Methods Of Mathematical Physics Research Articles

Time-reversed electromagnetic fields in anisotropic media.

Electromagnetic time reversal is commonly used for field imaging and focusing. This Letter builds upon the concept of the time-reversal cavity, which constitutes the main theoretical framework of time reversal theory. We study the behavior of the fields using modern methods of mathematical physics involving Colombeau generalized functions. This approach allows for a direct expression of time-reversed electric and magnetic fields in anisotropic time-reversal-invariant and nonreciprocal media. Moreover, the results hold for any arbitrary localized source and can readily be applied beyond the dipole approximation. Finally, a general result allows the prediction of the quality of focusing of the time-reversed fields as a function of the electrical permittivity and the magnetic permeability tensors in homogeneous anisotropic media, which contributes to the understanding of time reversal in complex media such as super-resolution enabling metamaterials.

SOLUTION OF A MIXED BOUNDARY VALUE PROBLEM FOR THE LAPLACE EQUATION BY THE METHOD OF DIVIDING AN OBJECT IN A RECTANGULAR PARALLELEPIPED

The aim of the study is to solve the boundary value problem for the Laplace equation in a rectangular parallelepiped by the method of separation of variables and to estimate the obtained Fourier variable separation constants.
 
 Materials and methods. Methods of mathematical physics were used to solve the boundary value problem for the Laplace equation. The initial problem was divided into three standard ones, in which the inhomogeneous boundary conditions were given only on two parallel sides, for the rest of the problem they being assumed to be equal to 0.
 
 Results. The boundary value problem for the Laplace equation in a rectangular parallelepiped has been broken down into three problems. Partial solutions to these problems under given boundary conditions have been obtained. The Fourier variable separation constants are estimated.
 
 Findings. The solution to the Laplace’s equation for a parallelepiped is the sum of the solutions to three partial problems. The boundary functions of a parallelepiped are odd periodic over two variable functions whose periods are equal to the lengths of the corresponding sides of the parallelepiped. The Fourier constants of partial solutions to the problem are the coefficients of the expansion of the boundary periodic functions of two variables into a trigonometric Fourier series. In two-dimensional series of the solution to the Laplace’s equation for odd-numbered harmonics and for a set of simultaneously even and odd harmonics, the Fourier constants differ only in signs.

OUTER CYLINDRICAL SPACES OF A HOLLOW CYLINDER OF FINITE LENGTH

Introduction. The solution of the Laplace equation by the method of separation of Fourier variables for cylindrical domains makes it possible to study electromechanical devices in 3D format.
 
 The purpose of the study is to obtain analytical expressions for magnetic potentials and inductions of external three-dimensional spaces of electromechanical devices in a cylindrical coordinate system.
 
 Materials and methods. The unknown constants in the expressions for the regions under consideration are found from the boundary conditions of the magnetic field at the common boundaries of the electromechanical device with its outer space: the scalar magnetic potentials and magnetic inductions are the same, the current sheets experience a jump. When obtaining analytical expressions, the methods of mathematical physics were used.
 
 Research results. To find the Fourier constants, the outer three-dimensional spaces of classical electromechanical devices (generators, motors) are considered, which in a cylindrical coordinate system are represented by both hollow and solid cylinders joining the active regions of the devices with their cylindrical or end surfaces. General expressions for magnetic potentials and inductions of external spaces are obtained based on the solution of the Laplace equation as the Sturm–Liouville problem by the Fourier method of separation of variables. The magnetic fields of the outer spaces of the original hollow cylinder are analyzed: an outer cylinder with an infinitely large radius; internal solid cylinder; end cylinders of finite length. These data make it possible to increase the required number of equations to find the required number of Fourier constants in the analytical calculation of electromechanical devices by the method of separation of variables.
 
 Findings. Analytical expressions for the boundary values of magnetic potentials and inductions of the external space adjacent to the active regions of electromechanical devices with air gaps of both radial and axial form are obtained.

On the time-optimal control problem for a heat equation

In previous works, we have considered some control problems for parabolic type equations, namely, control problems for parabolic type equations were studied as boundary value problems of the first type, and the weight function was expanded into a Fourier series by sines. In this paper, we consider boundary control problem for a heat equation on the interval. In the part of the bound of the given domain it is given value of a solution and it is required to find a control to get the average value of the solution. By the mathematical-physics methods it is proved that like this control exists and the estimate of a minimal time for achieving the given average temperature over some domain is found.

THE USE OF MECHANICAL FILTER MODELS IN THE ANALYSIS OF FORMING AND COMPACTION PROCESSES OF FORMATION AND COMPACTION OF BUILDING/CONCRETE MIXTURES BY VIBRATING FIELD

The paper describes the use of various types of mechanical filter models, which are used for the analysis of the processes of formation and compaction of the construction/concrete mixtures of building/concrete mixtures by means of vibrating fields. The values of resonant frequencies and equivalent masses for different resonators modeling the propagation in the latter of vibrating-wave formations have been established. The analysis of the influence of a vibrating field on the processes of formation and compaction of concrete/concrete mixtures in this study is based on the methods of mathematical physics, classical variation calculus, physics of oscillations and waves and methodology of solution of ordinary differential equations and partial differential equations. The conditions and main integral characteristics of resonance phenomena, the possibility of occurrence of which is conditioned by: 1) the geometry of the initial boundary-edge problem (it is The so-called "geometric resonances" of the considered system with distributed parameters simulating the mixture to be processed); 2) the working rheological model of the mixture involved in the study (these are the so-called "rheological resonances"). The approach developed and scientifically substantiated in this work allows us to establish the main parameters and opportunities for the use of energy-saving modes of operation of vibration systems intended for the formation and vibration compaction of the above mixtures. The results obtained in the work The results obtained can be further used to clarify and Improvement of existing engineering methods of calculation of vibration systems for the formation and compaction of concrete/concrete mixtures in order to optimize the operating modes of their functioning both at the design stage and in the modes of real operation.

Analysis and Kinetics Modeling of the Isothermal Oxidation Behavior of Silicide Coatings

In this paper, an online apparatus was developed for isothermal thermogravimetric measurement of silicide coatings within a wide temperature range (from −180 °C to 2300 °C) based on thermogravimetric analysis. Firstly, the measuring principle and method regarding silicide coatings of this apparatus were studied. Secondly, on the basis of oxidation kinetics analysis, the intrinsic mechanism and kinetic parameters of three stages (oxidation, diffusion, and fall-off) of silicide coatings were studied, and the oxidation kinetics features were also analyzed. In addition, according to mathematical physics methods, a kinetics model of silicide coatings in different stages of oxidation was established, including parameters such as weight change, oxidation rate, oxidation time, etc. Finally, online isothermal experiments from −180 °C to 2300 °C werecarried out and analyzed. The results showed that the kinetic model established in this paper was in good agreement with the oxidation process of silicide coatings. In this paper, a complete kinetics model including different oxidation stages is proposed for the entire oxidation process of a silicide coating, revealing its oxidation mechanism. The research will play a significant role in the study of preparation technology improvement and high-temperature environment application. This paper studied two measuring methods: weight gain and weight loss measuring methods. Also, an experiment was carried out on the silicide coatings to explore the physical oxidation process between −180 °C and 2300 °C. The results proved the perfect consistency of the kinetics model proposed by this paper and the oxidation process of silicide coatings. This paper will play a significant role in the study of preparation technology enhancement and high-temperature environment application. It also provides a theoretical foundation for accelerated aging and life evaluation methods.

APPLICATION OF THE CASE METHOD IN THE PROCESS OF STUDYING THE COURSE “METHODS OF MATHEMATICAL PHYSICS”

The article considers the use of case-study method while studying of the topic “Equations with partial derivatives of the hyperbolic type” from the course “Methods of mathematical physics”. The requirements for the educational institutions are outlined in accordance with the concept of mathematical education in Ukraine. The emphasis is placed on the system-activity approach to the learning process. Attention is focused on the necessity to use the method of problem-based learning and one of its forms – the case method. There has been analyzed the concept of a case. There have been described the advantages of the case-study method and the system of requirements for it. There has been revealed the importance of choosing the topic “Equations with partial derivatives of the hyperbolic type” related to its applied nature. The article demonstrates the main stages of the research work of students of the Faculty of Physics and Mathematics in establishing the minimum conditions for the solvability of a mixed problem for a linear homogeneous hyperbolic equation. The paper also identifies and characterizes the main groups of results to be processed. The investigation indicates the selection criteria that help to establish the minimum conditions for the existence of a solution of a mixed problem for a linear homogeneous hyperbolic equation. The results of the research are presented in the form of a table. The research presents the main statements recognized as the best conditions corresponding to the set goal of the study. The originality of the method of proving statements and the skillful combination of two important topics of the course “Methods of mathematical physics” – “Equations with partial derivatives of the hyperbolic type” and “Integral equations” are noted. Application of the case-study method makes it possible to activate the learning process, to form positive motivation and to renew the creative potential of teachers. Proven research work within the course “Methods of mathematical physics” provides a solid basis for using the case method in the further professional activities of teachers. Along with this, the theoretical and methodical preparation of case technologies in institutions of higher education requires further improvement.

Boundary control problem for the heat transfer equation associated with heating process of a rod

In this paper, we consider a boundary control problem for a parabolic equation in a segment. In the part of the domain’s bound it is a given value of the solution and it is required to find controls to get the average value of the solution. The given control problem is reduced to a system of Volterra integral equations of the first kind. By the mathematical-physics methods it is proved that like this control functions exist over some domain, the necessary estimates were found and obtained.

DYNAMIC OPTIMIZATION OF LOADS, OCCURRING DURING THE BRAKING OF AN OVERHEAD CRANE TRAVEL GEAR: AN ANALYTICAL APPROACH

The paper proposes a model for determining the dynamic loads arising during thebraking of an overhead crane travel mechanism. By the methods of mathematical physics, classical variationalcalculus at application of the apparatus of the decision of the usual linear differential equations thelaws of movement of the mechanism of movement of the specified crane at which dynamic loadings in themechanism are minimized are analytically defined. This result gives an opportunity to carry out constructivemeasures for decrease in level of dynamic loading of the crane.In order to reduce the dynamic loads during the braking of the bridge crane movement mechanism,additional measures have been proposed, namely: 1) regulation of crane braking force; 2) reduction of bogieand crane weight by: a) application of steel structures in the form of trusses instead of solid wall ones; b) useof bent profiles for manufacturing of steel structures; c) application of lighter alloys; 3) reduction of the momentof inertia of the electric motor rotor of the crane travel mechanism by: a) using several motors with totalpower equal to the power of one drive motor, which reduces the moment of inertia of the rotor; b) usingmotors with lightweight rotors; 4) using couplings with variable stiffness.The results obtained in the work can be further used to refine and improve the existing engineeringmethods for calculating the braking systems of overhead crane travel mechanisms to minimize the dynamicloads in these mechanisms both at the stages of their design and in the modes of real operation.

Pengembangan Media Pembelajaran Fisika Matematika Materi Transformasi Koordinat Berbasis E-Modul

Learning physics mathematics is one of the subjects in the physics education study program at the University of Jambi. Based on the results of the questionnaire, students' needs regarding learning mathematics and physics are considered difficult to understand the material taught by lecturers. This is due to the less varied teaching materials and the books used are only sourced from the book Mathematical Methods in Physics written by Mary L. Boas with a thickness of 839 pages consisting of 15 chapters with the language used in English. This is why this research was conducted with the aim of being able to help students with problems when learning mathematics and physics, especially in the material of coordinate transformation using e-modules created using Flip PDF Professional software. The method used is a mixed method with an explanatory design where quantitative data in the form of questionnaire results are strengthened by qualitative data in the form of interviews. Based on the research results, the existence of e-modules can be an alternative in learning that can be used by lecturers and students who are used for independent study with a flexible design that can be used.

The commonly overlooked environmental tipping points

Climate models are developed based on well-established physical principles applied to past and recent climate changes. There is considerable confidence that the models can also provide estimates of some climate variables (i.e., surface temperature, CO2 levels, ocean heat content). Despite advanced mathematical developments in the field of climate modeling, the existing climate models suffer from the following major limitations: first, the models do not consider that their estimations will be highly unreliable when a tipping point is triggered; secondly, many of the environmental tipping points are already triggered, however their existence is overlooked; and third, the existing climate models do not consider the interrelations among the tipping points (i.e., one tipping point can trigger other tipping points to be tipped more rapidly). Our objective is to describe the importance of environmental “tipping points,” the importance of which is often ignored or downplayed in relevant literature. Our analysis, based on extensive multidisciplinary literature searches, reveals that there are many environmental tipping points which are overlooked in climate-modeling studies. We argue that climate modeling could be improved when the tipping points and their interrelations are all considered within the modeling process. We further discuss two other important issues regarding environmental tipping points: first, all tipping points might not be as impactful on the climate system, therefore their relative impacts should be ranked; second, it is in principle impossible to know the exact number of environmental tipping points, therefore even though it could be possible to devise improvements to the existing climate models with our suggestions, it may be impossible to achieve a perfect model to estimate the climate variables of the upcoming years. The remainder of this paper is structured as follows: In the background section, we introduce research on tipping points within commonly used climate models. We explain the aerosol masking effect and ocean dynamics with respect to their commonly overlooked roles as important contributors to environmental change. We introduce remote sensing and AI methods that serve as promising approaches for identification of currently unknown tipping points. We mention perturbation theory, a standard set of mathematical methods in physics that serves as a potentially systematic method to rank environmental tipping points according to their impact on extant climate models. In the discussion section, we make suggestions regarding further research on identifying the typically overlooked tipping points, and we make suggestions to improve climate models by considering additional information presented in the current paper. Finally, we conclude this article summarizing our chief methodological recommendations.

СКЛАДАННЯ МАТЕМАТИЧНОЇ МОДЕЛІ ВІБРАЦІЙНОГО СТОЛУ

In this article, a mathematical model is obtained, which combines the design and technological parameters of the technological set of equipment for the concrete goods production (vibrating table), in which the vibrating exciter is fixed on the lever vertically in the center under the plate compactor. This equipment is used for the manufacture of small-sized concrete products. The mathematical model is composedwith the help of Lagrange equations of the second kind, which are the most common method for solving problems about the motion of a mechanical system. Methods of mathematical physics and physical and mathematical modeling by methods of applied mechanics were used in the compilation.To determine the position and describe the free motions of the material bodies that make up the mechanical system under consideration, an orthogonal vi-brational reference system of three coordinate systems was used. Analyzing the kinematic diagram of the vibration table, it is determined that the position in space of all material bodies of the mechanical system, which simulates the specified vibrating table, can be uniquely set by the following independent parameters: Cartesian coordinates, vibration angles and the angle of rotation of debalance.Thus, the mechanical system in question has seven degrees of freedom with seven generalized coordi-nates.As a result, a mathematical model of a vibrating table for concrete products manufacturing in the form of seven second order differential equations system, which describes the vibrating table spatial motion surface, is obtained.The theoretical values of the vibrating table vibrations amplitude at variable lever length were also determined and compared with experi-mental data at the same initial parameters. The difference was 15%, which confirms the adequacy of the created mathematical model to the real technological process on the vibrating table, which is being studied.Analyzing the change in the vibration oscillations amplitude on the vibrating table working surface from the change in the lever length on which the vibrating exciter is fixed, it was found that the lever fixation allows an almost linear increase in the amplitude of vibration oscillations due to an increase in the lever length. This, in turn, makes it possible to reduce energy consumption when compacting the concrete mixture.

A new strict decay rate for systems of longitudinal $ m $-nonlinear viscoelastic wave equations

<abstract><p>Recent years have been marked by a significant increase in interest in solving nonlinear equations that arise in various fields of natural science. This trend is associated with the creation of a new method of mathematical physics. The present study is devoted to the analysis of the propagation of $ m $-nonlinear viscoelastic waves equations in an unbounded domain. The physical properties are determined by the equations of the linear theory of viscoelasticity. This article shows the main effect and interaction between the different weak and strong damping terms on the behavior of solutions. We found, under a novel condition on the kernel functions, an energy decay rate by using an appropriate energy estimates.</p></abstract>

On the propagation of intrinsic viscoelastic waves in a cylindrical shell

The article illustrates that viscoelastic wave in an extended cylindrical shell of circular cross-section. To solve boundary value problems used classical methods of mathematical physics. The unwavering quality of the comes about of the work is guaranteed by the rightness of the detailing of issues, the utilize of the classical device of scientific material science and other demonstrated strategies for tackling issues of wave engendering, the consistent legitimacy of the conclusions and the great assentation of the comes about with the uncommon cases known from the references.

On the Characterization of a Finite Random Field by Conditional Distribution and its Gibbs Form

In this paper, we show that the methods of mathematical statistical physics can be successfully applied to random fields in finite volumes. As a result, we obtain simple necessary and sufficient conditions for the existence and uniqueness of a finite random field with a given system of one-point conditional distributions. Using the axiomatic (without the notion of potential) definition of Hamiltonian, we show that any finite random field is Gibbsian. We also apply the proposed approach to Markov random fields.

МАТЕМАТИЧНА МОДЕЛЬ ДЛЯ АНАЛІЗУ ПЕРЕХІДНИХ ПРОЦЕСІВ В ДВООБВИТКОВИХ ТРАНСФОРМАТОРАХ У МЕТОДІ ПРЯМИХ

A mathematical model of a two-winding transformer for research of fast transient processes in windings in the form of an electromagnetic circuit, taking into account the main magnetic flux, own and mutual interturns and interwinding dissipation fluxes, transverse turns and longitudinal interwinding capacitances of the winding, is developed by using the straights method. The created model simplifies engineering calculations of fast transient processes in windings by applying the methods of the theory of electromagnetic circuits instead of the methods of mathematical physics, which are difficult for practical application. Transient processes and voltage distribution in transformer windings, under the action of pulse overvoltage in the form of a standardized pulse, are researched. References 8, figures 4.

ДИСПЕРСІЙНИЙ АНАЛІЗ ПОЗДОВЖНІХ ІМПУЛЬСІВ ДЕФОРМАЦІЙ У КАНАТАХ МІНІМАЛЬНОЇ МАСИ ВАНТАЖОПІДЙОМНИХ МЕХАНІЗМІВ КРАНІВ

Dispersion analysis of longitudinal deformation pulses in minimum mass ropes of cranes load lifting mechanisms (which satisfy the conditions of equal strength in each specific cross section) is carried out in the article. The phase and group waveform velocities occurring in the ropes of this type are determined. The impulse dispersion by the method of lifting the load "with the pickup"/"from the base" ("from the ground") was investigated using the stationary phase method. Such methods are used in this work as: 1) classic methods of mathematical physics; 2) complex Fourier transform over time; 3) stationary phase method for the calculation of the integral which characterizes the deformations in the remote field during its asymptotic schedule. It is shown that the asymptotic schedule reduces the analysis of stress and deformation wave fields arising in the rope to the using of the Airy function with a complex argument. The sign of this argument is determined by the sign of the third derivative of the frequency by the wave vector for mentioned above fields near stationary points of the group velocity of wave formations (hear, in accordance, the second derivative is equal to zero). It is substantiated that Airy function is responsible for reproducing the nature of perturbations before and after the wave front moving inside the rope system. The phase velocity with which the constant phase plane of the exponential multiplier is carried in the rope is determined. This multiplier the spatial-temporal dependence of the emerging waveform describes. Near the pulse front (load/deformation of the rope), its duration increases, but the amplitude decreases in proportion to the cubic root of the distance to the observation point. The formula is established, according to which the asymptotic of this problem solution should be built, in the case of the curve extreme of waveform propagation group velocity in the rope. A similar approach gives the possibility to explore the main patterns of no stationary wave fields generated in the ropes of cranes load lifting mechanisms. The results obtained in the work can be used for establishing cranes rope systems loads and deformations quantities estimates under the condition of their transience. Such loads/deformations usually cause of the above systems overstrain when lifting or lowering loads with different types cranes (particularly, bridge, gantry, portal), which work in operation modes (real operation mode) and can lead to emergencies (for example, breaks in rope systems).

THE STATE OF THE PROBLEM OF THE JOINT MOVEMENT OF FLUID IN THE PORE SPACE

This article discusses the problems of studying the issue of joint motion of liquids in the porous space. The article provides the construction of a mathematical model of the theory of ltration, which describes phase transitions. The main diculty in constructing this model is associated with the fact that free interphase boundaries create regions that change over time, and it is required to nd the temperature or concentration elds of substances in them. In this case, the coordinates of the considered phase boundaries are not initially specied and must be calculated already in the process of solving. For this, a derivation of the averaged equation for the problem of nding the rupture surface during the movement of two incompressible viscous liquids in the pores of the soil skeleton was proposed. The article deals with the case when the skeleton is an absolutely rigid body. The rationale was given for the choice of an averaged ltration model instead of a microscopic one. The main research methods are classical methods of mathematical physics, functional analysis and computation methods of the theory of partial dierential equations, as well as dierence methods. The formulation of the problem is given, and the denition of a generalized solution for solving the problem is provided. Next, an averaged model is derived and the existence of at least one generalized solution to the problem is proved.

Строгие и приближенные выражения для коэффициентов взаимной индукции и электродинамических сил соосных витков и плоских (дисковых) катушек с током

Object and purpose of research. The paper is concerned with the mutual inductance coefficients of co-axial circuits in a planar coil system. Known modifications for mutual inductance and electrodynamic forces in a system of co-axial circuits are written, and relations for their calculation are compared. Materials and methods. For this purpose the methods of mathematical physics and theoretic electric engineering are used. Main results. Alternative expressions are given for mutual inductance coefficients of co-axial coils and electrodynamic forces between the live coils. Curves of mutual inductance are plotted for planar coils with the distance between planes equal to 2 cm and the distance between coil turns Р equal to 1 сm and the distance between coil turns N equal to 0.5 сm, 0.4 сm & 0.3 сm. Conclusion. Relations have been derived for calculation of mutual inductance coefficients and electrodynamic forces at current passage in planar co-axial coils, which are very important for wireless transfer of power.

Inverse Radon transform without singularity for rays with a point focus

Background: The need to reconstruct images from their projections has been repeated many times as a means of studying an object in a wide variety of fields of science, technology and medicine. The basis of this method is the penetration through object of beams under different angles and the acquisition of a shadow image on the surface sensitive to these rays. Since that the tissues and bones have different absorption densities from such projections it is possible to get a reasonably accurate idea of the presence of pathological formations or lesions. Despite all the advantages of tomography, it has disadvantages. One of the most serious is the exceedance of radiation for humans. Therefore, there are several options for reducing the burden on the patient, one of which is the use of tomosynthesis instead of CT scanning. Tomosynthesis occupies an average position in informativeness between computer tomography and radiology not much higher than the dose of the latter. As a result, work is underway worldwide to improve the tomosynthesis method to increase the number of diseases that can be diagnosed by this method and to improve the quality of the images obtained. Objectives: Defines the efficiency of the image recovery method when using the Radon transform in general form compared to classical methods. Materials and methods: Mathematical physics methods for reconstructing medical images, namely the Fourier transform in the polar coordinate system and the Radon transform. Results: As a result of the work done, an expression for the reverse Radon transform was obtained and the work of the tomograph with point focus of X-ray rays was modeled. When comparing different reconstruction techniques with the original image, the Radon transform is generally more accurate. Conclusions: Despite the considerable increase in the time of reconstruction computation, about three times, this method has practical advantages in the case of detection of pathologies that only develop and have no clear boundaries or in the case of detection of small objects.