Course detail

Mathematical Analysis III

FSI-SA3Acad. year: 2024/2025

The course provides an introduction to the theory of infinite series and the theory of ordinary differential equations. These branches form the theoretical background in the study of many physical and engineering problems. The course deals with the following topics:
Number series. Function series. Power series. Taylor series. Fourier series.
Ordinary differential equations. First order differential equations. Higher order linear differential equations. Systems of first order linear differential equations. Stability theory.

Language of instruction

Czech

Number of ECTS credits

7

Mode of study

Not applicable.

Guarantor

prof. RNDr. Jan Čermák, CSc.

Department

Institute of Mathematics (ÚM)

Entry knowledge

Linear algebra, differential and integral calculus of functions in a single and more variables.

Rules for evaluation and completion of the course

Course-unit credit is awarded on the following conditions: Active participation in seminars. Fulfilment of all conditions of the running control of knowledge. At least half of all possible 40 points in both check tests (the first test takes place in 7th week of the semester, the second one in 12th week of the semester). If a student does not fulfil this condition, the teacher can set an alternative one.

Examination: The examination tests the knowledge of definitions and theorems (especially the ability of their application to the given problems) and practical skills in solving of examples. The exam is written and oral, the written part (90 minutes) consists of 12 examples.

Topics of the written part: Number, function, power and Fourier series, ODEs and their properties, solving of ODEs via the infinite series method and the Laplace tranform method, orthogonal trajectories, stability, autonomous systems.

The final grade reflects the result of the written and oral part of the exam (maximum 100 points). 
Grading scheme is as follows: excellent (90-100 points), very good (80-89 points), good (70-79 points), satisfactory (60-69 points), sufficient (50-59 points), failed (0-49 points).


Attendance at lectures is recommended, attendance at seminars is obligatory and checked. Lessons are planned according to the week schedules. Absence from seminars may be compensated for by the agreement with the teacher.

Aims

The aim of the course is to explain basic notions and methods of solving ordinary differential equations, and foundations of infinite series theory. The task of the course is to show that knowledge of the theory of differential equations can be utilized especially in physics and technical branches. Moreover, it is shown that foundations of infinite series theory are important tools for solving various problems.
Students will acquire knowledge of basic types of differential equations. They will be made familiar with differential equations as mathematical models of given problems, with problems of the existence and uniqueness of the solution and with the choice of a suitable solving method. They will master solving of problems of the convergence of infinite series as well as expansions of functions into Taylor and Fourier series.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Fichtengolc, G.M.: Kurs differencialnogo i integralnogo isčislenija, tom II, Moskva, 1966.
Fichtengolc, G.M.: Kurs differencialnogo i integralnogo isčislenija, tom III, Moskva, 1966.
Hartman, P.: Ordinary differential equations, New York, 1964.

Recommended reading

Čermák, J., Nechvátal, L.: Matematika III, Brno, 2016. (CS)
Čermák, J.: Sbírka příkladů z Matematické analýzy III a IV, Brno, 1998.
Kalas, J., Ráb, M.: Obyčejné diferenciální rovnice, Brno, 1995.
Ženíšek, A.: Vybrané kapitoly z matematické analýzy, Brno, 1997.

Classification of course in study plans

  • Programme B-FIN-P Bachelor's 2 year of study, winter semester, compulsory
  • Programme B-MAI-P Bachelor's 2 year of study, winter semester, compulsory

  • Programme C-AKR-P Lifelong learning

    specialization CZS , 1 year of study, winter semester, elective

Type of course unit

 

Lecture

39 hod., optionally

Teacher / Lecturer

prof. RNDr. Jan Čermák, CSc.

Syllabus

1. Number series. Convergence criteria. Absolute and non-absolute convergence.
2. Function and power series. Types of the convergence and basic properties.
3. Taylor series and expansions of functions into Taylor series.
4. Fourier series. Problems of the convergence and expansions of functions.
5. ODE. Basic notions. Initial and boundary value problem.
6. Analytical methods of solving of 1st order ODE. The existence and uniqueness of solutions.
7. Higher order ODEs. Properties of solutions and methods of solving of higher order homogeneous linear ODEs.
8. Properties of solutions and methods of solving of higher order non-homogeneous linear ODEs.
9. Systems of 1st order ODEs. Properties of solutions and methods of solving of homogeneous linear systems of 1st order ODEs.
10. Properties of solutions and methods of solving of non-homogeneous linear systems of 1st order ODEs.                                11. Laplace transform and its use in solving of linear ODEs.
12. Boundary value problem for 2nd order ODEs.                                  13. Method of infinite series in solving of inital and boundary values problems for ODEs. 

Exercise

33 hod., compulsory

Teacher / Lecturer

prof. RNDr. Jan Čermák, CSc.
doc. Ing. Luděk Nechvátal, Ph.D.

Syllabus

1. Limits and integrals - revision.
2. Infinite series.
3. Function series.
4. Power series.
5. Taylor series.
6. Fourier series.
7. Analytical methods of solving of 1st order ODEs.
8. Applications of 1st order ODEs.
9. Higher order linear homogeneous ODEs.
10. Higher order linear non-homogeneous ODEs.
11. Applications of higher order linear ODEs.
12. Systems of 1st order linear homogeneous ODEs.
13. Systems of 1st order linear non-homogeneous ODEs.

Computer-assisted exercise

6 hod., optionally

Teacher / Lecturer

prof. RNDr. Jan Čermák, CSc.

Syllabus

The course is realized in computer labs. The MAPLE software is utilized to illustrate and complete the following topics: 1. Function series - graphical illustrations of types of the convergence (with a special emphasize on Taylor and Fourier series). 2. ODEs - graphical methods of solving (direction fields), geometrical interpretation of solutions (the phase portrait), the Taylor series method, geometrical applications (orthogonal trajectories and others).