Magistracy
Speciality Code:
7M05403
Speciality Name:
Applied mathematics and control theory
Faculty:
Mechanics and Mathematics
Qualification:
 Scientific and pedagogical direction  Master of Natural Sciences
 Model of graduating student
 Mandatory disciplines
 Elective disciplines
 Professional
ON1. Form the ability of abstract and critical thinking, analysis, synthesis and readiness for selfdevelopment, selfrealization, the use of creative potential.
ON2. Develop effective mathematical methods for solving applied problems of mathematics, physics, mechanics, economics and management.
ON3. Create mathematical models of objects, systems, processes and technologies and explore them with mathematical methods.
ON4. Create a software package for solving applied problems of natural science with modern methods of highperformance computing technologies, the use of modern supercomputers in the research.
ON5. Develop software algorithms, tools on the subject of ongoing scientific research projects.
ON6. Develop a package of scientific and technical documentation, draw up scientific and technical reports, reviews, publications based on the results of completed studies.
ON7. Conduct an analysis of modern and scientific development, domestic and foreign developments in the field of applied mathematics.
ON8. Competently use linguistic and cultural linguistic knowledge for communication and integration in the multilingual and multicultural society of the Republic of Kazakhstan and in the international arena, to have the opportunity to work in a team.
ON9. Implement modern achievements in the theory of optimal control methods and mathematical physics in science, industry and economics.
ON10. Conduct classes of mathematical disciplines in higher and secondary specialized educational institutions using digital technologies, to develop their methodological support.
ON11. Solve nonlinear problems of differential equations and mathematical physics with modern methods of convex analysis and minimization.
ON12. Solve applied problems of natural science with modern and classical methods of functional analysis.
ON2. Develop effective mathematical methods for solving applied problems of mathematics, physics, mechanics, economics and management.
ON3. Create mathematical models of objects, systems, processes and technologies and explore them with mathematical methods.
ON4. Create a software package for solving applied problems of natural science with modern methods of highperformance computing technologies, the use of modern supercomputers in the research.
ON5. Develop software algorithms, tools on the subject of ongoing scientific research projects.
ON6. Develop a package of scientific and technical documentation, draw up scientific and technical reports, reviews, publications based on the results of completed studies.
ON7. Conduct an analysis of modern and scientific development, domestic and foreign developments in the field of applied mathematics.
ON8. Competently use linguistic and cultural linguistic knowledge for communication and integration in the multilingual and multicultural society of the Republic of Kazakhstan and in the international arena, to have the opportunity to work in a team.
ON9. Implement modern achievements in the theory of optimal control methods and mathematical physics in science, industry and economics.
ON10. Conduct classes of mathematical disciplines in higher and secondary specialized educational institutions using digital technologies, to develop their methodological support.
ON11. Solve nonlinear problems of differential equations and mathematical physics with modern methods of convex analysis and minimization.
ON12. Solve applied problems of natural science with modern and classical methods of functional analysis.

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.

Actual problems of differential equations and control theory
 Number of credits: 5
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The aim of the course is the familiarization of master studentsmathematicians with common methods of equation theory and extremum theory.  Know and understand the general theory of differentiation as a relation between the theory of equations and the theory of extremum;  Justify optimization methods for systems with distributed systems;  Apply numerical methods for solving optimal control problems;  critically evaluate the current state of the theories of differential equations and extremum;  To teach special courses in the theory of equations and the theory of extremum. The general theory of differentiation. Algebraic equations, ordinary differential equations, partial differential equations as necessary conditions for an extremum for various problems of minimizing functionals. Gradient methods for the approximate solution of equations and extremum problems.

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 models and methods for intelligent systems
 Number of credits: 5
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: Purpose: The purpose of the course is to teach undergraduates with mathematical modeling of various physical phenomena, to study the basic research methods, mathematical problems arising in this process, to clarify the physical meaning of the solutions obtained. In the course of studying the course to form the abilities: Create mathematical models of physical, chemical, biological, etc. processes and explore them with mathematical methods.  Apply modern methods of the theory of differential equations and control theory to solve applied problems;  Analyze and understand the actual problems of the theory of applied problems and methods for their solution;  Develop effective mathematical methods for solving applied problems in various fields of science;  To have fundamental knowledge in modern sections of mathematical modeling and numerical solution.  Perform scientific work on current problems of differential equations and control theory. As part of the course, students become familiar with the features of complex solution of nonlinear physical problems in the field of continuum mechanics, with the correct formulation of additional conditions at the boundaries of the computational domain, mastering modern numerical methods for solving problems of nonstationary dynamics of compressible media.

Organization and planning of research work
 Number of credits: 5
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline is to study basic concepts and definitions from the field of planning, organization and management of scientific research and innovation activities in IT, mathematics, computer science, mechanics and modelling. The role and importance of organization, planning and management of scientific research, innovation activities in industrial enterprises, research institutes and design organizations at the present stage. In the course of the study course is to form undergraduates' abilities: Study of the principles and stages of applying for research, patent applications. Knowledge methods of inventions, normative documents for registration of dissertations and reports. Planning and conducting scientific and experimental research.Writing and design of scientific works. Normative documents and State Standarts for registration of dissertations and reports. Latex for scientific articles.Methods of the invention, Methods of research. Style of scientific writing. Publication of scientific results. Databases of scientific results Selfmotivation. Commercialization of scientific 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 problems of functional analysis and its applications
 Number of credits: 5
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: Purpose: to acquaint magistrates with modern methods, current problems and areas of application of functional analysis. To teach theories and methods of functional analysis to solve problems of classical functional analysis, applied problems of natural science, and also to solve it using methods to solve problems related to other fields of knowledge; To form abilities for abstract thinking, analysis, synthesis and independent study of new sections of the basic sciences. In the course of the study course is to form undergraduates' abilities: To possess and know the fundamental knowledge of modern sections of functional analysis; To own and know classical and modern methods of functional analysis; To be able to apply the methods of functional analysis to the solution of applied problems of natural science; To be able to perform scientific work on applied problems of natural science. The discipline "Modern Problems of Functional Analysis and Its Applications" is the main part of the curriculum in the direction of training specialists in applied mathematics and control processes. The course examines the basics of functional analysis, modern methods and the possibility of its application to applied problems. Course content: basic statements of functional analysis, metric, linearly normalized, Hilbert, Banach and separabel spaces. Floor spaces. It's full. Convergence. Open and closed sets. Open and closed sets, their properties. Neighborhood points, their properties. Convergence in metric space. Full metric spaces. Separable spaces. Compact sets. Theory of continuous linear operators and functionals. Compactness criterion. Completeness of the space of linear continuous operators. Compression operator, Banach theorem. Banach – Steinhaus theorem. Generalized functions. Weak and strong convergence. Conjugate and selfadjoint operators. The problem of eigenvalues and eigenfunctions.

Numerical analysis and parallel calculations of the equation of mathematical physics
 Number of credits: 5
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: Purpose: The purpose of mastering the discipline "Numerical Analysis and Parallel Computing of the Equation of Mathematical Physics" is to prepare undergraduates for the development and software implementation of effective computational algorithms for solving boundary value problems for partial differential equations. To acquaint undergraduates with the basic tasks of mastering the discipline, modern numerical methods for solving applied problems of natural sciences and the principles of constructing iterative algorithms for solving linear and nonlinear systems of equations. In the course of the study course is to form undergraduates' abilities: To have fundamental knowledge in modern sections of mathematical modeling and numerical solution; To be able to develop efficient computational algorithms for solving applied problems of mathematical physics and output analysis;  Master the use of modern methods of numerical analysis to solve applied problems of natural science; To be able to explore complex systems with modern methods of highperformance computing technologies, the use of modern supercomputers in the research; Owning to research and develop mathematical models, algorithms, methods, research projects and the preparation of scientific and scientifictechnical publications. The course represents the conceptual and practical aspects of numerical solutions of partial differential equations. This course covers advanced numerical methods that are used in largescale scientific and engineering calculations and simulations. Mathematical modeling of physical processes Topics include finite difference, volume, elemental and spectral methods, and explicit / implicit methods for integrating over time, convergence, consistency, stability, CourantFriedrichsLevy criteria, Lax equivalence theorem, error analysis, Fourier stability analysis von Neumann. Discretization of the domain of continuous change of arguments. Numerical methods for solving differential and integro  differential equations. Methods for solving multidimensional equations. Variational methods. The finite element method.

Convex analysis in Banach space
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: Purpose: A systematic study of the basis of differential calculus in a Banach space, the mathematical apparatus of the theory of extremal problems in a Banach space. In the course of studying the course to form the ability of undergraduates: Develop a master's skills in convex analysis apparatus, subdifferential calculus; To master the elements of the convex analysis used in other mathematical and applied disciplines, the study of which is provided for the basic and working curricula; Be able to apply convex analysis methods when analyzing problems and solving applied problems in economic and social systems; Master the Lagrange method and its justification for solving convex extremal problems; Critically evaluate the current state of the subject area in the context of the latest scientific theories and concepts; Quickly find, analyze and correctly process scientific and technical, naturalscience and general scientific information contextually, leading it to the problemproblem form; Apply modern methods for independent research and interpret their results; Apply skills to write reviews, reports and research articles. The course “Convex analysis in a Banach space” is a fundamental course devoted to the study of general properties of convex sets and convex functions in Banach spaces, which makes it possible to investigate solutions to various problems of finding minima of convex functions defined on convex sets. Course contents: convex sets and convex functionals and their properties; the convex hull of a set, convex combinations of points of this set, their connection; Carathéodory's theorem on the convex hull of sets in Rn; Hausdorff metric for sets, its properties; global minimum theorems and optimality conditions; fundamentals of the theory of separability of convex sets, the Lagrange multiplier method, conditions for weak bicompactness of convex sets, weak lower semicontinuity of functionals, Weierstrass theorem in a Banach space.

Integrodifferential equations and their applications
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: Aim: to study the basic problems of the theory of initial and boundary problems for singularly perturbed integrodifferential equations and methods for solving such problems. During the study of course, undergraduates should be competent in:  Have a modern theoretical understanding of the role and place of singularly perturbed problems;  Freely possess asymptotic methods for solving singularly perturbed integrodifferential equations and clarify the scope of these methods;  the skills of mathematical modeling of applied problems described by singularly perturbed integrodifferential equations and the interpretation of the results obtained;  To conduct intensive research work and publicly present their own new scientific results;  Work in a team and to defend the correctness of the choice of solving the problem;  To critically evaluate yourself activities, the activities of the team, and to be capable of selfeducation and selfdevelopment. nitial and boundary value problems with an initial jump for singularly perturbed integrodifferential equations. Initial and boundary functions of singularly perturbed homogeneous differential equations. Analytical formula and asymptotic estimate of the solution of initial and boundary value problems for singularly perturbed linear integrodifferential equations. Estimation of the difference between solutions of singularly perturbed and unperturbed problems. Uniform asymptotic expansion of solutions of initial and boundary value problems.

Perturbation Methods in an Applied Problem
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: Aim: indepth study of some sections of the asymptotic theory of differential equations, the formation of knowledge necessary for the effective use of asymptotic methods for constructing and analyzing solutions of ordinary differential equations and partial differential equations with a small parameter for higher derivatives and the ability to apply these methods in the study of fundamental and applied problems. During the study of course, undergraduates should be competent in:  Use fundamental knowledge in the field of mathematical analysis, complex and functional analysis, differential equations in future professional activities;  Freely possess asymptotic methods for solving singularly perturbed ordinary and partial differential equations and clarify the scope of these methods;  Use the methods of mathematical modeling in solving theoretical and applied problems;  Conduct intensive research work and publicly present their own new scientific results;  Work in a team and to defend the correctness of the choice of solving the problem;  To critically evaluate yourself activities, the activities of the team, and to be capable of selfeducation and selfdevelopment. Dependence of solutions on parameters; the method of small parameter for solving differential equations; finding periodic solutions of linear differential equations; asymptotic integration; ordinary differential equations with a small parameter at the derivative; limit transition theorem; asymptotic behavior of solutions of differential equations with respect to a small parameter; singularly perturbed partial differential equations; a singularly perturbed first boundary problem for systems of linear hyperbolic equations.

Identification methods for systems and their applications
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The aim of the course " Identification methods for systems and their applications " is formulation of inverse problems of mathematical physics and the development of practical methods for its solving. In the course of studying the course to form the ability of doctoral students:  To know and understand the problem of identifying mathematical models and formulations of inverse problems of mathematical physics;  To know the concept of wellposedness of problems of mathematical physics and understand the features of illposed problems;  To be able to reduce inverse problems to extreme problems;  Develop numerical methods for solving inverse problems;  Critically evaluate the current state of the subject area in the context of the latest scientific theories and concepts;  To teach special courses on methods for solving inverse problems of mathematical physics. The course content is aimed at analyzing the study of the formulation of inverse problems of mathematical physics, identifying their qualitative features, developing practical methods for solving inverse problems of mathematical physics and evaluating the degree of their effectiveness.

Methods for minimizing functionals in a Banach space
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: Purpose: A systematic study of the theoretical foundations of minimization of functionals in a Hilbert space. In the course of the study course is to form undergraduates' abilities:  To be able to apply methods for minimizing functionals in a Banach space to solve applied problems;  Have fundamental knowledge on the basic methods of minimization in the Banach space;  To acquire practical skills in solving the minimization problem on computers;  To possess knowledge in the field of natural and technical sciences where methods of minimization are applied;  To carry out scientific work on methods of minimizing functionals in a Banach space. The gradient of the functional on the set of solutions of ordinary differential equations; The gradient of the functional on the set of solutions of a parabolic equation; The gradient of the functional on the set of solutions of a hyperbolic equation; minimization methods in Banach space: gradient method, gradient projection method, conditional gradient method.

Methods for solving problems of mathematical physics
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The course studies elliptic, parabolic, and hyperbolic equations with boundary and initialboundary conditions. First, we give the necessary definitions and known theorems from the theory of functional spaces and partial differential equations. Applicability of theories of generalized functions for problems of partial differential equations. Fundamental solutions and their properties are constructed for some boundary problems of equations of mathematical physics. Methods for obtaining a priori estimates in functional spaces are given. In the course of studying the course to form undergraduates' abilities:  Explain the key concepts of generalized functions in the context of the relevant theory;  Calculate problems (generalized solutions, ordinary differential operators, inverse scattering problems, soliton solutions) using modern methods of the theory of generalized functions;  To prove the solvability of applied problems using the theory of generalized functions;  Solve theoretical and applied problems of physics, mechanics, etc .;  Describe the solution of the problems of nonlinear equations of mathematical physics by the methods of the theory of generalized functions and the theory of functional spaces.  Design the process of studying an applied problem using the methods of the theory of generalized functions;  To work in a team, reasonably defend the correctness of the choice of a solution to a problem. As a result of training, undergraduates should know: generalized functions and functional spaces, their properties and the formulation of common boundary value problems. Methods for solving problems for equations of mathematical physics. To be able to: apply the wellknown modern methods for solving the boundary value problem of mathematical physics. Possess: solve the set boundary and initialboundary problems; Solve specific applications using the above objectives.

Methods for solving nonlinear problems in mathematical physics
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline is devoted to the study of nonlinear problems of mathematical physics from the point of view of modern functional analysis. Therefore, the concept of nonlinearity prevails throughout the course. The following modern methods for studying initialboundary value problems for equations of mathematical physics are considered here: a priori estimates method, variational methods, monotonicity and compactness methods, a parameter continuation method, a regularization method, and numerical methods. In the course of studying the course to form undergraduates' abilities:  Explain the basic concepts of generalized, analytical and approximate solutions of initialboundary value problems;  Identify weak and strong generalized solutions for nonlinear problems of mathematical physics, using modern methods of theories of functional analysis;  To prove the solvability of applied problems using the method of a priori estimates, variational methods, methods of monotonicity and compactness, the method of continuation with respect to a parameter, the method of regularization and numerical methods;  Solve theoretical and applied problems of physics, mechanics, etc .;  Describe the unique solvability of nonlinear problems of mathematical physics;  Design the process of studying an applied problem using modern methods for solving nonlinear problems of mathematical physics;  To work in a team, reasonably defend the correctness of the choice of a solution to a problem. As a result of training, undergraduates should know the theory of generalized functions and its application in boundary value problems for nonlinear equations. Namely the Fourier Method. The solution of the simplest nonlinear equations by the method of characteristics. The inverse problem method for the Korteweg de Vries equation. Soliton solutions. Solution of the inverse scattering problem by the GelfandLevitanMarchenko method. The method of compactness in combination with the methods of monotony and Galerkin. Leray – Schauder method. Classic solutions. Generalized solutions.

Nonlinear problems of mathematical physics
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline: The objectives of the development of the discipline "Nonlinear problems of mathematical physics"  acquaintance with the methods of mathematical modeling of physical processes;  acquaintance with modern analytical methods for studying nonlinear problems of mathematical physics; acquisition of skills to use the modern mathematical apparatus in research and applied activities. In the course of studying the course to form undergraduates' abilities:  Explain the key concepts of generalized functions in the context of the relevant theory;  Calculate problems (generalized solutions, ordinary differential operators, inverse scattering problems, soliton solutions) using modern methods of the theory of generalized functions;  To prove the solvability of applied problems using the theory of generalized functions;  Solve theoretical and applied problems of physics, mechanics, etc .;  Describe the solution of the problems of nonlinear equations of mathematical physics by the methods of the theory of generalized functions and the theory of functional spaces.  Design the process of studying an applied problem using the methods of the theory of generalized functions;  To work in a team, reasonably defend the correctness of the choice of a solution to a problem. As a result of training, undergraduates should know: the physical meaning of the classical nonlinear equations of mathematical physics; Basic ideas and methods of the spectral theory of ordinary differential operators; The main ideas of the inverse scattering method. To be able to: Build mathematical models of physical and other phenomena; Use the ideas of the inverse problem method to study solutions of nonlinear equations of mathematical physics. Possess: Skills of joint application of various mathematical methods; The skills of combining analytical and approximate methods in the study of complex mathematical and applied problems.

Practical methods of solving of optimization control problems
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The aim of the course "Practical methods of solving of optimization control problems" is the analysis of the difficulties encountered in the practical solving of optimization problems, and the development of ways to overcome these difficulties. In the course of studying the course to form the ability of master students:  Know and understand general methods for solving optimization problems;  Develop numerical methods for solving optimization problems;  To carry out diagnostics of computer solving results for optimization problems;  To find ways to overcome the identified difficulties;  critically evaluate the current state of the subject area in the context of the latest scientific theories and concepts;  To carry out teaching of special courses on methods for solving extreme problems. The course content is aimed at analyzing the immediate difficulties encountered in the computer solution of optimal control problems; identifying possible causes of unsatisfactory results and developing practical recommendations for overcoming the difficulties encountered.

Modern methods for solving extremal problems
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the course: undergraduates must have fundamental knowledge of the theory of extremal problems in Banach space for partial differential equations, possess the basics of convex analysis, practically have the skills to calculate the gradient of a parabolic equation that is functional on a set of solutions, a hyperbolic equation, be able to apply elastic flexible strings to solve applied problems of heat conduction; know the methods of minimization of functionals in a Banach space. During the course study, form undergraduates' abilities: On the basis of deep system knowledge in the subject area to solve modern problems and problems in mathematics. Critically evaluate the current state of the subject area in the context of the latest scientific theories and concepts. Quickly find, analyze and correctly process scientific and technical, naturalscience and general scientific information contextually, leading it to the problemproblem form. Apply modern methods for independent research and interpret their results; Apply skills to write reviews, reports and research articles. Independently formulate the problem and select the appropriate mathematical model. Contents of the discipline: To solve theoretical problems of mathematical physics, natural sciences, new mathematical methods of optimal control of the processes described by partial differential equations are needed. In this discipline, we study the basic methods of minimizing functionals in the form of multiple integrals on a set of partial differential solutions; methods of minimizing functionals: gradient method, gradient projection method, conditional gradient method, NewtonKantorovich method, integral functionals method.

Theory of boundary value problems of optimal control
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The purpose of the discipline is to acquaint undergraduates with new results on methods for solving boundary value problems of optimal control for the processes described by ordinary differential equations, that differring from the known methods based on the Lagrange principle. Masters must have a high theoretical background and be able to apply for solving applied problems on computers: – Elicitation of controlability theory role in optimal control; – Existence criteria test methods statement for boundary value optimal control problems with different kinds of constraints; – Solution construction methods statement for boundary value optimal control problems with linear and quadratic performance criteria; – Constructive optimal control theory application in solving applied problems ways description; – To work in a team, to defend the correctness of the choice of the solution of the problem. Content of the discipline: The basis of the proposed new method for solving boundary value problems of optimal control is the principle of immersion. The idea of the principle of immersion is that the solution to the original complex boundary value problem was made possible by finding the usual solution, a class of Fredholm integral equations of the first kind. The existence of a solution to a boundary value problem is reduced to the construction of minimizing sequences and the definition of the lower bound of the functional. The construction of an optimal solution is carried out by narrowing the range of admissible controls.

Energy methods for solving the problem of physics
 Type of control: [RK1+MT1+RK2+Exam] (100)
 Description: The objectives of mastering the discipline "Inverse problems in hydrodynamics" are  to acquaint undergraduates with the theory and new results on methods for solving inverse problems hydrodynamics; study and skill of the basic settings of inverse problems; study of the features of their solution, some algorithms for their solution. In the course of studying the course to form undergraduates' abilities:  Know and explain the basic concepts and methods for solving the inverse problem of hydrodynamics;  Estimate and research the current state and achievement of the area of the inverse problem of hydrodynamics; Apply theoretical knowledge to solve applied problems of natural science. To be able to independently set the formulation of tasks and select effective methods for solving them; To analyze the results and compare with other modern results of the researcher Basic concepts of the theory of inverse problems. Formulations of direct and inverse problems of hydrodynamics. Classification of inverse problems. Known results on the direct problem of hydrodynamics. The main methods for solving inverse problems. Inverse problems for Stokes equation. Inverse problems for linearized and nonlinear NavierStokes equations. Inverse problems of heat convection, magnetic hydrodynamics. Inverse problems for nonNewtonian fluids.

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.