Mathematical Model Of Problem Research Articles

A new method of parameter identification for proton exchange membrane fuel cell based on hybrid particle swarm optimization with differential evolution algorithm

With the characteristics of high energy conversion efficiency, high energy den?sity and low operating temperature, the proton exchange membrane fuel cells (PEMFC) have become one of the green energy sources with broad prospects. The establishment of accurate mathematical model of the PEMFC is the basis of simulation and control strategy. At present, some intelligent algorithms have certain drawbacks, and can hardly find the balanced point between precision and computational time. In this study, a novel parameter identification approach com?bining the hybrid particle swarm optimization (PSO) algorithm with differential evolution, i.e. hybrid DEPSO, is proposed to obtain the unknown parameters in the PEMFC mathematical model and solve the problems of premature convergence of PSO and poor global search ability of differential evolution. Six benchmark functions are applied to verify the performance of the algorithm. The results prove that the hybrid DEPSO can evade local optimum preferably while having swifter convergence rate. Two PEMFC stacks are investigated and modeled. In order to evaluate the accuracy of model, the sum of squared errors between the measured voltage and the estimated output voltage are examined. Numerical results show higher accuracy of the hybrid DEPSO-based model comparing with other recently published optimization approaches. Furthermore, the simulation results indicate that the accuracy of the PEMFC model optimized by the hybrid DEPSO algorithm improves 0.19-1.86%, which can provide a new solution the multi-objective opti?mization problem and promote the practical application of the PEMFC.

Восстановление порядка дробной производной в задаче математического моделирования накопления радона в избыточном объеме накопительной камеры по данным Петропавловск-Камчатского геодинамического полигона

Восстановление порядка дробной производной в задаче математического моделирования накопления радона в избыточном объеме накопительной камеры по данным Петропавловск-Камчатского геодинамического полигона

METHODS OF MATHEMATICAL SIMULATION AND MACHINE IDENTIFICATION OF ANOMALOUS DIFFUSION PROCESSES

For the class of anomalous diffusion processes, the mathematical models of which are formalized in the form of variational inequalities in partial derivatives, a method of mathematical modeling based on the optimization procedure is proposed. The method is considered in relation to the generalized mathematical model of the studied class of anomalous diffusion processes. Which made it possible to ensure the principle of unification and typification in the application of this method, as well as the correctness of using the generalized mathematical model in applied problems of mathematical modeling of known industrial and practically important natural cases of anomalous diffusion processes. At the same time, the task of implementing mathematical models of anomalous diffusion processes based on the proposed method is reduced to finding the maximum of the Hamiltonian function defined in the state space of the processes under consideration. A method of parametric identification of mathematical models of anomalous diffusion processes in the formulation of the problem of optimal control is also proposed. The method is reduced to the use of the optimization procedure of the gradient projection method. The possibility of solving the problem of parametric identification in cases of both linear and non-linear mathematical models of anomalous diffusion processes is proved. Moreover, the nonlinear formulation of the parametric identification problem does not lead to computational implementation complications, since the solution is based only on finding the gradient projection of the state function of the anomalous diffusion process. The proposed methods are presented in strict compliance with the provisions of functional analysis, which ensures their correctness and adequacy in solving a wide range of applied problems.

СОПОСТАВЛЕНИЕ КОМПЬЮТЕРНЫХ РЕАЛИЗАЦИЙ РАВНОВЕСНОЙ И КОНТИНУАЛЬНОЙ МЕТОДИК МОДЕЛИРОВАНИЯ МНОГОФАЗНЫХ СРЕД НА ПРИМЕРЕ ОДНОМЕРНОГО НЕСТАЦИОНАРНОГО ТЕЧЕНИЯ ГАЗОВЗВЕСИ

The work is devoted to the problems of mathematical modeling of hydrodynamic processes in inhomogeneous media. The purpose of the study is to compare the techniques for modeling the flows of inhomogeneous media. The continuum and equilibrium methods of modeling the dynamics of a gas suspension – a gas with dispersed inclusions – were compared. Comparisons were made between calculations carried out by an equilibrium mathematical model of the dynamics of gas suspensions with numerical implementations of continual mathematical models of the dynamics of gas suspensions with an ideal and viscous heat-conducting carrier medium. Various parameters of the simulated processes obtained by various mathe¬matical models were compared. Materials and methods. The paper presents two methods for modeling the dynamics of multiphase media. In the equilibrium method, an inhomogeneous medium is described as an ideal gas with the calculation of the correction coefficients of gas-dynamic functions, in the continuum modeling method for particles, the concept of “average density” is introduced; thus, for both the carrier phase – gas and the dispersed phase, a system of equations is solved consisting of the density conservation equation, for of the dispersed phase, the “average density” conservation equation, the momentum conservation equations of the mixture components, and the energy conservation equations of the gas and dispersed phase. The systems of equations of motion of single-phase and two-phase media were integrated by a numerical finite-difference method of the second order of accuracy. To suppress numerical oscillations at each time step, the method of nonlinear correction of the grid function was used. The software package for mo¬deling the dynamics of a gas suspension consisted of several components: a subroutine for setting the boundary conditions, a subroutine for the formation of a finite-difference partition of a physical domain, and the main program for calculating the dynamics of an inhomogeneous medium. Results. The discontinuity decay process in a homogeneous gas and in a gas suspension was modeled. Numerical calculations of pressure discontinuity decay in a homogeneous gas and numerical calculations of discontinuity decay in a gas suspension, obtained by the equilibrium modeling technique, were compared with analytical solutions known from the literature. It is determined that the highest propagation velocity of a disturbance in a gas suspension is observed in an equilibrium model, a lower propagation velocity in a continuum model with an ideal medium, and the lowest velocity is observed in a gas suspension with a viscous heat-conducting medium. It was found that for flows with different initial pressure discontinuity intensities, the presence of a dispersed phase in the mixture has a different effect. Conclusions. The regularities obtained demonstrate the features of each of the methods for modeling the dynamics of gases, which can be used in the development of computer models of the dynamics of multiphase media.

Mathematical modeling of the mass transfer of the drying process of a multilayer thermal insulation coating

Polymer thermal insulation materials are widely used in modern industry and technological production of energy carriers. Thermal insulation with polymer coatings is one of the main ways to protect thermal equipment from temperature effects, corrosion, cavitation, erosion, and other influences, reducing the consumption of expensive materials. However, although polymeric materials can significantly reduce the cost of heat losses, their use is kept at a relatively low level. This is due to the low level of culture in the construction industry and the desire to save on projects, even at the expense of quality. The important issue of forming a reliable system “polymer sheet – adhesive film – environment” is given minimal attention, which, as a result, greatly affects the performance and efficiency of the operation of power facilities. In this paper, we studied the problem of mathematical modeling of the mass transfer of the process of drying a multilayer thermal insulation coating on a polymer basis. The proposed method for calculating the concentration and temperature fields allows for optimizing the drying process and improving the quality and reliability of the technological process.

The main directions for optimizing traffic flows to increase their throughput

This paper analyzes various problems of mathematical modeling of traffic flows that are relevant for optimizing and reducing the congestion of road networks in large cities and metropolitan areas. Various types of relationships between traffic lights on a controlled section of the road have been studied, formulas have been proposed for calculating the optimal joint mode of operation and minimizing the formation of traffic jams in front of the intersection. A model for calculating the optimal traffic flow on a section of the road network and its implementation at the program level are proposed.

Optimization of regional emergency supplies distribution vehicle route with dynamic real-time demand.

Given the particular characteristics of a sudden outbreak of an epidemic on a regional scale and considering the possible existence of a latent period process, this paper takes the distribution of regional emergency supplies as the research object. Form the proposes a dynamic vehicle path problem from the perspective of real-time demand changes. First, when there is a sudden outbreak of a small-scale epidemic, there is uncertainty about demand in the epidemic area. The objective functions of minimizing the vehicle travel route cost of emergency vehicles, the late arrival penalty cost of emergency vehicles, and the fixed cost of emergency vehicles, as well as the objective function of minimizing the total distance traveled by vehicles, are established. Second, a mathematical model of the dynamic real-time demand vehicle route problem is built using the actual vehicle routing problem as a basis. The model is then solved using the SFSSA method. Finally, the computational results demonstrate that the SFSSA algorithm can effectively reduce transportation cost and distance when solving the constructed mathematical model problem, providing a solution to the problem of optimizing the route of emergency material distribution vehicles for a regional scale.

MATHEMATICAL MODELING OF THE QUASI-STATIONARY PROCESSES OF VISCOUS MIXTURE MIXING IN A RECTANGULAR AREA BY THE R-FUNCTIONS METHOD

Mixing processes are found in the chemical, pharmaceutical and food industries. Fluid mixing is one of the fundamental scientific problems associated with modern concepts of regular and chaotic dynamics. The paper considers the problem of mathematical modeling of the quasi-stationary process of mixing a viscous mixture. This problem consists of two sub-problems: determination of the velocity field in the flow region (Eulerian formalism) and investigation of the trajectories of individual fluid particles (Lagrange formalism). To solve the first subproblem, it is proposed to jointly use the principle of superposition, the structural method (method of R-functions) and the Ritz variational method. The methods of nonlinear dynamics and qualitative theory of differential equations are used to solve the second subproblem. A plane quasi-steady flow is considered in a rectangular region and it is assumed that the side walls are at rest, and the upper and lower walls move alternately according to the given laws. According to the method of R-functions, the structures of the solutions were built and the use of the Ritz variational method for the approximation of the uncertain components of the structures was justified. The operation of the proposed method is illustrated by the results of a computational experiment, which was conducted for different modes of wall motion. The practical interest of the considered regimes is due to the fact that they lead to the emergence of chaotic behavior when mixing occurs most efficiently. Using the methods of nonlinear dynamics, the location of periodic (hyperbolic and elliptical) points was investigated and the Poincaré section was constructed. Further research with the help of the method proposed in the work can be related to the consideration of flows in more geometrically complex regions and more complex mixing regimes, as well as in the application to the calculation of industrial problems.

Studying the Features of Streamer Formation and Development in Air by Using Computation Experiments

The aim of the work is to estimate the relevance and current state of the problem of mathematical modeling of streamer discharges in air, as well as to identify the features pertinent to formation and initial development stage of two-head streamers in air by using computational experiments. The streamer discharge mathematical model developed at the National Research University Moscow Power Engineering Institute Department of High Voltage Engineering and Electrophysics was used as the research tool. It is shown that mathematical modeling of a streamer discharge can be effectively used in its studies and in training specialists in the field of electric power engineering and high-voltage electrical technologies. The article presents the results of computational experiments on studying the occurrence of streamers in a discharge gap filled with dry air with a uniform electric field (EF) under standard atmospheric conditions. The initial conditions were chosen so that a single-avalanche-streamer formation mechanism was realized. It is shown that in the considered range of initial conditions, the distance between the anode and initial inhomogeneity has a weak, within 1%, influence on the minimum and maximum values of electric field strength in the streamer. A different picture is observed on the anode surface. When a two-head streamer is formed in the discharge gap depth, the field strength at the anode remains close to the average strength in it. As the negative streamer head approaches the anode, the field strength on its surface increases to values close to those achieved in the positive head. After that, the field at the anode weakens rapidly to a level at which effective impact ionization becomes impossible. The article will be of interest for specialists and graduate students working in the field of electric power engineering and high-voltage electrical technologies.

Reverse Mathematical Modelling

Background: Research in mathematics education shows that mathematical modelling is a practice that seeks to “translate” problem situations into possible mathematical models without, however, explaining the complexity involved in the reverse formulation, starting from the mathematical model to delimit the type of situation. Objective: To highlight the problem of reverse mathematical modelling, in the sense of a reverse formulation that goes from the mathematical model to the situation. Design: For this, a course of study and research was carried out guided by the investigative cycle of mathematical modelling that is based on theoretical-methodological resources of the anthropological theory of the didactic. Setting and Participants: Pre-service teachers of a teaching degree at a public institution were faced with a problem in an unusual context about the decimal number system and, more broadly, the positional number system. Data collection and analysis: We present an empirical approach based on research carried out by Ferreira (2020) with teachers in initial training. Results: The empirical results observed confirm the hypothesis of the existence of the problem of reverse mathematical modelling, even in the face of normative models, in the sense that they can faithfully describe a real situation. Conclusions: Ultimately, the study of a type of problem in an unusual context, in addition to highlighting the encounter of teachers with different objects of knowledge, revealed the remarkable difficulty in delimiting the type of quantification situation that can be associated with the mathematical model of the number and as stimulate future research on the teaching of reverse mathematical modelling.

Mathematical Models of Technological Processes of Oil Refining and Their Qualitative Analysis Based on the General Concept Of Models

Theorems of existence and uniqueness of the decision of system of the equations in the private derivatives, representing the generalized mathematical model of processes and devices of preprocessing of crude hydrocarbons are formulated and proved. Generalization gives the chance to apply the principle of unification and typification when de-veloping a method of numerical realization of mathematical models of a class of processes (devices) of preprocessing of crude hydrocarbons, and the proof of the corresponding theorems (an essence —the qua-litative analysis) provides a correctness of application of the generalized model in applied problems of mathematical modeling of studied pro-cesses (devices). Proofs of the formulated theorems are strict, logically true and are consistently executed within terms of the functional ana-lysis. Practical applicability of theorems of existence and uniqueness of the decision as component of the qualitative analysis, is defined by pos-sibility of research on their basis of adequacy of algorithmic means of mathematical modeling of a studied class of processes (devices).

On the peculiarities of using sparse matrices in problems of mathematical modeling

Mathematical modeling and the related computer experiment are now one of the main means of studying objects, processes and phenomena of various nature: in science, engineering, economics, society, etc. A significant improvement in the quality of mathematical modeling in many areas of science and engineering is possible only with the use of fundamentally new three-dimensional models, the transition from computer simulation of individual components and assemblies to the calculation and optimization of the product as a whole. It is obvious that the consideration of problems in such a formulation leads to discrete mathematical models of super-large sizes. Existing supercomputers of different parallel architectures make it possible to efficiently solve such problems. However, the time for solving problems on parallel computers consists of the time of the actual solution and the time of performing additional operations, that are necessary for the exchange of information between computing devices, that is overhead costs. This is especially true for problems of linear algebra with different structures of sparse matrices of large volumes, that arise in the mathematical modeling of processes. Sparse matrix compaction schemes, decomposition of data arrays between processors are one of the main factors for the effective solution of these problems on parallel computers. The paper considers efficient methods for processing sparse matrices of arbitrary structure for the purpose of effective mathematical modeling of structural strength problems on parallel computers. Various methods of regularization and decomposition of sparse matrices of arbitrary structure, efficient data storage schemes, technology for studying the conditionality of a matrix with approximate data on a computer are proposed. This way of using sparse matrices in mathematical modeling ensures more efficient use of computing resources and reliability of computer results. Problems of mathematical modeling are presented, where the considered methods of processing sparse matrices were effectively applied.

Melting phenomenon in the flow of dusty nanofluid over a stretching sheet in the presence of single walled carbon nanotubes

The present research is based on the thermal and flow properties of the dusty liquid past a stretching sheet. Further, the liquid flow is embedded with Methanol base fluid suspended by single walled carbon nanotubes as nanoparticles. To crack the current mathematical model problem, the numerical Runge-Kutta-Fehlberg fourth-fifth order method was used. With the aid of graphs, the effect of physical parameters on the fluid flow and thermal profile were discussed. To achieve the most accurate results, the correct initial assumptions and several non-dimensional parameter values are carefully chosen and well-adjusted. Results reveal that, the rise in values of temperature interaction parameter improves the fluid phase heat transfer but declines the dust phase heat transport. The increase in melting parameter values improves the both fluid and dust phase heat transport. The fluid phase thermal distribution drops faster than dust phase heat transfer for growing values of gamma. The growth in values of melting parameter drops the rate of heat transport.

Excitation-based fully connected network for precise NIR-II fluorescence molecular tomography.

Fluorescence molecular tomography (FMT) is a novel imaging modality to obtain fluorescence biomarkers' three-dimensional (3D) distribution. However, the simplified mathematical model and complicated inverse problem limit it to achieving precise results. In this study, the second near-infrared (NIR-II) fluorescence imaging was adopted to mitigate tissue scattering and reduce noise interference. An excitation-based fully connected network was proposed to model the inverse process of NIR-II photon propagation and directly obtain the 3D distribution of the light source. An excitation block was embedded in the network allowing it to autonomously pay more attention to neurons related to the light source. The barycenter error was added to the loss function to improve the localization accuracy of the light source. Both numerical simulation and in vivo experiments showed the superiority of the novel NIR-II FMT reconstruction strategy over the baseline methods. This strategy was expected to facilitate the application of machine learning in biomedical research.

On fixed point theorems in modular spaces characterized by $C^*$-class functions

Generally, finding a solution to a theoretical mathematical modelling problem is equivalent to finding a fixed point for a suitable operator. Accordingly, fixed point theory is therefore a very important and crucial in many areas, such as mathematics, sciences, and engineering. A very popular and important fixed point theory is those formulated by Stefan Banach in 1922. The theory is related to a complete normed space and known as the Banach fixed point theory. Recently there have been numerous generalization of the Banach fixed point theory. One of them is a fixed point theory in modular spaces. In this paper, we will formulate some fixed point theorems in modular spaces by using $C^*$-class functions. The obtained results generalize and improve some results in [Supama, On Some Common Fixed Point Theorems inModulared Spaces, International Mathematical Forum, Vol. 7, no. 52, (2012), 2571 - 2579.].

Flow and thermal analysis of Oldroyd 8 constant fluid in a porous channel

AbstractThe present research is based on the thermal and flow properties of the viscoelastic Oldroyd 8 constant fluid in an upright microchannel. The energy and momentum equations were solved with the support of temperature Jump and velocity slip boundary conditions. To measure the irreversibility rate of the flow system, the acquired results of velocity and thermal equations were used. To crack the current mathematical model problem, the numerical Runge–Kutta–Fehlberg method was used. With the aid of graphs, the effect of physical parameters such as thermal radiation, thermal‐dependent heat source, Joule heating, fluid parameters, velocity slip, and temperature Jump parameters on the fluid flow, thermal energy, and system entropy generation was discussed. Fluid parameters have different effects on the velocity profile. The Grashof and Hartmann numbers demonstrate opposite effects on the momentum field. The thermal energy of the system reduces with thermal radiation and temperature Jump factor. The thermal radiation, Hartmann number, and temperature Jump parameters reduce the system's irreversibility rate. With the Brinkman number and temperature Jump parameter, the irreversibility ratio increases.

A rubric for assessing mathematical modelling problems in a scientific-engineering context

Abstract Mathematics modelling is a vital competency for students of all ages. In this study, we aim to fill the research gap about valid and reliable tools for assessing and grading mathematical modeling problems, particularly those reflecting multiple steps of the modelling cycle. We present in this paper the design of a reliable and valid assessment tool aimed at gauging the level of mathematical modelling associated with real-world modeling problems in a scientific-engineering context. The study defines and bases the central modelling processes on the proficiency levels identified in PISA Mathematics. A two-dimensional rubric was developed, reflecting the combined assessment of the type and level of a modelling process. We identified criteria that enable a clear comparison and differentiation among the different levels across each of the modelling processes. These criteria allow for concrete theoretical definitions for the various modelling processes, introducing a well-defined mathematical modelling framework from a didactical viewpoint, which can potentially contribute to promoting modelling competencies or the understanding of modelling by teachers and students. Theoretical, methodological and practical implications are discussed.

STUDY OF THE INFLUENCE OF THE PASSAGE OF HEAVY TRAINS ON THE THROUGHPUT AND CARRYING CAPACITY OF RAILWAY SECTIONS WITH ELECTRIC TRAIN TRACTION

The criterion for the effectiveness of the organization of heavy traffic on the sections of the Russian railway network is the presence of large correspondences of bulk goods presented by the consignor for transportation. An obstacle to the development of a promising cargo traffic are areas with limited throughput and carrying capacity, the overcoming of which is the organization of heavy traffic. All countries that have railways, at a certain stage of development, face capacity constraints. Summarizing the world experience, we can conclude that the priority in the development of increasing volumes of “heavy cargo” is increasing the weight norms of freight trains. The problem of passing heavy trains for Russian Railways is urgent, because the capacity of some sections has exhausted itself completely. The very timely appearance of heavy traffic becomes one of the options for solving the problem of increasing the carrying capacity of sections and directions and, as a result, creating a reserve of carrying capacity. In this article, a study of the influence of their movement on the carrying capacity and carrying capacity has been carried out.
 Aim. To consider the effectiveness of the formation and passage of heavy trains. The influence of the organization of the movement of such trains on the throughput and carrying capacity of railway sections.
 Methods. To solve the problem, graphic and analytical methods of mathematical modeling, scientific methods of collecting and processing statistical data, modern achievements in terms of general principles and methods of risk management was used.
 Results. The authors have developed a formula for determining the rate of removal of freight trains. The analysis of the influence of the values of the inter-train intervals of freight trains and heavy trains on the removal rate is given.
 Practical relevance. As the main alternative to increasing the size of traffic on the main cargo-intensive directions, with a shortage of traffic capacity, it is proposed to use the technology of heavy traffic. In modern market conditions, an increase in the weight and length of a freight train is one of the main reserves for increasing throughput and carrying capacity. By passing heavy trains, we get an increase in carrying capacity. In the article, we prove the consistency of these measures. The increase in carrying and carrying capacity is considered in the context of increasing unified weight standards.

MATHEMATICAL MODEL OF THE MARKET OF ONE PRODUCT WITH OPTIMAL DELIVERY TO THE MARKET UNDER CONDITIONS OF DELAY

MATHEMATICAL MODEL OF THE MARKET OF ONE PRODUCT WITH OPTIMAL DELIVERY TO THE MARKET UNDER CONDITIONS OF DELAY

MODELING OF SEISMIC WAVES STRESSES IN A HALF-PLANE WITH A VERTICAL CAVITY FILLED WITH WATER (THE RATIO OF WIDTH TO HEIGHT IS ONE TO TEN)

The problem of mathematical modeling of unsteady seismic waves in an elastic half-plane with a ver-tical rectangular cavity filled with water is considered. The problem of modeling problems of the transition peri-od is an actual scientific problem. A quasi-regular approach is proposed to solve a system of linear ordinary dif-ferential equations of the second order in displacements with initial conditions and to approximate the region un-der study. The method is based on the schemes: a point, a line and a plane. An algorithm and a set of programs for solving flat (two-dimensional) problems that allow obtaining a stress-strain state in complex objects have been developed. To assess the reliability of the developed methodology, algorithm and software package, the problem of the effect of a plane longitudinal wave in the form of a Heaviside function on an elastic half-plane was solved. The numerical solution corresponds quantitatively to the analytical solution. The problem of mathe-matical modeling of unsteady elastic stress waves in a half-plane with a cavity filled with water (the ratio of width to height is one to ten) under seismic influence is solved. A system of equations consisting of 8016008 un-knowns is solved. Contour stresses and components of the stress tensor are obtained in the characteristic areas of the problem under study. A cavity filled with water, with a width-to-height ratio of one to ten, reduces the amount of elastic contour stress.