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Course title 
Mathematical modeling of dynamic processes 

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Year of study, study programme 
4 131 03 09 Computer mathematics and systems analysis


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Semester of study 
7 

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Number of credits 
3 

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Lecturer 
Gromak Valery Ivanovich 

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Course Objective 
Formation of students’ skills and abilities in conducting mathematical research on the basis of symmetry analysis, teaching basic methods of group classification of equations of mathematical physics. As a result of the training, the student must know: – basic concepts of the theory of continuous Lie groups; – the basic methods for studying the symmetry properties of differential equations; – the main types of applied problems that can be effectively solved using the theory of Lie groups. be able to: – to carry out research of dynamic models on the basis of groups of symmetries, – to construct Lie algebras and universal invariants, – find exact solutions or reduce the order of differential equations that admit local oneparameter Lie groups; 

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Prerequisites 
Course materials 16 semesters: “Algebra and Number Theory”, “Geometry”, “Mathematical Analysis”. “Differential equations”, “Computer mathematics”; materials of 7 semesters of the course “Mathematical modeling of dynamic processes”, 

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Course content 
Mathematical models of dynamic systems. Modeling of the dynamic system. A dynamical system described by a finite system of differential equations. Classification of the behavior of dynamic systems. Oneparameter Lie groups of transformations, infinitesimal group operator, group invariants. Point and contact transformations. Lie group of point transformations. Tangent vector field. Infinitesimal operator. Invariants, invariant manifolds, and criteria for the invariance of the Lie group of transformations. Groups of transformations and differential equations. The theory of the continuation of a group and an infinitesimal operator. Differential invariants. Finding the group of transformations allowed by the differential equation Determination of the most general differential equation admitting a given group. Methods for the integration of ordinary differential equations admitting a known group of transformations. Lie algebras of symmetries. Commutator of infinitesimal operators. The Lie algebras of symmetries and their computation. Generalization of Lie groups of point transformations to the multidimensional case. Group classification of mathematical physics equations. Group classification of the equations of mathematical physics: Burgers, Kortewegde Vries, heat conductivity, NavierStokes, Boussinesq, sineGordon, etc. Invariant and partially invariant solutions of the equations of mathematical physics


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Recommended Literature 
Основная:


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Teaching Methods 
Lecture with laboratory works, using elements of distance learning and electronic materials. 

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Teaching language 
Russian 

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Requirements, current control 
Presentation and discussion of laboratory results. Tests. 

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Method of certification 
Credit 