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Course title

Mathematical modeling of dynamic processes

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Year of study,

study programme

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1-31 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 one-parameter Lie groups;

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Prerequisites

Course materials 1-6 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.

One-parameter 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, Korteweg-de Vries, heat conductivity, Navier-Stokes, Boussinesq, sine-Gordon, 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