Mathematical Model Of Problem Research Articles

Open modelling problems: cognitive barriers and instructional prompts

Open mathematical modelling problems that can be solved with multiple methods and have multiple possible results are an important part of school curricula in mathematics and science. Solving open modelling problems in school should prepare students to apply their mathematical knowledge in their current and future lives. One characteristic of these problems is that information that is essential for solving the problems is missing. In the present study, we aimed to analyze students’ cognitive barriers while they solved open modelling problems, and we evaluated the effects of instructional prompts on their success in solving such problems. A quantitative experimental study (N = 263) and a qualitative study (N = 4) with secondary school students indicated that identifying unknown quantities and making numerical assumptions about these quantities are important cognitive barriers to solving open modelling problems. Task-specific instructional prompts helped students overcome these barriers and improved their solution rates. Students who were given instructional prompts included numerical assumptions in their solutions more often than students who were not given such prompts. These findings contribute to theories about solving open modelling problems by uncovering cognitive barriers and describing students’ cognitive processes as they solve these problems. In addition, the findings contribute to improving teaching practice by indicating the potential and limitations of task-specific instructional prompts that can be used to support students’ solution processes in the classroom.

The concept of a complex digital twin of a furnace for metal heat processing

A number of mathematical models, digital twins and diagnostic systems are considered, which are used for individual processes and elements of flame furnaces. An idea was put forward about the expediency of creating a complex digital twin, which simultaneously takes into account the main processes in metal heat processing (heating for rolling or heat treatment) in a flame furnace. A conceptual definition of the structure and interconnection of the elements of a complex digital twin of a furnace for metal heat processing is carried out. It is substantiated that the data to ensure the operation of the digital twin are divided into two groups: constantly or periodically measured values and data that are unchanged. The minimum set of quantities, the measurement of which is necessary for the operation of the proposed complex digital twin of the furnace, is determined. A list of objective data about the unit and the technological process, which are unchanged over a long period of operation, is also compiled. At the same time, a scheme for organization the interaction of special computing units has been developed, which allows to achieve high accuracy in solving the traditional problem of mathematical modeling of metal heat processing ‒ determining the metal temperature field. To achieve this result, computing units are used to identify the parameters of all processes that affect the solution of the problem: the value of air suction into the furnace chamber and along the smoke path, the value of the recovery coeffi-cient, the oxygen consumption for the oxidation of the metal surface, the average heat flux density absorbed by the metal surface. Special blocks are also provided for diagnosing the state of the heat exchanger, fan, smoke path, chimney and elements of the mechanical equipment of the furnace. Ultimately, top-level blocks are provided that allow, based on the processing of information from all the basic blocks of the complex digital twin, to analyze the energy efficiency of the furnace unit, obtain the technical and economic characteristics of its operation, as well as to issue recommendations to clarify the current parameters of technological processes and maintenance of equipment elements.

Comparative Evaluation of the Accuracy of Numerical Methods on Reference Solutions

When organizing a mass practical solution of computational problems of gas dynamics with the help of mathematical modeling, information on the comparative accuracy of the numerical methods used is now increasingly in demand. As a rule, calculators need information not only for a particular combination of the defining gas-dynamic parameters of the problem (characteristic Mach, Reynolds numbers, etc.), but also for the variation of these parameters in certain ranges. This work presents numerical studies devoted to a comparative assessment of the accuracy of numerical methods for a number of problems with reference solutions. The calculation of these problems is carried out for ranges of characteristic numbers using various numerical methods. The results obtained are compared with the reference solution and make it possible to estimate the error for each of the numerical methods. Calculations are carried out using the construction of a generalized computational experiment. A generalized computational experiment is a computational technology that combines the solution of mathematical modeling problems, parallel technologies and visual analytics technologies. The results of the generalized computational experiment are multidimensional arrays, where the dimension of the arrays corresponds to the defining parameters. Analysis and visual representation of the obtained results provide information on the comparative accuracy of the numerical method for the selected class of problems.

Learning statistics using realistic mathematics education: A case of fifth-grade students of elementary school

This study aims to describe realistic mathematics education (RME) in helping students' understanding of concepts in statistical material about data collection. This research uses descriptive qualitative research methods with a design research approach. The subjects of the study were 29 fifth-grade students of a public school in Pekanbaru, Riau, Indonesia. The results of this study show that RME is applied in learning with the following steps, namely first, providing and explaining contextual problems to be solved by students; second, students solve contextual problems for mathematical models in their own way; third, discuss and compare each problem solving given by students; Fourth, students conclude and summarize the learning concept. From these activities, students can develop knowledge and skills in collecting and presenting data.

Mathematical Modeling Problem Solving Viewed from Students’ Mathematical Self-Concept on Means-Ends Analysis Based on Blended Learning

Mathematical Modeling Problem Solving Viewed from Students’ Mathematical Self-Concept on Means-Ends Analysis Based on Blended Learning

Mathematical model for the dynamics of bacterial meningitis (Meningococcal meningitis): a case study of Yobe State Specialist Hospital, Damaturu, Nigeria

A model for bacterial meningitis was created by adding a class of transporters to the basic Susceptible Carrier Infected and Recovered (SCIR) model, since vaccination and treatment are the best methods of controlling the transmission of most overpowering sicknesses. Immunization assists helpless people with building either a drawn out invulnerability or transient resistance while treatment decreases the quantity of sickness actuated passing and the quantity of irresistible people locally or country. This study comprises of a mathematical model for bacterial meningitis dynamics that can be used to a wide range of mathematical modeling problems. In this exploration, a nonlinear deterministic model with time reliance controls has been proposed to depict the elements of bacterial meningitis in a populace. We discovered that the (EEP) and (DFE) are locally asymptotically stable in our study. We now advise the researcher to determine whether it is globally asymptotically stable in order to achieve optimal disease control. The presence of an endemic harmony and the calculation of the reproduction number . The mathematical arrangement shows that vaccinating vulnerable people will prompt disposal of the infection in the public and furthermore it will lessen the weight on wellbeing suppliers. Numerical simulations were presented to explain the parameters in the end path in the model were carried out by using MAPLE software. Most likeable of this research work shows that the rate at which treatment and vaccination rate increases to the higher value the recovered compartment increases to the peak point. This means treatment and vaccination has an impact on reducing the case of bacterial meningitis in a population. Now we conclude that a high infection transmission rate requires a high vaccine and treatment rate on the effect of vaccination against meningitis.

Mathematical modeling of movement of multi-rotor type aircraft

The problem of determining the features and setting the problem of mathematical modeling of multirotor aircraft (LA) is considered. Differences of their mathematical models from the classical mathematical models of aircraft and single-rotor helicopter type aircraft are considered. The analysis and substantiation of the forces and moments acting on a multirotor aircraft are carried out, taking into account the peculiarities of considering the corresponding coordinate systems necessary for studying the spatial motion of the aircraft. The problem of controlling the trajectory motion of an aircraft is formulated taking into account the rotation of its structure around the center of mass. Based on the consideration of the structural diagram of one of the most common four-rotor aircraft (quadcopter), a scheme for creating control forces and moments acting on the aircraft, under the influence of which the aircraft’s trajectory in space changes, is substantiated. The main mathematical dependences characterizing the kinematics of the motion of a multi-rotor aircraft are given. On the basis of the analysis carried out, a generalized block diagram of the control process for such an aircraft is substantiated and presented. To test the performance and adequacy of the mathematical model, a study was made of the movement of a quadrocopter in a vertical plane between given points in space in accordance with the law of forced control, which ensures the movement of an aircraft in space with maximum speed. The computer simulation of the obtained analytically mathematical dependences showed that this approach is applicable to the construction of mathematical models of the motion of multirotor type aircraft of various design layouts.

Students’ errors in solving climate change context mathematical modeling problems

This research is a descriptive study that aims to describe the errors of students of the Faculty of Education and Teacher Training (FKIP), Sriwijaya University, in solving Climate Change context mathematical modeling problems, to minimize errors made in solving mathematical modeling problems, especially in the context of climate change in the future. No one has researched student errors in solving mathematical modeling problems in the Climate Change context using the Newman error procedure. Data collection techniques used in this study included tests and semi-structured interviews. The data analysis technique used for test data is by analyzing errors using the Newman procedure. The results of this study with 52 participants were types of errors by FKIP Sriwijaya University in solving Climate Change context mathematical modeling problems as follows for question 1 and question 2, and respectively there were 23.1% and 15.4% errors in understanding the problem, 46.2% and 96.2% transformation errors, 32.7% and 96.2% writing. The causes of errors are errors in reading, errors in determining the problem and existing information, errors in making assumptions, errors in identifying variables, and errors in making mathematical models.

A comprehensive approach to double inverted pendulum modelling

The problem of mathematical modelling and indication of properties of a DIP has been investigated in this paper. The aim of this work is to aggregate the knowledge on a DIP modelling using the Euler-Lagrange formalism in the presence of external forces and friction. To indicate the main properties important for simulation, model parameters identification and control system synthesis, analytical and numerical tools have been used. The investigated properties include stability of equilibrium points, a chaos of dynamics and non-minimum phase behaviour around an upper position. The presented results refer to the model of a physical (constructed) DIP system.

Development of mathematical modeling methods filtration flows of multicolored liquids

Today the relevance of the issues of environmental consequences of technological human intervention in the world around us is beyond doubt. The rapid development of technique and technology entails the development of new ways and methods of environmental protection. At the same time, issues of theoretical research, the creation of mathematical, and then computer models of the studied phenomena, play an important role. The article considers the influence of pollution of stationary filtration flows of an incompressible multicolored liquid on the environment. The problems of mathematical modeling of the filtration flows of multicolored liquids are solved. It is assumed that pollution does not significantly change the properties of the filtered liquid, its density and viscosity. Liquids do not mix and have a clear interface, which does not have a harmful effect on the environment.

Data-driven nonlinear MIMO modeling for turbofan aeroengine control system of autonomous aircraft

The mathematical modeling problem of a general turbofan aeroengine control system used in autonomous aircraft is investigated in this paper. Unlike the thermodynamics or physical mechanism-based modeling methods in the existing literature, a pure data-driven modeling approach is presented with high accuracy via online and offline data only. This is achieved by using the nonlinear autoregressive neural network with exogenous inputs (NARX) neural network. In comparison with the existing NARX-based MIMO modeling methods, the series–parallel structure and the parallel structure are integrated for the neural network training. Faster convergence and stability as well as higher modeling accuracy are thus achieved. Another feature of this approach is that it is independent of all the components’ dynamics in the turbofan aeroengine. Simulation results are given to verify the effectiveness of the proposed modeling approach. This is further validated by experimental testing with the method applied to mini turbofan aeroengine testbed.

A Comparative Study on Solution Methods for Fractional order Delay Differential Equations and its Applications

Fractional calculus is one of the evolving fields in applied sciences. Delay differential equation of non-integer order plays a vital role in epidemiology, population growth, physiology economy, medicine, chemistry, control, and electrodynamics and many mathematical modeling problems Fractional Delay differential equations usually lacks analytic solutions and some of these equations can only be solved by some numerical methods. In this review article we present a comparative study on some standard numerical methods applied to solve linear fractional order differential equations with time delay. Fractional finite difference method (FFDM), Predictor-corrector method (PCM) with new and extended versions has been discussed in this article. All above mentioned methods use the Caputo type fractional differential operator to define fractional derivatives. Solution of a real-life problem formulated by FDDEs has been discussed under these methods. Results have been presented in tabular and graphical form to analyze the efficiency and scarcity of mentioned methods. These graphical and numerical comparisons are provided to illustrate and corroborate the similarity and differences between these methods.

Modeling of thermal power plants in the study of reliability of power supply and energy security

RELEVANCE. Maintaining the required level of energy security (ES) and reliability of fuel and energy supply is one of the priority tasks in the management of the energy sector. In the functioning of the fuel and energy complex various threats are possible, which can lead to the violation of the reliability of fuel and energy supply and the emergence of shortages ofvarious types of energy resources. To assess the level of energy security, approaches are considered to form a mathematical model of the FEC, which combines all sectors of the energy sector. Despite the trends of transition to low-carbon energy sources associated with energy transformation, thermal power plants remain one of the main sources of heat and electricity.Due to the ability to generate large amounts of heat only thermal power plants can be a source of centralized energy supply of large territories. In this regard, when modeling the fuel and energy complex, it is necessary to develop mathematical models of TPPs, which will correctly reflect the technological processes that affect the reliability of energy supply. OBJECT. Development of mathematical models of interaction of electric power, heat supply and fuel systems within the framework of thermal power plants designed to study energy security and reliability of fuel and energy supply. METHODS. Mathematical modeling of fuel consumption dependences on electrical and thermal load at thermal power plants is used as methods. To obtain analytical dependencies, the least squares approximation of the consumption characteristics of thermal power plants was carried out. RESULTS. As part of the work, analytical dependences of fuel consumption on thermal and electrical load for various types of thermal units were obtained. For all considered steam turbines, linear dependences of theamount of heat supplied to the turbine on its load (electrical and / or heating depending on the type of turbine) were obtained. For boilers, an approximation of the dependence of efficiency on thermal load was carried out. According to the results of the approximation, a linear dependence is assumed. For gas turbine installations, an approximation of the dependence of the efficiency on the load was carried out, according to the results of which a second-order polynomial was obtained. CONCLUSION. The paper investigates the problem of mathematical modeling of the interconnected operation of fuel, heat and power and electric power systems within the framework of the functioning of the energy sector. Analytical dependences are obtained for different types of generating equipment, including boilers, steam turbines and gas turbine installations. The obtained dependences are necessary for the analysis of energy security and reliability offuel and energy supply. The models of thermal power plants presented in the paper are the most suitable for the analysis of energy security and reliability of fuel and power supply, as they provide the necessary accuracy of calculations and take into account the specifics of various generating equipment, without dropping to the level of microparameters.

A Novel Mutation-based Differential Evolution Algorithm for Solving Real-World Optimization Problems

In today's environment, evolutionary algorithms play a key role in solving optimization problems in mathematical models and applications. This article introduces a multi-objective differential evolution algorithm that extends the single-objective differential evolution (DE) algorithm to solve optimization problems in mathematical models and applications. Many existing algorithms face issues with diversity loss and convergence rate. To address these problems, this article proposes a novel mutation operator called ubiquitination mutation for DE, which has been added to the DE method. The proposed approach was tested on three real-world optimization problems: the next release problem, the multiline distance minimization problem, and data clustering. Results indicate that the proposed mutation operator outperformed state-of-the-art algorithms. In addition, the proposed approach provided better solutions in both single- and multi-objective platforms for various real-world problems.

Geodynamics of the Caucasus – Anatolian-Arabian region and Turkey-Syria Earthquakes 2023

The activation of natural disasters in the world requires the development of new approaches to the study of geological processes, in particular, at the boundaries of lithospheric plates, characterized by earthquakes, increased seismicity, volcanism, landslide processes, tsunamis and other dangerous natural processes and hazards. Earthquake of M 7.8 struck south -eastern Turkey and north - western Syria on 6 February 2023. The M 7.8 earthquake is the largest in Turkey since the 1939 Erzincan earthquake, and the second-strongest recorded in the country, after the 1668 North Anatolia earthquake. More than 52,800 deaths were confirmed: more than 46,100 in Turkey, and more than 6,700 in Syria. It is the deadliest natural disaster in Turkey's modern history. The earthquakes caused over US$100 billion in damages. The geodynamic models construction for the deep structure of natural hazards regions is an important contribution to the study of active continental margins, which is necessary for the earthquake forecast, prediction and prognosis, assessing geoecological risks and preparing population actions in the event of natural disasters and catastrophes. The Caucasus - Arabian region is a complex highly-stressed geodynamic structure, characterized by increased heat flow, high seismicity, magmatism and volcanism. The geodynamics of the Caucasus - Arabian region is determined by the collision of the Eurasian and Arabian lithosphere plates, as well as the complex history of the development of the Alpine-Himalayan belt and surrounding territories. The problem solution of geological structures formation and evolution in various complex geodynamic settings and natural hazards forecast and prognosis requires an analysis of all available geological - geophysical data, as well as the formulation and solution of problems of mechanical and mathematical modeling. Slow lithospheric deformations are simulated by models of viscous flow in multi-layered, incompressible, high-viscosity Newtonian fluid, using Navier-Stocks equation and discontinuity equation. The solution of the inverse problem of geodynamics by the direct method is developed. The first inverse problem of geodynamics was solved - the restoration of the velocity fields, pressures and stresses at the depth of the lithosphere according to the available data on the velocities on the surface. The second inverse problem of geodynamics has been posed and solved - the determination of the movement of boundaries at the depth of the lithosphere based on the given movements of the surface. The solutions obtained can be used to analyze deep geodynamic problems, and together with geothermal modeling, geological and geophysical methods and seismic tomography can serve as a reliable apparatus for studying deep geodynamics due to the formation and evolution of geological structures and the lithosphere stress-strain state researches. The solution of the problem is analyzed on the example of the Caucasus - Arabian region geodynamics. The Geodynamic concept of geoenvironment has been developed. Geodynamic models of the regions of hazardous natural processes in order to predict and prevent natural disasters and catastrophes are constructed. An algorithm for creating monitoring systems is suggested.

RESEARCH OF COMPUTATIONAL MATHEMATICAL MODELS FOR TECHNICAL SYSTEMS

In the theory of analysis and synthesis of technical systems, mathematical modelling and optimization of multilayer systems containing sources of physical fields occupy an important place. This is due to the fact that their state is described by means of boundary value problems with multidimensional differential equations. To solve the boundary value problems and implement the process of optimizing the technical parameters of the modelled systems, it is necessary to conduct interdisciplinary studies of computational and applied optimization mathematical models. Fulfilment of the conditions for the existence of a single solution to boundary value problems by default is possible only when the object of study is a single-layer material under the action of load sources. If it is necessary to calculate and optimize the technical parameters of a multilayer material subjected to load sources, then it is impossible to immediately guarantee the correctness of the calculated and applied optimization mathematical models, since it is necessary to obtain the conditions for the existence and uniqueness of solutions to boundary value problems with systems of differential equations. Maximizing the technical parameters of load sources and averaging the characteristics of material layers will lead to approximate values of the objective function and technical parameters of the modelled system, which leads to irrational consumption of energy and heat resources and uncontrolled losses, and useless losses of the test material in the technological process. The article presents the conditions for the correctness of multipoint boundary value problems with multidimensional differential equations describing the state of a multilayer material under thermal action. It is advisable to use these studies to substantiate the correctness of other technical and biotechnological systems, which will increase the accuracy of the implementation of applied optimization problems of economic and mathematical modelling.

On the Bilinear Second Order Differential Realization of an Infinite-Dimensional Dynamical System: An Approach Based on Extensions to M2-Operators

Considering the case of a continual bundle of controlled dynamic processes, the authors have studied the functional-geometric conditions of existence of non-stationary coefficients-operators from the differential realization of this bundle in the class of non-autonomous bilinear second-order differential equations in the separable Hilbert space. The problem under scrutiny belongs to the type of non-stationary coefficient-operator inverse problems for the bilinear evolution equations in the Hilbert space. The solution is constructed on the basis of usage of the functional Relay-Ritz operator. Under this mathematical problem statement, the case has been studied in detail when the operators to be modeled are burdened with the condition, which provides for entire continuity of the integral representation equations of the model of realization. Proposed is the entropy interpretation of the given problem of mathematical modeling of continual bundle dynamic processes in the context of development of the qualitative theory of differential realization of nonlinear state equations of complex infinite-dimensional behavioristic dynamical system.

The Effect of Mathematical Modelling Activities on Students' Mathematical Modelling Skills in the Context of STEM Education

This study was conducted to examine the effect of mathematical modelling activities on the mathematical modelling skills of secondary school students in the context of STEM education. The study was designed according to the embedded design, one of the mixed research methods. The study group of research consists of 66 eighth-grade students studying in a public school in the central district of a large province in the south of Turkey in the 2020-2021 academic year. While the criterion sampling method, one of the purposeful sampling methods, was used to determine the quantitative study group of the research, the maximum variation sampling method was used to determine the qualitative study group. On the other hand, in the context of STEM education, mathematical modelling problems, evaluation rubric and semi-structured interview forms were used as data collection tools in the research. As a result of the research; It was concluded that mathematical modelling activities in the context of STEM education positively improved the mathematical modelling skills of secondary school students. In addition, it has been concluded that the students who receive education with mathematical modelling activities applied in the context of STEM education gain different interdisciplinary perspectives, experience positive developments in their thinking skills, adapt to group work more easily, and increase their interest in engineering and technology.

Multiphysics Simulations of the Process of Neuromuscular Activation

In this Master's thesis, we consider the problem of mathematical modelling and computer simulations of neuromuscular activation. We describe the biological and biochemical processes that result in a muscle contraction. For each of them, we derive a mathematical model, established in the literature. In particular, we consider: the Hodgkin-Huxley model of neural transmission; a reaction-diffusion model for the process of neurotransmitter release in the synaptic gap between a nerve and a muscle cell; an ODE system, based on chemical reaction schemes, proposed by Williams, for the process of calcium dynamics inside the muscle cell; Hill’s model for the generation of muscle force, triggered by calcium dynamics. We study the models numerically to illustrate the behaviour of the model solutions and to interpret them from a biological point of view. For the model of calcium dynamics, we also make qualitative analysis and obtain original results for the asymptotic behaviour of the model solutions. Further, we propose a framework for coupling the models, mentioned above, so that we can obtain new integrated multiphysics simulations of the whole process. Our initial motivation for the study is the future application of the proposed approach for modelling neuromuscular diseases. Therefore, the framework we propose is based on the idea of modelling micro-scale processes and studying their effect on the macro-scale muscle action.

From Multidimensional Ornstein - Uhlenbeck Process to Bayesian Vector Autoregressive Process

The main purpose of  this paper is to make the connexion between stochastic analysis, the Bayesian Statistics, and time series analysis for policy analysis. This approach solves the problem of mathematical modelling - the presence of uncertainties in the models and parameters -  that reduces the  policy analysis and forecasting   effectiveness. By using the multiple It\^o  integral, the multidimensional Ornstein - Uhlenbeck process can be written as a Vector Autoregressive with lag 1 (VAR(1)) that is the generalization of Vector Autoregressive process. The  limit of this approach is in fact it requires  the strong foundations of stochastic analysis, the Bayesian Statistics, and time series analysis.