Mathematics III (FSI-3M-A)

Academic year 2023/2024
Supervisor: prof. Mgr. Pavel Řehák, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: English
Aims of the course unit:
The aim of the course is to explain basic notions and methods of solving ordinary and partial 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.
Learning outcomes and competences:
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.
Prerequisites:

Linear algebra, differential and integral calculus of functions of a single variable and of more variables.
Course contents:

The course provides an introduction to the theory of infinite series and the theory of ordinary and partial 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.
Partial differential equations. Classification. Modelling with differential equations.
Basic numerical methods for solving differential equations with a suitable software (e.g. Matlab).
Teaching methods and criteria:
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 (this concerns also the seminars in computer lab). At least half of all possible points in each of the both tests.

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 has written and oral part. The written exam consists in particular of the examples on the following topics:
Number and function series, the expansion of a function into Taylor series, solving of first order ODEs, solving of higher order linear ODEs, solving of system of first order linear ODEs, Fourier series, solving of ODEs via the infinite series and the Laplace tranform method, boundary value problems, basics of PDEs theory. Some theoretical queries concerning basic concepts can be included in the written part as well.
Topics of practical part: The expansion of a function into Taylor series, solving of first order ODEs, solving of higher order linear ODEs, solving of system of first order linear ODEs.
The final grade reflects the result of the written part of the exam (maximum 80 points) and the result of the oral part (maximum 20 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.
Controlled participation in lessons:
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:
    Lecture  13 × 3 hrs. optionally                  
    Exercise  13 × 3 hrs. compulsory                  
    Computer-assisted exercise  13 × 1 hrs. compulsory                  
Course curriculum:
    Lecture 1. Number series. Basic notions. Convergence criteria.
2. Operations with number series. Function series. Basic properties.
3. Power series. Taylor series and expansions of functions into power series.
4. Trigonometric Fourier series. Problems of the convergence and expansions of functions.
5. Ordinary differential equations (ODE). Basic notions. The existence and uniqueness of the solution to the initial value problem for 1st order ODE. Analytical methods of solving of 1st order ODE.
6. Higher order ODEs. Basic notions. The existence and uniqueness of the solution to the initial value problem for higher order ODEs. Genral solutions of homogeneous and nonhomogeneous linear equations. Methods of solving of higher order homogeneous linear ODEs with constant coefficients.
7. Methods of solving of higher order non-homogeneous linear ODEs with constant coefficients.
8. Systems of 1st order ODEs. Basic notions. The existence and uniqueness of the solution to the initial value problem for systems of 1st order ODE. General solution of homogeneous and non-homogeneous systems of 1st order ODE.
9. Methods of solving of homogeneous systems of 1st order linear ODEs.
10. Methods of solving of non-homogeneous systems of 1st order linear ODEs.
11. The Laplace transform and its use in solving of linear ODEs. The method of Taylor series in solving of ODEs.
12. Stability of solutions of ODEs and their systems. Boundary value problem for 2nd order ODEs. Partial differential equations. Basic notions. The equations of mathematical physics.
13. Mathematical modelling by differential equations.
    Exercise 1. Limits and integrals - revision.
2. Infinite series.
3. Function and power series.
4. Taylor series.
5. Fourier series.
6. Analytical methods of solving of 1st order ODEs.
7. Analytical methods of solving of 1st order ODEs (continuation).
8. Higher order linear homogeneous ODEs.
9. Higher order non-homogeneous linear ODEs.
10. Systems of 1st order linear homogeneous ODEs.
11. Systems of 1st order linear non-homogeneous ODEs.
12. Systems of 1st order linear ODEs - continuation.
13. Laplace transform method and series method of solving of ODEs.
    Computer-assisted exercise The course is realized in computer labs with utilizing a suitable software (e.g. Matlab). This part of the course is focused mainly on demonstration of the use of the software in numerical methods in differential equations and related topics.
Literature - fundamental:
2. Hartman, P.: Ordinary Differential Equations, New York, 1964.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
B-STI-A full-time study --- no specialisation -- Cr,Ex 8 Compulsory 1 2 W
B-STI-Z visiting student --- no specialisation -- Cr,Ex 8 Recommended course 1 1 W