Course: Ordinary Differential Equations

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Course title Ordinary Differential Equations
Course code KMA/ODR
Organizational form of instruction Lecture + Seminar
Level of course Bachelor
Year of study 3
Semester Winter
Number of ECTS credits 6
Language of instruction Czech
Status of course Compulsory, Compulsory-optional, Optional
Form of instruction Face-to-face
Work placements This is not an internship
Recommended optional programme components None
Lecturer(s)
  • Čepička Jan, Ing. Ph.D.
  • Nečesal Petr, Ing. Ph.D.
  • Cibulka Radek, Ing. Ph.D.
  • Looseová Iveta, Ing.
  • Tomiczková Světlana, RNDr. Ph.D.
Course content
Week 1.-3.: Differential equations in physics. Basic notions. Cauchy problem in equations and in system of equations. Picard theorem. Linear equation, fundamental system, method of variation of parameters. Special cases. Systems of linear differentioal equations, fundamental system, method of variation of parameters. Week 4.-5.: Equations and system of equations with constant parameters. Symbolic method. Power series method. Week 6.-7.: Theory of stability Week 8.-12.: Boundary value problems. Basic notions. Sturm?Liouville theory. Eigenvalue problem. Homogeneous and non-homogeneous problem. Solution methods (Fourier, calculus of variations, method of variation of parameters). Green function. Week 13.: Brush-up

Learning activities and teaching methods
Lecture supplemented with a discussion
  • Contact hours - 65 hours per semester
  • Preparation for comprehensive test (10-40) - 25 hours per semester
  • Preparation for formative assessments (2-20) - 15 hours per semester
  • Preparation for an examination (30-60) - 55 hours per semester
prerequisite
professional knowledge
popsat a vysvětlit elementární metody řešení obyčejných diferenciálních rovnic (v rozsahu předmětu KMA/SDR)
popsat a vysvětlit základní numerické metody řešení diferenciálních rovnic (v rozsahu předmětu KMA/NM)
popsat a vysvětlit základní pojmy diferenciálního a integrálního počtu (v rozsahu předmětů KMA/M1 a KMA/M2)
popsat a vysvětlit základní pojmy lineární algebry (v rozsahu předmětu KMA/LA)
professional skills
pro zadanou matici vypočítat vlastní čísla a vlastní vektory
pro zadanou vektorovou funkci vypočítat derivaci a integrál
vypočítat Jakobiovu matici zobrazení z Rn do Rm
ovládat aritmetické operace s vektory a maticemi
vyřešit základní typy obyčejných diferenciálních rovnic metodou přímé integrace nebo metodou separace proměnných
vyřešit obyčejnou diferenciální rovnici druhého řádu s konstantními koeficienty
general eligibility
N/A
N/A
learning outcomes
professional knowledge
definovat a vysvětlit pojem řešení okrajové úlohy včetně úlohy na vlastní čísla pro Sturmův-Liouvilleův operátor
formulovat a vysvětlit základní věty o existenci a jednoznačnosti řešení Cauchyovy úlohy pro soustavu nelineárních diferenciálních rovnic prvního řádu
definovat a vysvětlit základní pojmy teorie stability včetně základních typů bifurkací
vysvětlit využití teorie obyčejných diferenciálních rovnic při modelování jednoduchých problémů ve fyzice, biologii, mechanice a ekonomii
vysvětlit variační formulaci okrajové úlohy a použití Ritzovy a Galerkinovu metody, resp. metody konečných prvků k jejímu numerickému řešení
professional skills
načrtnout a interpretovat fázový portrét a bifurkační diagram pro jednoduché dynamické systémy ve dvou dimenzích
v alespoň jednom SW prostředí (Maxima, Matlab, Mathematica, Maple) analyzovat zadanou úlohu a implementovat vybrané numerické metody
rozhodnout o stabilitě a asymptotické stabilitě klidového stavu lineárních, skorolineárních a nelineárních dynamických systémů
aplikovat na základní okrajové úlohy Fourierovu metodou a variační přístup
pro Sturmův-Liouvilleův operátor vypočítat vlastní čísla a vlastní funkce
pro zadanou soustavu lineárních diferenciálních rovnic prvního řádu nalézt fundamentální systém a obecné řešení metodou variace konstant
general eligibility
N/A
N/A
N/A
teaching methods
professional knowledge
Interactive lecture
Lecture supplemented with a discussion
professional skills
Practicum
One-to-One tutorial
general eligibility
One-to-One tutorial
Self-study of literature
assessment methods
professional knowledge
Oral exam
professional skills
Written exam
general eligibility
Oral exam
Recommended literature
  • server TRIAL.
  • Kufner, Alois. Obyčejné diferenciální rovnice. 1. vyd. Plzeň : Západočeská univerzita, 1993. ISBN 80-7082-106-X.
  • Míka, Stanislav; Kufner, Alois. Okrajové úlohy pro obyčejné diferenciální rovnice. 2. upr. vyd. Praha : SNTL - Nakladatelství technické literatury, 1983.
  • Nagy, Jozef. Soustavy obyčejných diferenciálních rovnic : Vysokošk. příručka pro vys. školy techn. směru. 2., nezm. vyd. Praha : SNTL, 1983.


Study plans that include the course
Faculty Study plan (Version) Branch of study Category Recommended year of study Recommended semester
Faculty of Applied Sciences Computer Modelling (2016) Special and interdisciplinary fields 3 Winter
Faculty of Applied Sciences Geomatics (2015) Construction industry, geodesy and cartography 1 Winter
Faculty of Applied Sciences General Mathematics (2012) Mathematics courses 3 Winter
Faculty of Applied Sciences Training Teachers of Mathematics at Higher Secondary Scholls (2017) Pedagogy, teacher training and social care 1 Winter
Faculty of Applied Sciences Geomatics (2016) Construction industry, geodesy and cartography 1 Winter
Faculty of Applied Sciences Geomatics (1) Construction industry, geodesy and cartography 1 Winter
Faculty of Applied Sciences Training Teachers of Mathematics at Higher Secondary Scholls (2015) Pedagogy, teacher training and social care 1 Winter
Faculty of Applied Sciences Mathematics for Business Studies (2015) Mathematics courses 3 Winter
Faculty of Applied Sciences Mathematics and its Applications (2016) Mathematics courses 3 Winter
Faculty of Applied Sciences Mathematics and its Applications (2015) Mathematics courses 3 Winter
Faculty of Applied Sciences Computer Modelling (2014) Special and interdisciplinary fields 3 Winter
Faculty of Applied Sciences Financial Informatics and Statistics (2007) Economy 3 Winter
Faculty of Applied Sciences Financial Informatics and Statistics (2014) Economy 1 Winter
Faculty of Applied Sciences Training Teachers of Mathematics at Higher Secondary Scholls (2016) Pedagogy, teacher training and social care 1 Winter
Faculty of Applied Sciences Training Teachers of Mathematics at Higher Secondary Scholls (2017) Pedagogy, teacher training and social care 1 Winter
Faculty of Applied Sciences Mathematics for Business Studies (2011) Mathematics courses 2 Winter
Faculty of Applied Sciences Scientific Computing and Modelling (2011) Mathematics courses 2 Winter
Faculty of Applied Sciences Mathematics and its Applications (2017) Mathematics courses 3 Winter
Faculty of Applied Sciences Financial Informatics and Statistics (2017) Economy 3 Winter
Faculty of Applied Sciences Geomatics (2017) Construction industry, geodesy and cartography 1 Winter
Faculty of Applied Sciences Applied and Engineering Physics (2016) Special and interdisciplinary fields 3 Winter
Faculty of Applied Sciences Mathematics and Management (2011) Mathematics courses 3 Winter
Faculty of Applied Sciences Mathematics and its Applications (2017) Mathematics courses 3 Winter
Faculty of Applied Sciences Financial Informatics and Statistics (1) Economy 3 Winter
Faculty of Applied Sciences Training Teachers of Mathematics at Higher Secondary Scholls (2015) Pedagogy, teacher training and social care 1 Winter
Faculty of Applied Sciences Geomatics (1) Construction industry, geodesy and cartography 1 Winter
Faculty of Applied Sciences Mathematics for Business Studies (2017) Mathematics courses 3 Winter
Faculty of Applied Sciences Mathematics for Business Studies (2017) Mathematics courses 3 Winter
Faculty of Applied Sciences Applied and Engineering Physics (2012) Special and interdisciplinary fields 3 Winter
Faculty of Applied Sciences Mathematics and its Applications (2015) Mathematics courses 3 Winter
Faculty of Applied Sciences Financial Informatics and Statistics (2016) Economy 3 Winter
Faculty of Applied Sciences Geomatics (2016) Construction industry, geodesy and cartography 1 Winter
Faculty of Applied Sciences Financial Informatics and Statistics (2014) Economy 1 Winter
Faculty of Applied Sciences Training Teachers of Mathematics at Higher Secondary Scholls (2016) Pedagogy, teacher training and social care 1 Winter
Faculty of Applied Sciences Training Teachers of Mathematics at Higher Secondary Scholls (2014) Pedagogy, teacher training and social care 1 Winter
Faculty of Applied Sciences General Mathematics (2012) Mathematics courses 3 Winter
Faculty of Applied Sciences Mathematics for Business Studies (2016) Mathematics courses 3 Winter
Faculty of Applied Sciences Financial Informatics and Statistics (2017) Economy 3 Winter
Faculty of Applied Sciences Financial Informatics and Statistics (2016) Economy 3 Winter
Faculty of Applied Sciences Mathematics and its Applications (2016) Mathematics courses 3 Winter
Faculty of Applied Sciences Geomatics (2017) Construction industry, geodesy and cartography 1 Winter
Faculty of Applied Sciences Financial Informatics and Statistics (2011) Economy 1 Winter
Faculty of Applied Sciences Mathematics for Science (2012) Mathematics courses 3 Winter
Faculty of Applied Sciences Computer Modelling (2017) Special and interdisciplinary fields 3 Winter
Faculty of Applied Sciences Mathematics for Business Studies (2016) Mathematics courses 3 Winter
Faculty of Applied Sciences Applied and Engineering Physics (2017) Special and interdisciplinary fields 3 Winter