6 semester


Course title

Mathematical modeling of dynamic processes


Year of study,

study programme

1-31 03 09 Computer Mathematics and Systems Analysis


Semester of study



Number of credits




Kozlov Ilya Igorevich, Senior Lecturer


Course Objective

Systematization and consolidation of students’ methods and techniques of using knowledge obtained in general mathematical courses, on real production tasks and tasks of research and development. As a result of the training, the student must


–      sources of mathematical models, types of models, approaches to their construction and analysis, typical results of analysis of known mathematical models;

be able to:

–      decompose complex processes for modeling, to identify deterministic and stochastic parts, to construct mathematical models in continuous and discrete, one-dimensional and multidimensional cases, with the presence of delay and without it, to find equilibrium positions and analyze the behavior around them;

–      have knowledge of methods of linear approximation, mitigation, estimation, analogies and hierarchies in mathematical modeling.



Course materials of 1–6 semesters: «Algebra and Number Theory», «Mathematical Analysis», «Differential Equations», «Equations of Mathematical Physics», «Numerical Methods», «Computer Mathematics».


Course content

Introduction to mathematical modeling (concepts, classification of models, examples).

Sources of mathematical models (fundamental laws of nature, variational principles, analogy method, hierarchical approach, nonlinear models as a consequence of mitigation of assumptions).

Mathematical models of oscillatory occurrences (oscillation of Saturn’s rings, weight on the spring, model of Predator-Victim of Volterra, model of chemical reaction of Lotka, salary and employment, mathematical pendulum, addition of oscillations, nonlinear variants).

Modeling methods leading to DEPD (multidimensional mathematical models, transport equation, continuity equation, Boussinesq equation, Korteweg-De Vries equation, von Foerster model).

Direct and co-ordinated tasks (the model for the spread of air pollution in the atmosphere, the equation of turbulent diffusion, the optimization of the location of pollution sources and emissions in a tight quota policy).

Mathematical models with delay (logistic model, linear analysis of the behavior near the equilibrium position, the model of the restoration of the number of blood cells, the model of the arrhythmia of breathing Shane-Stokes).

Discrete mathematical models (discrete analogues of the logistic model, oscillatory and chaotic solutions, analytical coding).

Modeling of complex processes (strategy of making assumptions in the construction of models, compartmental models and models of the dynamics of epidemics, cellular automata, the model of the dynamics of relations).


Recommended Literature

1.    Самарский А.А., Михайлов А.П. Математическое моделирование: идеи, методы, примеры / Москва, Наука. 2002.

2.    Петросян Л.А., Захаров В.В. Математические модели в экологии / СПб. Изд-во СПбГУ. 1997.

3.    Амелькин В.В., Садовский А.П. Математические модели и дифференциальные уравнения / Минск, БГУ. 1982.

4.    Murray J.D. Mathematical Biology / Oxford Press. 2004.


Teaching Methods

Lectures with laboratory works, using elements of distance learning and electronic materials.

52 classroom hours, 18 hours of which are lectures and 34 are laboratory works.


Teaching language



Requirements, current control

Presentation and discussion of laboratory results, tests.

Examination estimate consists of: current estimate – 40%, verbal examination – 60%.


Method of certification