Course detail

Seminar on Applied Mathematics

FSI-0AMAcad. year: 2024/2025

The course follows topics in Mathematics I, II, III and will introduce the students to the possibilities of using the basic mathematical apparatus in mathematical modelling in physics, mechanics and other technical disciplines. In seminars, some problems  will be selected that students have previously encountered, and these will be discussed in more detail from a mathematics point of view. Furthermore, mathematical modelling using differential equations as well as methods of analysis of the equations obtained will be shown.

Language of instruction

Czech

Number of ECTS credits

2

Mode of study

Not applicable.

Guarantor

doc. Ing. Jiří Šremr, Ph.D.

Department

Institute of Mathematics (ÚM)

Entry knowledge

Linear algebra, differential calculus, integral calculus, solving of linear ordinary differential equations and their systems.

Rules for evaluation and completion of the course

Condition for awarding of the course-unit credit: Active participation in seminars.

Absence tolerated based on an agreement with the teacher.

Aims

Aim of the course: The aim of the course is to show the students in more detail the application of the basic mathematical apparatus in physics, technical mechanics and other fields. The objective is to teach the students to solve analytically selected problems for partial differential equations, and to analyze non-linear ordinary differential equations and their systems, which appear in some mathematical models.

Acquired knowledge and skills: After completing the course, the students will be able to solve analytically selected problems for partial differential equations and understand the relations with problems from other areas of mathematics. They will be able to determine stability and types of the equilibria of non-linear autonomous differential systems and behaviour of solutions in their neighbourhoods. On selected problems from physics, mechanics and other disciplines, the students will be familirized with the possibilities of mathematical modelling using ordinary differential equations and with the analysis of equations obtained.

Study aids

Not applicable.

Prerequisites and corequisites

Basic literature

L. Perko, Differential Equations and Dynamical Systems, Text in Applied Mathematics, 7, Springer-Verlag, New York, 2001, ISBN 0-387-95116-4. (EN)
M. Levi, Classical Mechanics With Calculus of Variations and Optimal Control: An Intuitive Introduction.Student Mathematical Library 69, American Mathematical Society, 2014.ISBN 978-0-8218-9138-4. (EN)
P. Drábek, G. Holubová, Parciální diferenciální rovnice [online], Plzeň, 2011, dostupné z: http://mi21.vsb.cz/modul/parcialni-diferencialni-rovnice. (CS)
P. Hartman, Ordinary differential equations, John Wiley & Sons, New York - London - Sydney, 1964. (EN)

Recommended reading

J. Kalas, M. Ráb, Obyčejné diferenciální rovnice, Masarykova univerzita, Brno, 1995, ISBN 80-210-1130-0. (CS)
L. Perko, Differential Equations and Dynamical Systems, Text in Applied Mathematics, 7, Springer-Verlag, New York, 2001, ISBN 0-387-95116-4. (EN)
P. Drábek, G. Holubová, Parciální diferenciální rovnice [online], Plzeň, 2011, dostupné z: http://mi21.vsb.cz/modul/parcialni-diferencialni-rovnice. (CS)

Classification of course in study plans

  • Programme B-OBN-P Bachelor's 1 year of study, summer semester, elective

Type of course unit

 

Exercise

26 hod., compulsory

Teacher / Lecturer

doc. Ing. Jiří Šremr, Ph.D.

Syllabus

After agreement with the students, some of the following topics will be selected:

  • First-order partial differential equations, transport equation.
  • Sturm-Liouville problem for second-order ordinary differential equations.
  • Heat equation, Diffusion equation.
  • Wave equation in the plane, characteristics, initial value problem.
  • Bessel equation, Bessel functions.
  • Vibrations of a string and a membrane.
  • Equation of catenary.
  • First-order implicit differential equations, envelope of a family of curves.
  • Euler differential equation in stress-analysis of thick-walled cylindrical vessels and analysis of deformation of shells.
  • Green functions of two-point boundary value problem in analysis of bending of beams.
  • Fredholm property for periodic problems and stability of compressed bars.
  • Planar autonomous systems of ODEs: Stability and classification of equlilibria, phase portrait.
  • Linear oscillators with one degree of freedom, different kinds of damping.
  • Duffing equation, Jacobi elliptic functions.
  • Non-linear oscillators with one degree of freedom.
  • Linear oscillations with two degree of freedom.
  • Mathematical modelling of a population dynamic.
  • Mathematical modelling of motions of dislocations in crystals.