Magistracy
Speciality Code:
7M06106
Speciality Name:
Mathematical and Computer Modelling
Faculty:
Mechanics and Mathematics
Qualification:
 Scientific and pedagogical direction  Master of technical sciences
 Model of graduating student
 Mandatory disciplines
 Elective disciplines
 Professional
ON1. Conduct research and obtain new fundamental and applied results, plan research in accordance with the approved direction of research in the field of specialization. Apply modern methods of mathematical modeling for scientific research.
ON2. Use the concepts of the essence, mechanisms and patterns of naturalphysical, chemicaltechnological, natural, biological, medical and random processes in the development of conceptual and theoretical models of solvable applied problems.
ON3. Use highperformance computing systems and methods of mathematical and computer modeling in the analysis and solution of applied problems.
ON4. To draft and develop recommendations for the implementation of the research and numerical experiments in the production and financial industry using mathematical and computer simulation methods. Use probabilistic methods to solve actual problems of mechanics, physics, and economics.
ON5. To carry out an indepth analysis of problems, to produce a substantiation of physical problems, to reveal their naturalscientific essence in the course of scientific and research activities, to apply the corresponding mathematical apparatus and a numerical algorithm to solve them.
ON6. Create mathematical models and apply numerical methods for solving problems of heat and mass transfer with a moving boundary, inverse and illposed problems of mathematical physics, biomedical processes, naturalphysical processes, chemical processes, financial processes, atmospheric processes, dynamics of multiphase turbulent flows.
ON7. Analyze, design and conduct numerical experiments of constructed mathematical models of industrial, physicotechnological, nonlinear nonstationary physical, chemical, biomedical, financial processes. Reproduce numerical solutions of engineering problems of hydrodynamics on highperformance systems
ON8. Conduct scientific research in the field of mathematical modeling of heat and mass transfer phenomena, continuum dynamics, thermal and complex processes, mechanical processes, thermodynamic processes in gas dynamics, thermodynamic and electrical processes in solids, applied problems, problems of mathematical physics, and stability theory in economics and technology.
ON9. Apply mathematical and numerical apparatus to research in the field of financial mathematics, computational hydrodynamics, mechanics, turbulence modeling, physical, chemical, biomedical, nonlinear nonstationary physical processes and complex systems for solving applied problems.
ON10. To conduct educational and extracurricular work, have the skills of pedagogical activity. Conduct lectures, seminars and laboratory classes for undergraduate students on the profile of the specialty. To master and introduce new innovative technologies and approaches in the field of education in teaching practice.
ON11. Conduct research to use the results obtained in the framework of the implementation of interstate programs in the field of mathematical modeling, mathematics, biomedicine, physics, chemistry and mechanics. Participate in scientific seminars and conferences.
ON12. To work in a team, tolerantly perceiving social, ethnic, confessional and cultural differences, critically evaluate one’s activity, team one’s activities, outline ways and choose means for selfdevelopment, improvement of one’s qualifications.
ON2. Use the concepts of the essence, mechanisms and patterns of naturalphysical, chemicaltechnological, natural, biological, medical and random processes in the development of conceptual and theoretical models of solvable applied problems.
ON3. Use highperformance computing systems and methods of mathematical and computer modeling in the analysis and solution of applied problems.
ON4. To draft and develop recommendations for the implementation of the research and numerical experiments in the production and financial industry using mathematical and computer simulation methods. Use probabilistic methods to solve actual problems of mechanics, physics, and economics.
ON5. To carry out an indepth analysis of problems, to produce a substantiation of physical problems, to reveal their naturalscientific essence in the course of scientific and research activities, to apply the corresponding mathematical apparatus and a numerical algorithm to solve them.
ON6. Create mathematical models and apply numerical methods for solving problems of heat and mass transfer with a moving boundary, inverse and illposed problems of mathematical physics, biomedical processes, naturalphysical processes, chemical processes, financial processes, atmospheric processes, dynamics of multiphase turbulent flows.
ON7. Analyze, design and conduct numerical experiments of constructed mathematical models of industrial, physicotechnological, nonlinear nonstationary physical, chemical, biomedical, financial processes. Reproduce numerical solutions of engineering problems of hydrodynamics on highperformance systems
ON8. Conduct scientific research in the field of mathematical modeling of heat and mass transfer phenomena, continuum dynamics, thermal and complex processes, mechanical processes, thermodynamic processes in gas dynamics, thermodynamic and electrical processes in solids, applied problems, problems of mathematical physics, and stability theory in economics and technology.
ON9. Apply mathematical and numerical apparatus to research in the field of financial mathematics, computational hydrodynamics, mechanics, turbulence modeling, physical, chemical, biomedical, nonlinear nonstationary physical processes and complex systems for solving applied problems.
ON10. To conduct educational and extracurricular work, have the skills of pedagogical activity. Conduct lectures, seminars and laboratory classes for undergraduate students on the profile of the specialty. To master and introduce new innovative technologies and approaches in the field of education in teaching practice.
ON11. Conduct research to use the results obtained in the framework of the implementation of interstate programs in the field of mathematical modeling, mathematics, biomedicine, physics, chemistry and mechanics. Participate in scientific seminars and conferences.
ON12. To work in a team, tolerantly perceiving social, ethnic, confessional and cultural differences, critically evaluate one’s activity, team one’s activities, outline ways and choose means for selfdevelopment, improvement of one’s qualifications.

Master’s dissertation preparation and defense (MDPaD)
 Number of credits: 12
 Type of control: Master Dissertation
 Description: The main purpose of "The implementation of a Master Thesis": the formation of master students in preparation for the defense of the thesis for the Master in specialty (by industry). During the study of course, master student's should be competent in: 1. demonstrate the progress of solving problems arising in the course of research activities and requiring indepth professional knowledge; 2. to argue for carrying out theoretical or experimental research within the framework of the tasks, including a mathematical (simulation) experiment; 3. can choose the necessary research methods, modify existing methods and develop new methods, based on the tasks of the specific study; 4. to use foreign languages for independent work on normative sources and scientific literature; 5. formulate the goals and objectives of the dissertation research, determine the scientific novelty and practical significance of the results of research activities; to develop a structurally methodological scheme for performing research.

Foreign Language (professional)
 Number of credits: 5
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: the acquisition and improvement of competencies in accordance with international standards of foreign language education, allowing the use of a foreign language as a means of communication for the successful professional and scientific activities of the future master. able to compete in the labor market, as new knowledge, technologies are available through a foreign language, mastering a professional foreign language serves as a tool in mastering new competencies

History and philosophy of science
 Number of credits: 3
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The course "History and Philosophy of Science" introduces the problem of science as an object of special philosophical analysis, forms knowledge about the history and theory of science; on the laws of the development of science and the structure of scientific knowledge; about science as a profession and social institution; оn the methods of conducting scientific research; the role of science in the development of society. The maintenance of a course includes detection of specifics and interrelation of the main problems, subjects of philosophy of science and history of science; studying consciousness of science in its social and philosophical foreshortenings; consideration of a phenomenon of science as professions, social institute and direct productive force; disclosure of disciplinary selfdetermination of natural, social and technical science, their communities and distinction.

Mathematical modeling in computer graphics
 Number of credits: 5
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: Thepurpose of the discipline: is forming the ability of students to develop and use basic mathematical and graphic tools of computer graphics for their successful application in professional activities related to graphic programming. During the study of course, master students should be competent in:  demonstrate the understanding and ability to apply the basic methodological principles of graphical programming for creating images of threedimensional scenes and objects;  determine the criteria for classification and organization of graphical systems and models;  combine and effectively use various tools of computer graphics to develop graphic programs and program modules;  apply the mathematical apparatus of computer graphics in their professional activities;  synthesize, interpret and critically evaluate various types of graphic information. During the study of the discipline master students will learn following aspects: OpenGL application programming interface. GLUT application structure. Standard 3D objects of the GLUT and GLU libraries. Clipping planes and stencils. Fractals. Their properties. Dynamic fractals. Control functions. Input devices. Simulation of light. Light sources and their types. Materials. Light effects. Textures. Types of textures.

Mathematical and computer modeling of medical and biological processes
 Number of credits: 5
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: to form knowledge in solving actual scientific and applied problems related to modeling processes occurring in living organisms and systems, processing and system analysis of experimental data, to form knowledge in the field of the theory of dynamic systems and nonlinear dynamics applied to the problems of physics of living systems. During the study of course, master students should be competent in:  to solve actual scientific and applied tasks related to the modeling of processes occurring in living organisms and systems;  to formulate the tasks of analytical and numerical research of dynamic systems;  choose adequate theoretical and numerical methods for their solution;  process and analyze experimental data. During the study of the discipline master students will learn following aspects: Classical examples of mathematical models of biomedical processes using the apparatus of linear and nonlinear dynamic systems reflecting the characteristic features of the studied processes will be studied. In particular, when simulating the course of chemical reactions, it is necessary to numerically solve rigid ODE systems. A wide class of processes in biology and medicine is modeled using nonlinear parabolic equations (reaction – diffusion). Methods for the numerical solution of such equations will be studied. A wide range of processes occurring inside a living organism are described on the basis of a system of partial differential equations of continuum mechanics. Methods for constructing mathematical models of biomedical processes will be shown, demonstrating the effectiveness of their use for understanding the mechanisms of functioning of systems. There will be considered numerical algorithms for solving the constructed models, modern ways of visualizing the obtained numerical results.

Pedagogy of higher education
 Number of credits: 5
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of disciplinemastering the basics of professional and pedagogical culture of higher school teacher, the formation of competencies, skills and teaching activities in universities and colleges. The following issues are studied: the role of pedagogical science in the system of Sciences; the system of higher professional education in Kazakhstan; methodology of pedagogical science; didactics of higher education; design of TLAstrategy of education, the use of traditional and innovative methods and forms of education.

Psychology of management
 Number of credits: 3
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline is to provide scientific training for highly qualified specialists on the basis of studying the fundamental concepts of management psychology, capable of understanding the current state of the theory and practice of management psychology in an amount optimal for use in subsequent professional activities; apply and describe psychological methods of studying individuals and social groups (communities) in order to increase management efficiency;

Modern methods of computational fluid dynamics
 Number of credits: 5
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: Thepurpose of the discipline: to form students' ability to apply modern methods of numerical solution of hydrodynamic equations, practical application of the main stages of mathematical modeling of hydrodynamic processes, including the physical formulation of the problem, the choice of a mathematical model and the formulation of the initialboundaryvalue problem, the construction of a network model, the choice and development of grid approximations, teach the construction of various algorithms for constructing finite difference and finite element meshes.. During the study of course, master students should be competent in: – applyingthe methods for the numerical solution of the NavierStokes equations in the case of an incompressible fluid on a spaced grid; – applying solutions to viscous compressible fluid; – building endtoend counting schemes, explicit and implicit methods for solving initial equations; – applying the McCormack method, the BimaWarming method, TVD schemes; – applying the high order schemes  ENO and WENO. During the study of the disciplines, master tudents will learn following aspects: methods for the numerical solution of the NavierStokes equations in the case of an incompressible fluid on a spaced grid. The solution methods for a viscous compressible fluid, hyperbolic systems, conservation laws, and problems of their solution are studied; crosscounting schemes, explicit and implicit methods for solving initial equations; McCormack method, BimWarming method, TVD schemes (monotonous reconstruction, slope limiters). High order methods. ENO and WENO.

Modern methods of mathematical modeling
 Number of credits: 5
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: to form the skills of applying theoretical and practical aspects of modern modeling methods for solving problems of applied mathematics, skills of using mathematics packages corresponding to tasks, to form skills to interconnect modules implemented in math packages with software implemented using highlevel programming languages (Python, Java). Modern mathematical packages will be studied that allow solving applied problems of natural science. During the study of course, master students should be competent in:  using modern mathematical packages that allow solving applied problems of mathematics and physics;  interconnecting the modules implemented in software math packages;  independently building the program modules for solving applied tasks.  knowing the methods of setting and solving problems numerically,  having the skills of constructing finitedifference approximations, conducting computer computational experiments;  having a deep knowledge of the basics of computational fluid dynamics. During the study of the discipline master students will learn following aspects: Modern mathematical packages will be studied that allow solving applied problems of natural science, like Ansys, which combines and connects many applications for calculating problems of fluid dynamics, mechanics of a deformable solid; Comsol, which allows to simulate almost all physical processes that are described by partial differential equations, SCAD is a computational complex for analyzing the strength of structures using FEM.

Modern technologies of parallel programming
 Number of credits: 5
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: learning the basic techniques of parallel programming in environments Open MP, MPI; the acquisition of skills for configuring a computing cluster in the operating systems Linux, Windows. During the study of course, master students should be competent in:  having indepth knowledge of parallel computing, basic parallel programming technologies;  understanding the systems with massive parallelism;  efficiently working on computer clusters;  using OpenMP, and MPI for programming languages Fortran, C++. During the study of the discipline, master students will learn following aspects: Debugging software, the use of basic functions OpenMP, MPI to solve complex problems of mathematical physics using programming languages C++ and Fortran 90; construction of topologies, subsystems; management of process groups, communicators.

High performance computing technologies
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: to form knowledge about the basic architectural concepts of building highperformance information processing facilities; macrostructure of largescale distributed computing systems (VS); functional structures and the most interesting industrial implementations of highperformance computing systems. During the study of course, master students should be competent in:  explain the key concepts and principles of the organization of parallel computing.  to demonstrate the main trends in the development of parallel architectures, factors affecting performance, criteria for choosing a software and hardware platform for solving computationallycomplex tasks of a given class.  to master the general methodology for developing parallel programs, methods for evaluating the effectiveness of parallel algorithms and the maximum attainable parallelism on the target computing architecture.  be able to use the means of remote access to computing resources for collective use and launch of parallel programs on computing clusters. the ability to generate new ideas and demonstrate the skills of independent research work and work in a research team;  the ability to conduct research and obtain new scientific and applied results;  ability to develop conceptual and theoretical models of solvable scientific problems and tasks;  the ability of indepth analysis of problems, formulation and justification of the tasks of scientific and designtechnological activities. During the study of the discipline, master students will learn following aspects: Problem, object and object orientation of highperformance computers and computing systems. Specific requirements for them. Singleprocessor and multiprocessor systems; multiprocessor systems with shared memory and local memory; parallel and distributed computing systems. Classification systems of supercomputers. Performance evaluation of highperformance computers and computing systems. Amdal's law, theoretical and real productivity growth in parallel computing. Principles of development of parallel methods. Simulation of parallel programs. Simulation of mass transfer processes.

Computer modeling and vizualization in graphic paсkages
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of teaching the discipline is to study various graphic packages for computer modeling and visualization. Course objectives  the study of undergraduates practical drawing techniques and types of drawings, such as construction, engineering and many others. The course covers the modeling of physical processes of the real world, using inverse kinematics, the creation of video editing effects, familiarity with the MaxScript language. During the study of course, students should be competent in: – know the inverse kinematics and the Character Studio module; – have the skills to work with atmospheric effects; – be able to simulate physical processes; – have the skills to work with global illumination; – demonstrate knowledge of the script language MAXScript; –  demonstrate knowledge of the possibilities of automating the development of design and design documentation.

Mathematical models of the theory of elasticity
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: formation the ability to correctly build mathematical models and algorithms for the study of dynamic processes in deformable solids and media. Acquaintance with physicomathematical models of deformable solids, methods for solving model boundaryvalue problems, with the basics of conducting various numerical experiments to study the dynamics of mechanical properties, and features of computer technology. During the study of course, master students should be competent in:  studying the concepts of deformations of the continuum, measures and stress tensors, their properties, concepts of geometrically linear and nonlinear approaches;  deriving stress tensors, moment voltages,  knowing the basics of nonequilibrium thermodynamics of the continuum, the concepts of material stability and design;  applying the basic concepts of nonlinear mechanics of continuous media for the formulation of mathematical formulation of problems in research and development activities.  analyzing the mathematical statement made, to linearize the task, to record the initial and boundary conditions; During the study of the discipline, master students will learn following aspects: Models for the study of dynamic processes in deformable solids and media. Physical and mathematical models of deformable solids, methods for solving model boundary value problems. The concepts of deformations of the continuum, measures and strain tensors, their properties, the basics of the thermodynamics of the continuum, the concepts of material stability and design.

Mathematical and computer modeling of meteorological problems
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The goal of mastering the discipline is to study the masters with modern methods of hydrodynamic modeling of atmospheric processes, based on integrating the system of atmospheric hydrodynamics equations, with the aim of shortand mediumterm weather prediction, to develop the skills of independent solution of theoretical and applied problems in the field of hydrodynamic modeling of natural processes using modern computational methods and devices. During the study of course, students should be competent in: – solve professional problems of hydrodynamic modeling; – know and be able to apply finitedifference and spectral models, methods of numerical integration of prognostic equations; – methods of parametrization of physical processes of subgrid scale; – apply knowledge when monitoring the natural environment, analyzing and forecasting the state of the atmosphere, assessing their possible change caused by natural and manmade causes.

Mathematical and Computer Modeling of Unsteady Nonlinear Physical Processes
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: to develop skills in solving nonstationary nonlinear physical processes problems by mathematical methods. During the study of course, master students should be competent in:  developing mathematical models of complex nonstationary nonlinear physical processes;  knowing and apply techniques and methods for solving complex problems of mathematical physics;  using different numerical methods to implement mathematical models of nonstationary nonlinear physical processes;  be able to use the scientific, reference, methodological literature on the subject;  writing the program code for the constructed mathematical model;  Creating a graph and perform animation for the results. During the study of the discipline master students will learn following aspects: Description of physical processes by mathematical equations. Construction of a mathematical model of the process. Closure of the system of equations by means of turbulent models. Selection of numerical methods. Construction of the difference equation of physical processes. Construction of a numerical algorithm for solving the difference equation. Creating code in one of the computer languages (Fortran, C ++, Java). Analysis of the results of numerical simulation of nonstationary nonlinear physical process. Graphical processing of numerical simulation results.

Mathematical and computer modeling of chemical processes
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline:is to improve the professional training of the student in the field of modeling of chemical and technological processes, includes master degree students knowledge in the field of modeling, compiling and optimizing mathematical models, using modern mathematical software packages in modeling; formation of professional skills in modeling chemical and technological processes, in the analysis and processing of data using modern information technologies. During the study of course, master students should be competent in:  building mathematical models of the systems under study;  carrying out analytical research and optimization of the developed mathematical model;  realizing the developed mathematical models in computer form;  applying the methods of computational mathematics to solve specific problems of the processes of chemical technology;  knowing methods of constructing a mathematical model of typical professional problems and a meaningful interpretation of the results obtained;  using packages of applied programs for modeling of chemical and technological processes. During the study of the discipline master students will learn following aspects: The course is designed to expand the knowledge of basic concepts, techniques and methods of mathematical and computer modeling, consideration of modern technologies for constructing and researching mathematical models for chemicaltechnological processes. The course discusses the principles of the formation of mathematical models, methods for constructing physicochemical models of chemicaltechnological processes, types of reactors and chemicaltechnological processes, methods for optimizing chemicaltechnological processes using empirical and / or physicochemical models.

Mathematical modeling of thermophysical processes in multilayer environments
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: is the development and research of a complex of mathematical models for solving heat and mass transfer problems in automatic process control facilities, developing control systems for thermophysical processes in such conditions for obtaining highquality, reliable products from composite materials and developing recommendations for improving technology. During the study of course, master students should be competent in:  identify specific requirements, opportunities and problems in the development of a mathematical model of thermophysical processes in multilayer environments;  systematize and interpret scientific theories and concepts of the latest trends in mathematical and computer modeling;  apply general programming knowledge in the field of computer modeling;  critically evaluate the results of scientific research, modern theories, problems and approaches, new trends in the study of thermal processes in multilayer environments;  to differentiate the priorities of educational and research activities, correlating their own scientific interests with social, ethnic values, the needs of production and society.  present the results of educational and research activities in the form of scientific reports, abstracts, abstracts of articles, physical and mathematical comments, master's theses. During the study of the discipline master students will learn following aspects: Identification of physicochemical factors that determine the temperature fields in the process of polymerization of composite materials and their accounting in mathematical models. Statement and solution of mathematical problems describing the temperature fields in the process of polymerization in the automatic recycling plant with regard to phase transitions and multistage.Development of finitedifference schemes and calculation of spatiotemporal distributions of temperature fields in the automated process control system.Analysis of the contribution of various processes to the temperature fields in the automated process control system and the development of algorithms and programs for managing thermophysical processes, the development of new installations (devices) for automatic control systems at all stages of product manufacturing. Development of a methodology for identifying during the process of manufacturing the sources of the degradation process. Identify the effeerature, time, heating rate and pressure on the quality of the material being manufactured.Development of recommendations for the management of the polymerization process in the automated process control system.

Finite element method in applied problems
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: to form the knowledge of the finite element method (FEM) and the development of skills for its practical application. During the study of course, master students should be competent in:  describing modern grid methods,  describing the finite element method (FEM);  describing the procedure for the approximation of the FEM, methods for its improvement;  solving twodimensional and threedimensional boundaryvalue problems using FEM;  describing the FEM data structures;  describing methods and algorithms for constructing finite element meshes;  using the basic principles of constructing modern finite element packages;  applying the basic methods of describing the design areas;  developing programs for the implementation of the FEM;  building finite element schemes of a higher approximation order. During the study of the discipline, master students will learn following aspects: various theoretical and practical aspects of FEM that will contribute to the development of skills for solving applied problems in various fields of the national economy: from the study of hydro and aerodynamics, soil seismic, physics problems, etc. to strength calculations of various structures and structures. The FEM is based on the discretization of an object, and its efficiency is especially obvious for problems with a complex configuration of the object under study and boundary conditions.

MonteCarlo methods and their applications
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: formation of understanding of Monte Carlo methods and their applications. During the study of course, master students should be competent in:  describe the mathematical formulation of the main problems solved by Monte Carlo methods.  explain the correctness of the mathematical formulation of the main problems of physics, chemistry, technology, biology.  to justify the algorithms of Monte Carlo methods for solving the main tasks of physics, chemistry, engineering, biology.  develop and analyze algorithms of Monte Carlo methods for solving basic problems.  solve on PC, analyze, explain the results. During the study of the discipline, master students will learn following aspects: Mathematical statements of the main problems solved by Monte Carlo methods. Correctness of mathematical formulation of the basic problems of physics, chemistry, engineering, biology. Algorithms of Monte Carlo methods for solving the main tasks of physics, chemistry, engineering, biology. Development and analysis of algorithms of Monte Carlo methods for solving basic problems. The decision on a PC, to analyze, to explain the results.

Methods of generalized functions and boundary integral equations in theproblems of thedynamics of elastic structure
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: is the formation of students: to formation of the knowledge from the theory of generalized functions necessary to solve problems of the dynamics of elastic media. The statements of nonstationary boundary value problems for secondorder differential equations are considered. Acquaintance with the construction of dynamic analogs of the Green and Gauss formulas in the space of generalized functions, obtaining their integral representations. During the study of course, master students should be competent in: – using the main types of special functions, their integral representations, asymptotics for solving problems of mathematical physics, –applying the theory of generalized functions to the solution of applied problems of mathematics and programming. – applying into practice the methods and techniques for solving the problems of the theory of elasticity, the theory of plasticity using different criteria of plastic flow; – using the laws of nonequilibrium thermodynamics of a continuous medium for the formulation and study of the problem statements of continuum mechanics; – knowing the practical techniques and methods of solving problems of mechanics of continuous media. During the study of the discipline students will learn following aspects: Boundary value problems for differential equations of the 2nd order. Construction of dynamic analogs of the Green and Gauss formulas in the space of generalized functions, obtaining their integral representations. Fundamentals of the integral Fourier and Laplace transforms in the space of generalized functions and methods for constructing solutions of partial differential equations. Generalized functions of one variable, operations with them. Generalized functions of many variables, operations with them.

Methods for solving inverse and illposed problems of mathematical physics
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of discipline: formation of undergraduates core competencies based on indepth study of research methods and inverse illposed problems. During the study of course, master students should be competent in:  knowing the concept of inverse and illposed problems of mathematical physics, their performances and applications,  knowing the main problems and methods of inverse and illposed problems, state of the art in this field;  be able to find and use scientific literature on the subjectoriented courses in algorithms and to select effective methods of inverse and illposed problems, investigate the properties and characteristics of the solutions of inverse inverse and illposed problems;  owning mathematical tools and skills to study inverse and illposed (unstable) problems in mathematical physics. During the study of the discipline, master students will learn following aspects: familiarity with the concept of students and research methods of inverse and illposed problems, the development of complex mathematical apparatus of ownership and the formation of skills and abilities to selfintensive research and scientific and exploration activities.

Modeling of heat and mass transfer processes in electrical contacts
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: to form knowledge in the field of the theory of boundary value problems for parabolic equations describing the processes of heat and mass transfer in bodies with variable cross sections. On the basis of the solution of the spatial problem of the Stefanov type, a mathematical model describing the processes of melting and welding of electrical contacts with through currents is presented. During the study of course, master students should be competent in: – understanding the processes describing heat and mass transfer in bodies with variable crosssection, – knowing the basics of the processes of melting and welding of electrical contacts with through currents – substantiating and choosing suitable methods for solving problems of heat and mass transfer in electrical contacts, – analyzing the mathematical statement made, to linearize the task, to record the initial and boundary conditions; – mastering practical techniques and methods for solving problems of continuum mechanics. During the study of the discipline, master students will learn following aspects: the theory of boundary value problems for parabolic equations describing the processes of heat and mass transfer in bodies with variable cross sections. A mathematical model based on the solution of a spatial problem of the Stefanov type, describing the processes of melting and welding of electrical contacts with through currents.

Modeling of the Turbulent Flows
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline is the formation of student’s knowledge about the methods of modeling turbulent flows. The problems of closure of Reynoldsaveraged Navier – Stokes equations are described. The concepts of Reynolds stress and turbulent viscosity are given. The main models of turbulence will be shown, in particular, models with one equation for the energy balance of turbulence, twoparameter models, and models with the equations of transfer of components of the Reynolds stress tensor. During the study of course, master students should be competent in: –averaging over time and space the NavierStokes equations. –deriving the transfer equations of secondorder pointlike correlation moments. –understanding the basic of semiempirical models of turbulence. –to be able to build models of turbulent viscosity and diffusion, models with one equation for the energy balance of turbulence. –to be able to build twoparameter models of turbulence. Models with differential equations for the transfer of components of the Reynolds stress tensor. During the study of the discipline students will learn following aspects: Reynoldsaveraged equations of motion for a viscous compressible fluid. Equations for Reynolds stresses. The equation for the kinetic energy of turbulent pulsations. Equation for isotropic turbulence dissipation. Algebraic models of turbulence. Model of the Prandtl way of mixing. Single equation models. Equation for turbulent viscosity. Models with two differential equations. Dissipative twoparameter model of turbulence. Modeling the terms of generation, dissipation and diffusion in the equation for isotropic dissipation. Twoparameter kε model of turbulence, kωWilcox model.

Modeling the stability of a deformable systems
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: to form the ability to mathematically model the problem of the stability of deformable systems, select an applied aspect in them, solve a model, analyze, interpretation of the result. During the study of course, students should be competent in:  demonstrate a systematic understanding of the process of modeling the stability of deformable systems;  critically evaluate the choice of criteria for static and dynamic stability of systems as applied to deformable media;  choose the methodology of analysis of static and dynamic stability of deformable systems and methods of solution;  to adjust the process of solving and visualizing the sustainability of the studied systems using modern software packages and determining the significance of the research for the technological development of society;  make recommendations in terms of the subject area of the studied subject.

Modeling of the physical processes in the heterogeneous environments
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: to form knowledge with the current state of the theory of singlephase and multiphase flows. The basics of classification of twophase flows are presented. The strategy of building a generalized mathematical model of multiphase flows using the Eulerian and Lagrangian approaches is described. Models of specific problems of the dynamics of multiphase media with phase transitions are shown. Derivation of the equations of motion and energy of a heterogeneous medium with phase transitions. During the study of course, master students should be competent in: –deriving the basic equations of multiphase flow. –usingthe Eulerian approach to describe the movement of continuous media in different aggregative states; –usingthe Lagrangian method to describe the motion of continuous media that are in a different aggregative state; –to be able to model specific tasks of the dynamics of multiphase media. –deriving the equation of motion and energy of a heterogeneous medium with phase transitions. During the study of the discipline, master students will learn following aspects: Multiphase flows in nature and technology, features of mathematical and physical modeling of heterogeneous flows. The main characteristics of heterogeneous flows. Collision of particles between themselves. Stokes numbers. Classification of heterogeneous turbulent flows. Mathematical modeling of turbulent gas flows with particles. Lagrange approach: advantages and limitations. Eulerian approach: advantages and limitations. The strategy of building a generalized model of twophase flows. Models of problems of the dynamics of multiphase media with phase transitions. Derivation of the equations of motion and energy of a heterogeneous medium with phase transitions.

Modeling of the Gas Dynamics
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: the formation of students' knowledge in the areas of theoretical and applied gas dynamics. The course examines the elements of thermodynamics, the equations of state of perfect and real gases, the laws of conservation and the ratio on strong discontinuities, the ratio of parameters on an oblique jump, the change in entropy. The method of characteristics for the equations of gas dynamics, onedimensional unsteady gas flow. During the study of the discipline master students will learn following aspects: –deriving the basic equations of gas dynamics in integral and differential forms. –understanding the equations of state of perfect and real gases. –explaining the surface of a strong and weak rupture on the basis of a onedimensional gas flow. –carrying out the derivation of the parameter ratios on an oblique jump, to explain the change in entropy. –using the characteristic method for twodimensional stationary supersonic gas flow. During the study of the discipline, master students will learn following aspects: Gas dynamics methods, gas dynamics model. Basic equations in integral and differential forms. The problem of the spread of sound. The speed of sound Mach number. Onedimensional stationary gas flow. Laval nozzle. Basic equations in integral and differential forms. Direct shock wave, change of parameters of gas at transition through a direct jump. Oblique shock wave. Ratios of parameters on an oblique jump, change in entropy. Loss of total pressure. dependence of the angle of inclination of the oblique jump on the angle of rotation of the flow. Twodimensional stationary supersonic gas flow. Characteristic method for irrotational flow. Small perturbation theory.

The turbulent flows, principles and applications
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: to form students' basic knowledge of the theory of turbulence and its principles. The concepts of turbulence are given, methods for describing the structures of turbulent flows are described. The basic principles of the Kolmogorov theory of developed homogeneous isotropic turbulence are given, in particular, concepts of the scale of turbulence, the spectrum of turbulent pulsations are given, the essence of the energycontaining and dissipative region of wave numbers. During the study of course, master students should be competent in: – understanding the processes of stability of a stationary fluid flow, the critical Reynolds number. –knowing the basic principles of the Kolmogorov phenomenological theory of developed homogeneous isotropic turbulence, – understanding the essence of the energycontaining and dissipative region of wave numbers. –deriving the NavierStokes equations for the motion of a viscous compressible fluid, to dimension the parameters that characterize the motion of a viscous fluid. – averaging the equations of a viscous fluid and deriving the Reynolds stress tensor. Understand closure problems of averaged equations. During the study of the discipline master students will learn following aspects: Fundamentalsof the theory of turbulence and its principles, concepts of turbulence, methods for describing the structures of turbulent flows. The main basics of Kolmogorov’s theory of developed homogeneous isotropic turbulence, the concepts of turbulence scales, the spectrum of turbulent pulsations. The essence of the energycontaining and dissipative region of wave numbers. The averaging of the equations of a viscous fluid and the derivation of the Reynolds stress tensor, the problem of the closure of averaged equations.

Direct methods for modeling turbulence problems
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: to form the ability to develop highorder accuracy methods for the numerical solution of a threedimensional turbulent flow system; direct numerical simulation of threedimensional transitional and developed turbulent flows; a detailed study of the mechanisms and evolution of transition and turbulent flows. During the study of course, master students should be competent in:  developing a class of stable difference schemes of high order of accuracy for the direct numerical solution of threedimensional turbulent flow.  simulating various physical processes with different schemes of calculation of the coefficient of turbulence;  making a comparison of calculations using the proposed model with observational data$  studying the structure and parameters of currents in the transition region and the region of developed turbulence, to obtain statistical characteristics of turbulence.  conducting detailed numerical studies of the structure and parameters of threedimensional transient and turbulent flows, as well as the stages of the turbulent flow evolution: vortex formation and associated oscillations of gasdynamic parameters, interaction of vortices in the flow, their dissipation and transition to the developed turbulent flow. During the study of the discipline, master students will learn following aspects: construction of a mathematical model for various physical processes; the correct choice of the algorithm of parametrization of turbulent exchange; construction of difference schemes and algorithms for solving problems; construction of flowcharts for numerical algorithm and program code; analysis of the results of numerical simulation of various turbulent flows.

Development and research of methods of modeling the behavior of complex processes
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: is the formation of knowledge and skills in creating and exploring mathematical simulation models of complex processes and systems. During the study of course, master students should be competent in:  to demonstrate deep knowledge of the current state and trends in the development of scientific knowledge in the field of mathematical modeling of complex processes;  to consider complexes of mathematical models of the behavior of complex processes;  use theoretical and experimental studies to simulate complex processes;  develop algorithms and software for managing complex processes based on research;  to generate the received scientific knowledge in own scientific research. During the study of the discipline master students will learn following aspects: Models of data structures; understanding of DBMS classification methods depending on implemented data models and methods of their use; studying the ways of storing data at the physical level, the types and methods of organizing file systems; detailed study of the relational data model and DBMS implementing this model, the language of SQL queries; An understanding of the problems and the main ways to solve them with the collective access to data; study of the capabilities of DBMS supporting various models of data organization, advantages and disadvantages of these databases in the implementation of various data structures, the means of these DBMS; understanding the stages of the life cycle of the database, support and maintenance; Obtaining an idea of specialized hardware and software tools oriented to building databases of large volumes of storage.

Development of mathematical models of control of multidimensional phase systems
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: the formation of knowledge about phase diagrams and their construction; forms the theoretical and methodological basis for understanding multidimensional phase systems; studying management of multidimensional phase systems, development of mathematical models of control, principles of construction of mathematical modelsis the formation of knowledge of the phase diagrams and their construction. During the study of course, master students should be competent in:  describe the characteristics of control models of multidimensional phase systems;  own modern research tools and various ways to build mathematical models;  use the developed mathematical models to control multidimensional phase systems:  integrate the knowledge of scientific theory of schools of directions in research practice;  to formulate problems and tasks of scientific research, choose the appropriate methodology, determine the stages of research, evaluate and interpret the results obtained; generate the scientific knowledge obtained in their own scientific research on mathematical and computer modeling. During the study of the discipline master students will learn following aspects: Classification of mathematical models. Principles of construction of mathematical models. Analysis of the object and the structure of its flows. Mathematical models of chemical transformations of material flows. Fluid dynamics models. Synthesis of mathematical models with distributed parameters. Basic concepts of the theory of identification. Adaptive identification algorithms. Mathematical models of complex systems.

Theory of Generalized and Special Functions
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline:to introduce students to the mathematical apparatus of the theory of generalized functions and various operations on them. Outlines the basics of the theory of generalized functions and operations on them. It also presents the basics of integral Fourier and Laplace transforms in the space of generalized functions and methods for constructing solutions of partial differential equations. During the study of course, master students should be competent in: – using the main types of special functions, their integral representations, asymptotics for solving problems of mathematical physics, – applying the theory of generalized functions to the solution of applied problems of mathematics and programming. – knowing the methodology of the study of functions, the rules of action in terms of calculating infinitely small quantities and the transition to integrated systems. – to be able to differentiate and integrate, investigate series and sequences for convergence, solve extremal problems, construct graphs of functions, solve equations and inequalities. – possessing the methods of formulation, analysis and solution of problems of differential and integral calculus, skills of independent solution of problems of theoretical and applied nature. During the study of the discipline master students will learn following aspects: Fundamentals of the theory of generalized functions and operations on them; fundamentals of integral Fourier and Laplace transforms in the space of generalized functions and methods for constructing solutions of partial differential equations. Generalized functions of one variable, operations with them. The main functions of many variables. The space of the main functions. Generalized functions of many variables, operations with them.

Formal methods of software development
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: to form the student a set of knowledge and skills related to the design and development of industrial software; the study of existing design methods and the current regulatory framework; the development of modern technologies for the development and implementation of software projects, as well as the fundamentals of project management. During the study of course, master students should be competent in:  methods of design and development of industrial and embedded software;  modern technologies for the implementation of software projects and related standards;  specifics of designing and developing software for highly efficient metamodels and multidisciplinary optimization systems;  develop software projects for industrial and embedded systems in compliance with domestic and foreign standards;  implement effective implementation of software projects in key executive and management positions;  apply model projects and practices as a basis for their own unique software solutions;  To make a risk assessment and consciously choose the best approaches and technologies for developing, testing and maintaining software;  skills of setting research tasks and skills of independent work. During the study of the discipline, master students will learn following aspects: general methodology and organizational and technical support of work. A brief overview of the history of technologies and software development methods in domestic practice and abroad. Classification of software systems: distinctive features, quantitative factors. Features of the design of industrial and embedded software. Software development based on formal methods (Cleanroom). Flexible development methodologies (Agile, Scrum, XP, MSF). Visual programming technologies (RAD). Software life cycle and its regulatory framework. The main stages of the design and development of software within the known models. Project Processes. Technical aspects of development. The choice between procedural, dataoriented and objectoriented approaches in the development of industrial and embedded software. Algorithmization and selection of an efficient algorithm, taking into account the characteristics of computational complexity. Use existing software code. Development of firmware and realtime software systems. The use of OpenMP and CUDA parallelization technologies in embedded systems.

Numerical methods for solution of the NavierStokes Equations
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: to form knowledge about the problems of the numerical solution of the NavierStokes equations. Methods for the numerical solution of the NavierStokes equations in the case of an incompressible fluid on a spaced grid will be shown. The methods for solution the variables in the function  vorticity parameters are shown. The solution methods for a viscous compressible fluid are studied, in particular, the McCormack method, the BimaWarming method, the Godunov method, the TVD scheme During the study of course, master students should be competent in: – deriving the NavierStokes equations of a viscous compressible flow, to made dimensionless the parameters characterizing the motion of a viscous fluid. – creating a mathematical model of hydrodynamic processes, including the physical formulation of the problem, –formulating initialboundary problems. – demonstrating knowledge of the basic finite difference, finite element and finite volume approaches to solving boundary value problems for the NavierStokes equations. – mading the program of the constructed numerical schemes for solving problems of hydrodynamics, to obtain results and to be able to interpret the mechanisms of the physical process. During the study of the discipline master students will learn following aspects: derivation of the NavierStokes equations, conservation laws, and basic hypotheses. Problems of numerical solution of equations. Numerical methods for solution of the NavierStokes equations in the case of incompressible fluid. The methods for solution the variables in the function  vorticity parameters are shown. Solutions for viscous compressible fluid, MacCormac method, BimaWarming method, Godunov method, TVD schemes, explicit implicit integration methods.

Elements of the theory of sustainability in economics and technology
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: to form fundamental knowledge in the theory of stability, theoretical and scientificpractical problems of sustainable development of economic systems. During the study of course, master students should be competent in:  describe the basic concepts and methods of the theory of stability  to demonstrate the idea of Lyapunov stability and asymptotic Lyapunov stability and asymptotic stability to investigate solutions for Lyapunov stability and asymptotic stability by linear approximation of solutions of differential and difference equations, to be able to depict phase portraits of a secondorder linearized system both in the case of a system of differential and in the case of a system of difference equations. During the study of the discipline, master students will learn following aspects: The concept of sustainability in the economy. The development of the theory of sustainability of economic systems. The main elements of economic sustainability. Introduction to the theory of sustainability in engineering. Stability of linear systems. Basic concepts of the theory of sustainability. General theorems on the stability of linear systems. Stability of linear systems with a constant matrix. Development of the conceptual concept of a particular type of sustainability under study. Definition of methodological features and methods for determining the stability of this type. Development and justification of a system of quantitative indicators determining the stability of the economic system. Evaluation of the specific features of the investigated object of sustainability, i.e., enterprises (taking into account its specialization, affiliation with a particular industry, regional characteristics of economic activity, size of economic potential, development opportunities, etc.). Carrying out the necessary calculations, analyzing the results obtained on the example of a real object and the adjustment of the originally developed methods. Testing the mechanism of managing the stability of the studied economic system.

Performance Doctor
 Type of control: Защита НИР
 Description: The main purpose of "The implementation of a Doctoral Thesis": the formation of doctoral students in preparation for the defense of the thesis for the Ph.D. in specialty (by industry). During the study of course, doctoral student's should be competent in: 1. demonstrate the progress of solving problems arising in the course of research activities and requiring indepth professional knowledge; 2. to argue for carrying out theoretical or experimental research within the framework of the tasks, including a mathematical (simulation) experiment; 3. can choose the necessary research methods, modify existing methods and develop new methods, based on the tasks of the specific study; 4. to use foreign languages for independent work on normative sources and scientific literature; 5. formulate the goals and objectives of the dissertation research, determine the scientific novelty and practical significance of the results of research activities; to develop a structurally methodological scheme for performing research. During the study of the discipline doctoral student will learn following aspects: Presentation and preliminary examination of the thesis. Registration of the applicant's case in the Academic Council of the University. Announcement of thesis defense. Publication and dispatch of the author's abstract. Registration of documents after the defense of the thesis.

Dissertation Writing
 Type of control: Защита НИР
 Description:

Research Seminar I
 Type of control: Защита НИР
 Description: The main purpose of "Research Seminar": the formation of master students in the skills of scientific research work. During the study of course, master student's should be competent in: 1. is able to competently substantiate the main directions of scientific research on the topic of dissertational work; 2. formulate a research problem, put a scientific problem and choose appropriate research methods; 3. can apply theoretical and experimental research methods in professional activity; 4. analyze the results of scientific research at each stage of the dissertation preparation; 5. are able to evaluate and draw conclusions on the main provisions of their research activities.

Research Seminar II
 Type of control: Защита НИР
 Description: The main purpose of "Research Seminar": the formation of master students in the skills of scientific research work. During the study of course, master student's should be competent in: 1. is able to competently substantiate the main directions of scientific research on the topic of dissertational work; 2. formulate a research problem, put a scientific problem and choose appropriate research methods; 3. can apply theoretical and experimental research methods in professional activity; 4. analyze the results of scientific research at each stage of the dissertation preparation; 5. are able to evaluate and draw conclusions on the main provisions of their research activities.

Research Seminar III
 Type of control: Защита НИР
 Description:

Scientific Internship
 Type of control: Защита НИР
 Description: The main purpose of "Scientific Internship": is the formation in the students of the ability to independently conduct research and development in the professional sphere using modern research methods and information and communication technologies on the basis of a foreign university. During the study of course, student should be competent in:  to substantiate the fundamentals of the methodology for performing scientific research, planning and organizing a scientific experiment, processing scientific data;  to argue methods of solving research and practical problems, including in interdisciplinary areas;  can analyze alternative solutions to research and practical problems and assess the potential benefits of implementing these options;  apply theoretical knowledge on methods of collecting, storing, processing and transmitting information using modern computer technologies;  choose the methods of presentation and methods of information transfer for different contingents of listeners.

Research practice
 Type of control: Защита практики
 Description: The main purpose of the discipline: the formation of pedagogical competence, the ability of pedagogical activity in universities and colleges based on the knowledge of the didactics of the higher school, the theory of education and management of education, analysis and selfassessment of teaching. During the study of course, master students should be competent in:  classify teaching methods based on criteria: traditionalistic  innovation; activity of cognitive activity; didactic goal and focus on results;  apply strategies and methods of training and education adequate to the goals;  develop research projects on topical issues of education and present the results in the form of presentations, articles, etc.;  describe different approaches to university management (university management  linear, structural, matrix): structure, quality, reputation;  evaluate and manage the processes of the organization of education, aimed at improving the structure, quality, reputation based on modern management approaches;  develop the provisions of the academic and research policy of the organization of education.

Teaching Internship
 Type of control: Защита практики
 Description: The purpose of teaching practice is to prepare for scientific and pedagogical activities in higher education, the acquisition and consolidation of practical skills for the implementation of the teaching and educational process in higher education, including the teaching of special disciplines, the organization of educational activities of students, scientific and methodological work on the subject.As a result of pedagogical practice, the undergraduate will have the skills of structuring and transforming scientific knowledge into educational material, oral and written presentation of the subject material, a variety of modern educational technologies, methods of drawing up tasks, exercises, etc.

Publication in the Proceedings of International Conferences
 Type of control: Защита НИР
 Description: The main purpose of "Publication in the Proceedings of International Conferences": is the formation of master candidates in the possibility of presenting the results of research work to the scientific community, receiving feedback, and exchanging experience in the field of professional activity. During the study of course, master student's should be competent in: 1. demonstrate current trends in scientific research; 2. to argue the annotated results of research in scientific journals, materials of international conferences and symposia; 3. they can apply new, scientifically grounded, theoretical or experimental results that allow solving a theoretical and applied problem; 4. analyze scientific results, the data of their colleagues and opponents in the sphere of the chosen professional activity; 5. generate ideas for the use of proposed developments in scientific research of the professional field of activity.