Mathematical Modelling by Differential Equations (FSI-SA0)

Academic year 2020/2021
Supervisor: prof. RNDr. Jan Čermák, CSc.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:
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.
Learning outcomes and competences:
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.
Course contents:
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.
Teaching methods and criteria:
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.
Controlled participation in lessons:
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.
Type of course unit:
    Lecture  13 × 2 hrs. optionally                  
Course curriculum:
    Lecture 1. Applications of ordinary differential equations (ODEs) in mechanics (basic problems).
2. Applications of ODEs in mechanics (linear oscillators).
3. Applications of ODEs in mechanics (special problems).
4. Applications of ODEs in flight dynamics (space velocities and related problems).
5. Applications of ODEs in flight dynamics (systems with a variable mass).
6. Geometric applications of ODEs (orthogonal trajectories).
7. Geometric applications of ODEs (some problems in optics).
8. Applications of ODEs in biology (logistic equation).
9. Applications of ODEs in biology (model predator-prey).
10. Applications of ODEs in chemistry.
11. Catenary curve problem.
12. Applications of ODEs in strength of materials.
13. Chaotic systems and their applications.
Literature - fundamental:
1. Perko, L.: Differential Equations and Dynamical Systems, Springer-Verlag, 1991. 
3. Fulford, G., Forrester, P., Jones, A.: Modelling with Differential and Difference Equations, New York, 2001.
Literature - recommended:
1. Strogatz, S.:  Nonlinear Dynamics and Chaos, With Applications To Physics, Biology, Chemistry, And Engineering (Studies in Nonlinearity), Avalon Publishing,  2014
2. Nahin, P.J.: Chases and Escapes: the mathematics of pursuit and evasion, Princeton University Press, Princetion, 2007.
3.  Rachůnková, I,  Fišer, J.: Dynamické systémy 1, UP  Olomouc,  2014
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
B-MAI-P full-time study --- no specialisation -- Cr 2 Elective 1 2 S