Course detail

Special Topics of Mathematical Analysis

FSI-SA0Acad. year: 2010/2011

The course provides an introduction to the theory of the Laplace transform and its applications and basics of the theory of difference equations. These branches form the theoretical background in the study of many physical and engineering problems. The course deals with the following topics:
The Laplace tranform and its calculation. Properties of the Laplace tranform and its applications. Essentials of the difference calculus. First order and higher order linear difference equations. Chaos and bifurcation.

Language of instruction

Czech

Number of ECTS credits

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 types of differece equations. They will be made familiar with difference equations as mathematical models of given problems and with the choice of a suitable solving method. They also will master solving differential equations by means of the Laplace transform.

Prerequisites

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

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations.

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 notions concerning the Laplace transform and the theory of difference equations. The task of the course is to show that knowledge of the theory of Laplace transform as well as difference equations can be utilized especially in physics and technical branches.

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 B3901-3 Bachelor's

    branch B-MAI , 2 year of study, summer semester, elective (voluntary)

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. RNDr. Jan Čermák, CSc.

Syllabus

1. Laplace transform and its properties.
2. Inverse Laplace transform. Convolution theorem.
3. Laplace transform and solving of differential equations.
4. Laplace transform and solving of systems of differential equations.
5. Applications of Laplace transform.
6. Fractional differential equations.
7. Laplace transform and solving of fractional differential equations.
8. First order difference equations.
9. Higher order difference equations. Properties of solutions of linear equations.
10. Methods of solving of linear higher order difference equations.
11. Chaos and bifurcation.
12. Applications of difference equations.
13. Dynamic equations on time scales.