Course detail

Mathematical Modelling by Differential Equations

FSI-SA0Acad. year: 2022/2023

The course provides basic applications of ordinary differential equations in technical and scientific branches. Various problems of mechanics, hydromechanics, flight dynamics, strength of materials, biology, chemistry and other areas are disussed in the framework of this course. Solvings of studied problems consist in forming of a differential equation as a corresponding mathematical model, finding its solution and analysis of this solution.

Language of instruction

Czech

Number of ECTS credits

2

Mode of study

Not applicable.

Guarantor

prof. RNDr. Jan Čermák, CSc.

Department

Institute of Mathematics (ÚM)

Learning outcomes of the course unit

Students will acquire knowledge of basic methods of mathematical modelling by means of ordinary differential equations. They also will master solving obtained differential equations.

Prerequisites

Differential and integral calculus of functions in a single and more variables, theory of ordinary differential equations.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline.

Assesment methods and criteria linked to learning outcomes

Course-unit credit is awarded on the following conditions: Active participation in lessons.

Course curriculum

Not applicable.

Work placements

Not applicable.

Aims

The aim of the course is to explain basic applications of the theory of differential equations. The task of the course is to demonstrate elementary procedures in mathematical modelling by means of ordinary differential equations, including finding and discussion of their solutions.

Specification of controlled education, way of implementation and compensation for absences

Attendance at lectures is recommended. Lessons are planned according to the week schedules. Absence from lessons may be compensated for by the agreement with the teacher.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Fulford, G., Forrester, P., Jones, A.: Modelling with Differential and Difference Equations, New York, 2001. (EN)
Perko, L.: Differential Equations and Dynamical Systems, Springer-Verlag, 1991.  (EN)

Recommended reading

Nahin, P.J.: Chases and Escapes: the mathematics of pursuit and evasion, Princeton University Press, Princetion, 2007. (EN)
 Rachůnková, I,  Fišer, J.: Dynamické systémy 1, UP  Olomouc,  2014 (CS)
Strogatz, S.:  Nonlinear Dynamics and Chaos, With Applications To Physics, Biology, Chemistry, And Engineering (Studies in Nonlinearity), Avalon Publishing,  2014 (EN)

Classification of course in study plans

  • Programme B-MAI-P Bachelor's 2 year of study, summer semester, elective

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. RNDr. Jan Čermák, CSc.

Syllabus

1. Applications of ordinary differential equations (ODEs) in mechanics (basic problems).
2. Basic pursuit problems and their ODEs models.
3. Calculations of escape velocities via ODEs.
4. The first Kepler's problem and its solving.
5. Geometric applications of ODEs (constructions of curves with special properties).
6. Applications of ODEs in hydromechanics.
7. Applications of ODEs in hydromechanics. (continuation).
8. Two special pursuit problems.
9. Basic models of systems with a variable mass.
10. Applications of ODEs in biology ( predator-prey model).
11. ODEs on graphs and their use.
12. Catenary curve problem.
13. Chaotic systems and their applications.