Mathematical Model Of Problem Research Articles

A Splitting Scheme for Diffusion and Heat Conduction Problems

The problem of mathematical modeling and optimization of nonstationary diffusion and heat conduction processes is considered. An approach that uses the idea of splitting and computation of the obtained difference schemes using explicit schemes of point to point computing is proposed for numerical solution of multidimensional diffusion and heat conduction initial–boundary-value problems. Construction of difference splitting schemes, approximation and stability on initial data are investigated. Differential properties of the quality functional are analyzed for the numerical solution of the optimal control problem for a parabolic equation. An iterative algorithm for finding the optimal control is proposed.

Models for combining facilities and resources of renewable energy in the process of cogeneration

The possibilities of the author’s methodology for managing the integration of resources and the imbalance of the goals of technological development of complex systems for studying the processes of combining objects and cogeneration of several types of energy are revealed. The object of research is a set of objects using a combination of traditional and renewable energy resources, and the subject is a mechanism for controlling the combination of objects and resources under the conditions of cogeneration of several types of energy. On the basis of the economic and mathematical model, mathematical and methodological problems were solved, which confirmed the increase in controllability of energy and environmentally efficient processes and the synergistic effect of combining objects and resources for the cogeneration of several types of energy.

Preface

The IV International conference “Supercomputer Technologies of Mathematical Modelling (SCTeMM’19) was organized by Steklov Mathematical Institute of the Russian Academy of Sciences in cooperation with Ammosov North-Eastern Federal University, Ivannikov Institute for System Programming of the Russian Academy of Sciences and Bauman Moscow State Technical University on June 19-21, 2019 in Moscow (Russia).The purpose of the conference is to share new achievements and create conditions for fruitful cooperation between researchers involved in mathematical modeling, high-performance computing. It is also aimed at discussing current problems of mathematical modeling of continuum mechanics, computational algorithms on multiprocessor systems, scientific computing, and prospects for development of parallel computing technologies.

The problem of mathematical finite element modeling of inhomogeneous deformable solids using scanning

Introduction.In the mathematical finite element modeling, an average value of the mechanical characteristics of the deformable solid material is used. In aircraft, machine building, construction engineering, medicine and other fields, polymer composite materials and materials of natural origin are increasingly used. In the latter case, the actual change in the mechanical characteristics differs significantly from the averaged change; therefore, when using the averaged parameters to build and analyze finite element models, the results can be significantly distorted. This paper describes the creation of mathematical methods for studying changes in the mechanical characteristics of a material of inhomogeneous deformable solids. The results obtained in this way are used to construct finite element models and analyze their stress-strain state.Materials and Methods. Naturally occurring materials and composites are considered as inhomogeneous deformable solids. To study the changes in the mechanical characteristics of the material, a method was developed based on the use of two components: the pixel characteristics of raster images scanned by a computer tomograph and the experimental data of field tests of standard samples.Research Results.A complex of mathematical methods has been developed for modeling the interpretation of scanning raster images by a computer tomograph, which allows for the study of any complicated structures of real deformable solids. The results are used in the construction of finite element models of such bodies considering the heterogeneity of the mechanical characteristics of the material. The analysis of the stress-strain state of finite element models of test samples has proved the accuracy and convergence of the numerical solution of the finite element method in modeling the property of heterogeneity of the mechanical characteristics of the material.Discussion and Conclusions. The developed approach can be applied to any physical principles of scanning (X-ray, ultrasound, laser, etc.) and for any types of materials if the data obtained as a result of scanning is developed in the form of a digital (raster) image.

Construction of models of students’ behavior based on matrixes of fuzzy relations considering their motivations to improve success in learning

Problem. In this article, the student behavior models are formalized, considering their motivation to improve the performance. The relevance of this topic is due to the need to introduce new information technologies into the education system of Ukraine, which are aimed at reforming the educational process in order to overcome the crisis in education and increase the level of professional education of university graduates. Goal. In the process of developing models, a large number of works were considered in which the problems of mathematical modeling and management of the learning process were discussed, with the aim of increasing student performance. The purpose of this article is to describe the behavior patterns of students for the development of information technologies aimed at improving academic performance. Methodology. To achieve this goal, the methods of cluster analysis, fuzzy sets and production modeling were used. Clustering parameters were determined as a result of processing statistical data obtained experimentally. According to the results of the experiments, the students were divided into typological groups depending on their performance, for this purpose a modified cluster analysis method was used. Considering poorly formalized factors, this task was presented as a decision-making task under conditions of uncertainty. The result of clustering was presented with the help of a fuzzy function, and production models of students' behavior during the test (exam) were built. Results. As a result, 5 models of students' behavior during the test (exam) were identified and described, and these models were formalized in the language of predicates. Matrices of fuzzy relations of students' preferences for evaluating their knowledge are built and formalized. The relations of strict preference for each model of student behavior are defined and a number of clearly non-dominant alternatives of student behavior that want to improve their academic performance are identified. Graphic representations of fuzzy possibilities of redistributing students to another typological group by academic performance, depending on their behavioral patterns, are constructed. Originality. This article proposes the production models of building up student behavior during the test (exam). Practical value. Formalization of student distribution models by typological groups depending on academic performance can be used in the development of an automated tool for creating information technology to form individual methods of teaching studentsProblem. In this article, the student behavior models are formalized, considering their motivation to improve the performance. The relevance of this topic is due to the need to introduce new information technologies into the education system of Ukraine, which are aimed at reforming the educational process in order to overcome the crisis in education and increase the level of professional education of university graduates. Goal. In the process of developing models, a large number of works were considered in which the problems of mathematical modeling and management of the learning process were discussed, with the aim of increasing student performance. The purpose of this article is to describe the behavior patterns of students for the development of information technologies aimed at improving academic performance. Methodology. To achieve this goal, the methods of cluster analysis, fuzzy sets and production modeling were used. Clustering parameters were determined as a result of processing statistical data obtained experimentally. According to the results of the experiments, the students were divided into typological groups depending on their performance, for this purpose a modified cluster analysis method was used. Considering poorly formalized factors, this task was presented as a decision-making task under conditions of uncertainty. The result of clustering was presented with the help of a fuzzy function, and production models of students' behavior during the test (exam) were built. Results. As a result, 5 models of students' behavior during the test (exam) were identified and described, and these models were formalized in the language of predicates. Matrices of fuzzy relations of students' preferences for evaluating their knowledge are built and formalized. The relations of strict preference for each model of student behavior are defined and a number of clearly non-dominant alternatives of student behavior that want to improve their academic performance are identified. Graphic representations of fuzzy possibilities of redistributing students to another typological group by academic performance, depending on their behavioral patterns, are constructed. Originality. This article proposes the production models of building up student behavior during the test (exam). Practical value. Formalization of student distribution models by typological groups depending on academic performance can be used in the development of an automated tool for creating information technology to form individual methods of teaching students

Identification of student’s metacognition level in solving modeling mathematical problem

This study aims at identifying the students’ metacognition level in solving mathematical modeling problem. Subjects of this study is eight seventh grade students in junior high school which selected based on their latest mathematics test exam score and communication ability test. The results of this study are obtained by using mathematical modeling problem test and interview. The results of this study are that students in the tacit use level is not know the reason why did they use the certain ways that they did to solve the problem, most of the time they just use ‘try and error’ method. Students in the aware use already know the reason why they use a certain ways to solve a certain problem. Students in the strategic use are capable on create their own strategy before solving the problem. Students in reflective use are not only building a plan before solving the problem, but also reflecting and evaluating during and after they obtain the result of the problem.

Математичне моделювання і задачі-моделі в контексті використання сучасних інноваційних комп’ютерних технологій

Математичне моделювання і задачі-моделі в контексті використання сучасних інноваційних комп’ютерних технологій

Modeling and sliding mode-based attitude tracking control of a quadrotor UAV with time-varying mass

In the practical applications, the mass of a quadrotor unmanned aerial vehicle (UAV) would be time-varying. This time-varying mass will deteriorate the control performance of UAV. To solve this challenge, the mathematical modeling problem of a quadrotor UAV with time-varying mass is first investigated in this paper. The nonlinear dynamics describing the six-degrees-of-freedom full motion is established. Based on the established model, taking attitude tracking control problem into consideration, a robust control scheme is then designed via the sliding mode control theory. Applying the developed proposed approach, the desired trajectory is followed with the attitude tracking error asymptotically stabilized. The proposed controller has the capability of rejecting the external disturbance and the uncertainties induced by the time-varying mass. The feasibility of the established model and the effectiveness of the presented control approach are validated through a simulation study.

Dynamics of a time-periodic two-strain SIS epidemic model with diffusion and latent period

Coinfection of hosts with multiple strains or serotypes of the same agent, such as different influenza virus strains, different human papilloma virus strains, and different dengue virus serotypes, is not only a very serious public health issue but also a very challenging mathematical modeling problem. In this paper, we study a time-periodic two-strain SIS epidemic model with diffusion and latent period. We first define the basic reproduction number R0i and introduce the invasion number Rˆ0i for each strain i(i=1,2), which can determine the ability of each strain to invade the other single-strain. The main question that we investigate is the threshold dynamics of the model. It is shown that if R0i⩽1(i=1,2), then the disease-free periodic solution is globally attractive; if R0i>1⩾R0j(i≠j,i,j=1,2), then competitive exclusion, where the jth strain dies out and the ith strain persists, is a possible outcome; and if Rˆ0i>1(i=1,2), then the disease persists uniformly. Finally we present the basic framework of threshold dynamics of the system by using numerical simulations, some of which are different from that of the corresponding multi-strain SIS ODE models.

Comparison of Sprinting With and Without Running-Specific Prostheses Using Optimal Control Techniques

SummaryThe purpose of our study was to get deeper insights into sprinting with and without running-specific prostheses and to perform a comparison of the two by combining analysis of known motion capture data with mathematical modeling and optimal control problem (OCP) findings. We established rigid multi-body system models with 14 bodies and 16 degrees of freedom in the sagittal plane for one unilateral transtibial amputee and three non-amputee sprinters. The internal joints are powered by torque actuators except for the passive prosthetic ankle joint which is equipped with a linear spring–damper system. For each model, the dynamics of one sprinting trial was reconstructed by solving a multiphase least squares OCP with discontinuities and constraints. We compared the motions of the amputee athlete and the non-amputee reference group by computing characteristic criteria such as the contribution of joint torques, the absolute mechanical work, step frequency and length, among others. By comparing the amputee athlete with the non-amputee athletes, we found reduced activity in the joints of the prosthetic limb, but increased torques and absolute mechanical work in the arms. We also compared the recorded motions to synthesized motions using different optimality criteria and found that the recorded motions are still far from the optimal solutions for both amputee and non-amputee sprinting.

Semi-Empirical Continuous Time Neural Network Based Models for Controllable Dynamical Systems

We discuss the problem of mathematical and computer modeling of nonlinear controllable dynamical systems with incomplete knowledge about the object of modeling and the conditions of its operation. The suggested approach is based on a merging of theoretical knowledge for the system with training tools of artificial neural network (ANN) field. We present an extension of previously proposed semi-empirical neural network modeling methods for the case of continuous time ANN-models, which makes it possible to expand the possibilities of this approach. The efficiency of this approach is demonstrated using the example of motion modeling for a maneuverable aircraft.

Modeling of the droplet surface dried on a solid substrate

The paper presents an approach to modeling the surface of a liquid droplet lying on a solid substrate. Receiving the real shape of the droplet surface plays an important role in modeling the process of drying droplets of liquid dispersed systems on solid diffusion-impermeable substrates. This relates to the strong influence of the characteristics of the droplet surface on the values of the coefficients of heat and mass transfer and the value of the surface area of evaporation. A description of the experimental setup for determining the characteristic parameters of the droplet surface is presented. The formulation and solution to the problem of mathematical modeling of the shape of the droplet’s surface, depending on the wetting angle of the liquid-substrate system, surface tension and droplet parameters, are given. The incorrectness of the assumption about the shape of a droplet as an element of a sphere segment is shown. Some results of visualization of the surface of a droplet for various wettability conditions are presented.

Problems of mathematical modelling of elastic boundary value in the stress-strain state of car body elements

This article considers mathematical modeling problems of elastic boundary value. The stress-strain state mathematical modeling is used to assess the adequacy of theoretical solutions. The conditions of stress-strain state elements of the study mathematical modeling of the car body are given. The obtained analytical dependences make it possible to simulate the stress-strain state of the car body elements in real operating conditions and to compare the experimental and calculated values of stresses, strains and displacements arising under load.

Enhancing metacognitive group strategies for modelling

Metacognitive strategies are needed to solve complex problems such as mathematical modelling problems in groups in a goal-oriented and successful manner. In this paper, the results of an intervention study aiming at enhancing students metacognitive group strategies for modelling are presented. Students from 18 different classes (grade nine and ten) worked on six different modelling problems over a period of 10 months. In a pre–post-test design, their metacognitive group strategies were measured by self-reports. Data analysis shows that up to 33% of the total variance of items lay between groups. Thus, a multilevel analysis was conducted with the student level as within level and a group level as between level. The comparison of different models revealed that the assumed structure of metacognitive strategies, divided into the three factors of strategies for proceeding, for regulating and for evaluating for the student level, can be assumed for the group level as well. In addition, the results show that the teaching unit resulted in a significant increase in evaluation strategies in the intervention group, but not in the control group, and not according the other kinds of strategies. Hypotheses to explain these outcomes are established in this report.

METHOD OF TWO-SIDED APPROXIMATIONS OF THE SOLUTION OF THE FIRST BOUNDARY VALUE PROBLEM FOR NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS BASED ON THE GREEN’S FUNCTION USE

Context. The questions of constructing a two-sided iterative process for finding a positive solution of the first boundary value problem for an ordinary second-order differential equation on the basis of the method of Green’s functions are considered. The object of the study is the first boundary value problem for an ordinary second-order differential equation The purpose of the paper is to develop a method of twosided approximations of the problem solution by using the methods of the nonlinear operators theory in semi-ordered spaces.Method. With the Green’s function help the original nonlinear boundary value problem for an ordinary differential equation is replaced by an equivalent integral equation, considered in the space of continuous functions, which is semi-ordered by means of the cone of nonnegative functions. The integral equation is represented as a nonlinear operator equation with a heterotone operator. For this equation a strongly invariant conic segment, the ends of which serve as initial approximations for two iterative sequences, is sought. The first of the sequences,monotonically increasing, approximates the exact solution of the problem from below, and the second one, monotonically decreasing, approximatesit from above. Two conditions for the existence of a unique positive solution of the boundary value problem under consideration and two-sided convergence of successive approximations to it are given. General recommendations on the construction of a strongly invariant conic segment are also given. The developed method has a simple computational implementation and a posteriori error estimate, convenient for use in practice.Results. The developed method was programmed and investigated in solving test problems. The results of the computational experiment are illustrated graphically and with the help of tables.Conclusions. The conducted experiments have confirmed the efficiency and effectiveness of the developed method and allow to recommend it for use in practice for solving the problems of mathematical modeling of nonlinear processes. The prospects for further research may include the development of two-sided methods for solving problems for partial differential equations and non-stationary problems using semi-discrete methods (for example, the Rothe’s method of lines).

Analysis and Simulation Methods of the Mechanisms for the Innovational Development

The paper dedicates in commemoration of K.A. Bagrinovsky, known scientist, doctor of economic sciences, professor. His thesis was theoretic problems of mathematical modeling and operation of economy. His works in the operations research, the methods making decision, the simulation were received in scientific world. The analysis and the modeling of the mechanisms for scientific and technological development for the production systems of different level in economic hierarchic both centrally controlled economy and making mechanism were conduced by Bagrinovsky in CEMI RAS. The paper presents the investigations (2001–2015) of the analysis and the simulation of the different mechanisms of the innovational activity. It also discusses the methods of the development the complex of the simulation models. In a sense simulation modeling is the science and the art as the selection of the salient parameters for the construction model, intake simplification, the computer experiment and the making decision based on scarcity of accuracy models rest on the heuristic power of men: the practical trial, the intelligence and the intuition. K.A. Bagrinovsky introduced the considerable endowment in the development of this direction for economic and mathematical investigation.The principal object was to show that the relationship between the innovational policy and the technological structure, scientific research sector and the introducing of the progressive production and the organizational structure is obtainable by the models. The character of these relationships may be to use in control of the parameters for the modernization economic. The construction simulation models and the experimental computation analysis were presented the investigations the different mechanisms of the innovational development ant the variants of the estimation have been accomplished on the modeling level by the computer experiment.

FUZZY-SETS MODELING OF THE FINANCIAL CONDITION OF THE ENTERPRISE

The existing models of the financial condition of the enterprise estimation is analyzed in the article, the disadvantages of these models in solving the set tasks isgived. A qualitatively new approach to modeling the financial condition of theenterprise based on fuzzy logic is proposed. It is established that in order to modelthe financial condition of an enterprise it is necessary to use not only quantitative (financial), but also qualitative (indicator) indicators of its activity. The method ofmodeling the financial condition of an enterprise by improving the efficiency of its management is improved. It is proved that unlike existing approaches, the proposed modeling technique allows to analyze the financial state of the enterprise, based not only on financial indicators of activity and specifics of enterprise management, but also allows to take into account the specifics of the country, the time period and the industry in which it operates. An analytical review of literary sources concerning the existing methods of modeling the financial state of the enterprises is in the article and the conclusion is made about the relevance of the development of a new model basedon the modern mathematical apparatus, namely, fuzzy sets. On the basis of the analysis of the object of research and the state of the problem of mathematical modeling of economic processes, the main drawbacks of existing methods and models are identified and the main direction of research is chosen. The basic research tasks are solved and formalized. The qualitative analysis of possible methods of solving the problem is carried out. The method of complex modeling of the financial state of the enterprise with the use of the apparatus of fuzzy logic is chosen and justified, which allows to form a model not only taking into account quantitative and qualitative factors of a particular enterprise, but also with the possibility of adjusting it in accordance with the specifics of the country, industry, period of time. Such models have the properties of flexibility and adaptability to the changing conditions of a market economy. Such models have the properties of flexibility and adaptability to the changing conditions of a market economy. And although the calculation of the indicators of financial levels of the enterprise is based on selected indicators and the existing set of sets of different levels, the proposed method provides an opportunity to flexibly adjust the model to the specific conditions and specifics of the company, which distinguishes our approach from previously developed similar approaches. The reliability of the calculations is due to the correct use of the selected research methods and is confirmed by the results of the analysis of the experiments conducted. The appropriateness of using the apparatus of fuzzy logic in conducting a comprehensive analysis of the enterprise. The studies conducted in this paper may be continued in order to develop the apparatus for the analysis of some other parties in the operation of economic systems, which would allow simultaneous analysis of many indicators of different levels. This would allow for a more flexible policy for the management of economic systems and, accordingly, would ensure their profitability and functioning at a higher level. The developed models are capable of conducting analysis on the basis of user-specified parameters. They can be used for analysis in other systems. Using such expert systems on the basis of effective mathematical models will help to obtain relevant recommendations for the formation of an optimal strategy for managing economic systems, which allows relying on the activation and increase of the efficiency of investment activity and the achievement of the most important goals of social policy. In order to increase the efficiency of the decision-making on the evaluation of the financial condition of the enterprise, it is proposed to form an appropriate multi-layered DSS. This will solve numerous problems of an economic and technical nature, among which the main is the need for a complete and accurate description in the model of the entity, identifying a large number of potential solutions, making an accurate decision on the evaluation of the financial condition of the enterprise in real time, the need for handling large arrays of input information, the need to take into account quantitative and qualitative evaluationparameters of the entity at the same time.

Investigation the effect discovery-based learning on students’ metacognition in solving rainbow 2-connection numbers

This study aims to investigate the success of the application of discovery based learning model to improve students’ metacognition in solving the problem of mathematical modelling rainbow 2-connection numbers. The research model used is a mixed method that is a combination of quantitative and qualitative models. The quantitative model used a t-test to test the student learning outcomes while questionnaires and interviews are used to achieve students’ metacognition skills. The results showed that the learning model of discovery based learning can improve students’ metacognitive skills in solving mathematical modelling problem of rainbow 2-connection number, as indicated by the difference of t-test between experimental class and control class pre-test score of two groups [[t(48) = 3,808, p > 0,005]] not significant. For the post-test class experimental was 65, 10 (SD = 11,875) and for control class was 53,85 (SD = 8,274). There is a difference between the two significant mean post-tests [t(48) = −11,25, p < 0,005]. The results of questionnaires and interviews also showed that the DBL model can improve students’ metacognition skills, especially in conducting activities to collect, compare, categorize, analyse, integrate, organize materials and make conclusions.

ANALISIS KEMAMPUAN KOMUNIKASI MATEMATIK SISWA SMP MUTIARA 1 BANDUNG PADA MATERI BENTUK ALJABAR

The purpose of this study was to describe the mathematical communication skills of students of SMP 1 Bandung Junior High School class VII and examine errors in solving test questions regarding the material in the form of algebra. The research method used is the description method. The subject of the research was conducted on 29 students of Grade 1 Pearl Junior 1 Bandung Bandung. The test questions given are in the form of 6 questions. The results showed that mathematical communication skills as a whole were moderate and there were 1 indicator that entered the criteria very well, namely for indicators to rewrite information in the form of images into the language of mathematics. (Well); indicators state the problem of mathematical models of story forms into mathematical symbols, (Good); the indicator states what is known and asked according to the mathematical formula (Good), the indicator completes what is known and what is asked daily events using mathematical formulas (Good). And 1 indicator which includes sufficient criteria, namely the indicator states the completion of the mathematical model of drawing with the right solution idea. And it can be seen that the average score of all questions from all students is included in the good criteria

ANALISIS KESALAHAN SISWA DALAM MENYELESAIKAN SOAL TRIGONOMETRI TIPE HIGHER ORDER THINKING SKILL DITINJAU DARI GENDER

This study aims to describe errors, factors that cause students to make mistakes made by students with high capability, medium, low, and the presence or absence of gender influence in solving Trigonometry question HOTS ( Higher Order Thinking Skill ) type . . This type of research is descriptive qualitative. The subjects of this study are six people, namely three male students and three female students, consisting of two highly capable students, two moderate capable students, and two low capable students. Data collection techniques in this study used the test, interview, and documentation methods. The data analysis technique is done in three stages, namely reduction data, presentation data, verification data, and conclusion. The analytical framework was developed based on Newman’s error category. The results of the study obtained five error types, namely (1) reading errors, (2) comprehension errors, (3) transformation errors, (4) process skills errors, (5) encoding errors. The most mistakes made by students are misunderstanding, transformation errors, and writing answers. General factors that cause these errors appeared because of students' lack of capability in interpreting the questions, level of understanding and students’ creativity who are low in identifying real problems in mathematical models. Research shows there are differences in the error type made by male and female subjects in solving problems which include five types of Newman's error. Keywords: Newman’s error, Trigonometry, HOTS, Gender