary waves. The transformed system of ordinary differential equations consists of four equations, corresponding to the real and imaginary parts of the initial system. Constraints on the parameters of the initial model are found. The system of differential equations corresponding to the real parts is considered together with the determined constraints. Compatibility conditions for the system are found. One of the equations corresponding to the real part is studied taking into account the constraints for the parameters obtained from compatibility conditions. Exact solutions in the form of solitary waves are found. Graphs of the solutions obtained at different parameter values of the initial system of equations with the determined constraints are constructed. The influence of the mathematical model parameters on the behavior of solutions is analyzed.
AUTOMATION AND ELECTRONICS
A physical prototype of a universal gas analyzer has been presented for online measuring the concentration of several gases (methane, propane, butane, carbon monoxide, hydrogen). Software for an Android smartphone has also been developed in order to perform the visualization and management of data received from the gas analyzer. The smartphone is used to display information, to control the modes of displaying information on the screen, to determine the location of the user of the gas analyzer, and to send operational information to a remote server. At the moment, we can visually observe changes in the concentration of detected gases, view the measurement locations on maps, and manage the program in a convenient interface. The proposed prototype has compact controls, a built-in power supply, which facilitates the operation of the device. The Central computing part of the device is an ATmega 328P microcontroller, which is chosen due to its availability, versatility, and the ability to reprogram the outputs. The design of the device provides a connector for easy reprogramming. The measuring functionality of the device is determined by a set of sensors connected to the microcontroller, and does not depend on the smartphone model. Each sensor can be replaced owing to the modularity principle. The proposed device is compact, durable, easy to use, has an ergonomic design, performs real-time measurements, has various monitoring capabilities, and can be used for industrial and domestic purposes to control hazards directly at the location of a person.
APPLIED MATHEMATICS AND INFORMATION SCIENCE
The possibility of applying the methods of technical diagnostics to monitor the technical condition of the VVER-1000 power unit refueling machine (RM) has been analyzed. Operational monitoring of the vibrational state of its mechanisms is complicated by the abundance of possible operating modes, differing in speed, direction of movement, and weight loads on the grips. Сluster analysis tools have shown which vibration parameters allow the best separation of the operating modes of mechanisms in the normalized feature space. Typical clusters of points corresponding to different operation modes of the RM mechanisms have been constructed on the data of an industrial experiment at the Rostov NPP. The necessity of using both traditional vibrational parameters and vibration acceleration kurtosis taking into account the form of vibration distribution has been shown. The analysis of the compactness profiles and the relationships between the distances inside and between the built clusters has revealed the regime parameters that mostly affect the vibrational state of the mechanisms.
Diagrams and algorithms for choosing the optimal variant of the technological process for decommissioning of nuclear facilities, as well as for determining the degree of stability of the chosen variant in conditions of uncertainty of the initial data, have been presented. Uncertainties of expert assessments and calculated values of some of these data give rise to uncertainties in the values of indicators for options for the technological process of decommissioning nuclear facilities. A scheme for selecting the optimal variant of the technological process and determining its stability in conditions of uncertainty of the indicators characterizing each considered variant of decommissioning (the list of possible indicators is presented below) is considered. The optimal variant of the technological process is chosen on the basis of a complex indicator that combines particular indicators in the form of a linear superposition for each considered variant of the technological decommissioning process. According to the proposed scheme, the optimal variant of the technological process is chosen according to the maximum for all considered decommissioning options, the average expected value of the complex indicators for these options, and the stability of the optimal variant of the technological process is tested against the complex indicator for the optimal variant of the technological process for the minimum possible value for each particular indicator.
The possibility of using the restricted Boltzmann machine (RBM) to solve the problem of author’s profiling of Russian texts has been studied on the example of determining the gender and age of an author. The restricted Boltzmann machine is used as a transformer that extracts useful features from documents, where words are encoded using morphological tags. The classification is carried out using a composite two-layer module, which includes MultinomialNB and LinearSVC. Within this task, four corpuses of documents are used, three of which are classified by the gender of the author, and the fourth one, by age. The experiments show that the constructed model successfully solves the tasks assigned to it, surpassing the baseline model (LinearSVC) on average (for all four corpuses) by 7.5 % in terms of f1-score. In addition, of the results are compared with the results of other models from the literature (in particular, using a complex model, based on a convolutional neural network and LSTM ). This comparison shows the efficiency of the constructed composite neural network based on the RBM and the stability of its results on a set of presented corpora.