Mathematical Analysis III
FSI-SA3Acad. year: 2020/2021
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.
Learning outcomes of the course unit
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.
Linear algebra, differential and integral calculus of functions in a single and more variables.
Recommended optional programme components
Recommended or required reading
Fichtengolc, G.M.: Kurs differencialnogo i integralnogo isčislenija, tom II, Moskva, 1966.
Kalas, J., Ráb, M.: Obyčejné diferenciální rovnice, Brno, 1995.
Fichtengolc, G.M.: Kurs differencialnogo i integralnogo isčislenija, tom III, Moskva, 1966.
Čermák, J., Nechvátal, L.: Matematika III, Brno, 2016. (CS)
Ženíšek, A.: Vybrané kapitoly z matematické analýzy, Brno, 1997.
Hartman, P.: Ordinary differential equations, New York, 1964.
Čermák, J.: Sbírka příkladů z Matematické analýzy III a IV, Brno, 1998.
Planned learning activities and teaching methods
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.
Assesment methods and criteria linked to learning outcomes
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 (75 minutes) consists of 12 test 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 part of the exam (maximum 60 points) and the results achieved in seminars (maximum 40 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).
Language of instruction
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.
Specification of controlled education, way of implementation and compensation for absences
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.
Type of course unit
39 hours, optionally
Teacher / Lecturer
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. Laplace transform and its use in solving of linear ODEs. The method of infinite series.
10. Boundary value problem for 2nd order ODEs.
11. Systems of 1st order ODEs. Properties of solutions and methods of solving of homogeneous linear systems of 1st order ODEs.
12. Properties of solutions and methods of solving of non-homogeneous linear systems of 1st order ODEs.
13. Stability of solutions of ODEs, autonomous systems, bifurcation, chaos.
33 hours, compulsory
Teacher / Lecturer
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.
6 hours, optionally
Teacher / Lecturer
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).