Mathematics III (FSI-3M)

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 plays an important role in physics and technical branches. Moreover, it is shown that the infinite series theory is a necessary tool 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 problems of the convergence of infinite series as well as expansions of functions into Taylor and Fourier series, including their applications.

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. Modelling with differential equations.
Basic numerical methods for solving differential equations with a suitable software (e.g. Matlab).

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 two tests should be obtained.

Examination: The exam tests both knowledge of concepts and understanding of fundamental principles, as well as practical skills in solving problems. The exam consists of a written part, followed by a possible oral discussion. The written exam is composed of computational problems primarily covering the following topics: solving first-order ODEs, solving higher-order linear ODEs, solving systems of first-order linear ODEs, Fourier series, solving ODEs using the method of infinite series and Laplace transforms, boundary value problems, numerical and power series, application of convergence criteria, Taylor series expansion of a given function and manipulation of this expansion. A smaller part of the written exam consists of theoretically oriented questions. The final grade is determined primarily by the result of the written exam, but the results of credit tests and the oral discussion may also be taken into account.

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.

- Number series. Basic notions. Convergence criteria. Operations with number series.
- Function series. Basic properties.
- Power series. Taylor series, expansions of functions into power series, and applications.
- Trigonometric Fourier series. Problems of the convergence and expansions of functions.
- Ordinary differential equations (ODE). Basic notions. The existence and uniqueness of the solution to the initial value problem.
- Analytical methods of solving of 1st order ODEs.
- Higher order ODEs. Basic notions. The existence and uniqueness of the solution to the initial value problem. The structure of general solutions.
- Methods of solving higher order homogeneous and nonhomogeneous linear ODEs with constant coefficients.
- Systems of 1st order ODEs. Basic notions. The existence and uniqueness of the solution to the initial value problem. The structure of general solutions.
- Methods of solving homogeneous and nonhomogeneous systems of 1st order linear ODEs.
- The Laplace transform and its use in solving of linear ODEs.
- Stability. Analysis in the phase plane.
- Boundary value problem for 2nd order ODEs.
- Partial differential equations - introduction.
- Mathematical modelling by differential equations.

- Limits and integrals - revision.
- Infinite series. Convergence tests.
- Function and power series.
- Taylor series.
- Fourier series.
- Analytical methods of solving 1st order ODEs.
- Higher order linear homogeneous and nonhomogeneous ODEs.
- Systems of 1st order linear homogeneous and nonhomogeneous ODEs.
- Laplace transform method and series method of solving of ODEs.
- Boundary value problems.

It is conducted using appropriate software (e.g., MATLAB) in a computer lab. The seminar focuses on demonstrating the use of computers, particularly for numerical methods in solving differential equations and also, for example, for the approximation of functions using infinite series.