Course detail

Selected Topics from Ordinary Differential Equations

FSI-0DRAcad. year: 2026/2027

The course covers advanced topics of the theory of ordinary differential  equations, which play a crucial role in the mathematical modelling of various processes in physics and other disciplines.

Language of instruction

Czech

Number of ECTS credits

4

Mode of study

Not applicable.

Guarantor

doc. Ing. Jiří Šremr, Ph.D.

Department

Institute of Mathematics (ÚM)

Entry knowledge

Linear algebra, differential calculus of functions of one and several variables, integral calculus of functions of one variable.

Rules for evaluation and completion of the course

Attendance at lectures and seminars is obligatory and checked. Absence may be compensated for based on an agreement with the teacher. 

Course-unit credit is awarded on the following conditions: Active participation at seminars.

Examination: The exam tests the knowledge of the theoretical apparatus and the ability to apply it to the given problems (depending on the discussed topics). Detailed information will be announced at the end of the semester.

Aims

Aim of the course: The course aims to introduce students to advanced topics of the theory of ordinary differential equations and discuss the potential applications of theoretical results in the mathematical modelling of various processes in physics and other disciplines.

Acquired knowledge and skills: The students will learn the fundamentals of selected topics of ordinary differential equations, understand their role in the mathematical modelling of specific processes, and thus be able to analyse given models more easily and draw the correct conclusions from them.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

G. N. Watson, A treatise on the Bessel functions, Cambridge University Press, Cambridge, 1966. (EN)
I. Kiguradze, Okrajové úlohy pro systémy lineárních obyčejných diferenciálních rovnic, Masarykova univerzita, Brno, 1997. (CS)
L. N. Hand, J. D. Finch, Analytical Mechanics, Cambridge University Press, Cambridge, 1998. (EN)
M. Levi, Classical mechanics with calculus of variations and optimal control: An intuitive introduction, Student Mathematical Library 69, American Mathematical Society, 2014. (EN)
P. Drábek, G. Holubová, Elements of partial differential equations, Berlin: De Gruyter, 2014. (EN)
P. Hartman, Ordinary differential equations, Philadelphia: SIAM, 2002. (EN)

Recommended reading

Not applicable.

Classification of course in study plans

  • Programme B-FIN-P Bachelor's 2 year of study, summer semester, elective
  • Programme N-MAI-P Master's 1 year of study, summer semester, elective

Type of course unit

 

Lecture

26 hod., compulsory

Teacher / Lecturer

doc. Ing. Jiří Šremr, Ph.D.

Syllabus

Depending on the level of knowledge and needs of the students registered, some of the following topics will be discussed:

Methods of solving first-order ordinary differential equations.
Methods of solving higher-order linear ordinary differential equations.
Methods of solving systems of first-order linear differential equations.
Fundamentals of the qualitative theory of systems of non-linear ODEs (initial value problem, existence and uniqueness of solutions, extendability of solutions, global solution, maximal and minimal solutions, well-posedness).
Fundamentals of the qualitative theory of boundary value problems for systems of linear ODEs (boundary conditions, Fredholm's alternative, Green's matrix, integral representation of solutions, well-posedness).
Sturm comparison and separation theorems.
Sturm-Liouville problem for second-order differential equations and other spectral problems.
Disconjugacy and oscillations of second-order linear differential equations.
Bessel equations and Bessel functions.
Variational principles of Lagrangian mechanics.
Mathematical view of the heuristic fundamentals of Hamiltonian mechanics.
Dirac distribution and its application.

Exercise

26 hod., compulsory

Teacher / Lecturer

doc. Ing. Jiří Šremr, Ph.D.

Syllabus

Supplementing remarks and examples concerning the topics discussed in the lectures.