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Course info
KMA / MMM
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Course description
Department/Unit / Abbreviation
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KMA
/
MMM
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Academic Year
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2023/2024
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Academic Year
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2023/2024
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Title
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Modern Mathematical Methods
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Form of course completion
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Exam
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Form of course completion
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Exam
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Accredited / Credits
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Yes,
6
Cred.
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Type of completion
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Combined
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Type of completion
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Combined
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Time requirements
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Lecture
4
[Hours/Week]
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Course credit prior to examination
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Yes
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Course credit prior to examination
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Yes
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Automatic acceptance of credit before examination
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No
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Included in study average
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YES
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Language of instruction
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Czech, English
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Occ/max
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Automatic acceptance of credit before examination
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No
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Summer semester
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2 / -
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0 / -
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0 / -
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Included in study average
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YES
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Winter semester
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0 / -
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0 / -
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0 / -
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Repeated registration
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NO
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Repeated registration
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NO
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Timetable
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Yes
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Semester taught
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Summer semester
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Semester taught
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Summer semester
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Minimum (B + C) students
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1
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Optional course |
Yes
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Optional course
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Yes
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Language of instruction
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Czech, English
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Internship duration
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0
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No. of hours of on-premise lessons |
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Evaluation scale |
1|2|3|4 |
Periodicity |
každý rok
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Evaluation scale for credit before examination |
S|N |
Periodicita upřesnění |
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Fundamental theoretical course |
No
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Fundamental course |
No
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Fundamental theoretical course |
No
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Evaluation scale |
1|2|3|4 |
Evaluation scale for credit before examination |
S|N |
Substituted course
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None
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Preclusive courses
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N/A
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Prerequisite courses
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N/A
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Informally recommended courses
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KMA/ODR and KMA/PDR
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Courses depending on this Course
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KMA/MMA
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Histogram of students' grades over the years:
Graphic PNG
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XLS
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Course objectives:
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The aim of this course is an introduction to modern methods of solving BVPs for ODEs and PDEs. The students will recall basic topological methods (Leray-Schauder degrese, fixed point theorems, upper and lower solutions) and will be acquainted with basic variational methods (critical point theorems, Mountain Pass theorem, Saddle Point theorem). The attention will be also paid to bifurcations of nontrivial solutions. These tools will be used to prove the existence, uniqueness or multiplicity of solutions on illustrative examples.
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Requirements on student
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Demonstrate knowledge of fundamental classical and modern methods of solving BVPs for ODEs and PDEs. Demonstrate knowledge of fundamental bifurcation theorems. Demonstrate knowledge of fundamental variational methods. The ability to apply theoretical results in solving problems on the topics in the syllabus.
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Content
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1. BVPs for ODEs, classical solutions, examples
2. BVPs for PDEs, classical solutions, examples
3. BVPs for ODEs, weak solutions, examples
4. BVPs for PDEs, weak solutions, examples
5. Bifurcation theory, motivation
6. Implicit function theorem and bifurcations
7. Leray-Schauder degrese and bifurcations
8. Potential bifurcation theorems
9. Variational methods
10. Mountain Pass theorem
11. Saddle Point theorem
12. Applications of variational methods
13. Summary and conclusion
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Activities
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Fields of study
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Guarantors and lecturers
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Literature
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Time requirements
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All forms of study
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Activities
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Time requirements for activity [h]
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Contact hours
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52
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Preparation for an examination (30-60)
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50
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Preparation for comprehensive test (10-40)
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30
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Total
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132
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Prerequisites
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Knowledge - students are expected to possess the following knowledge before the course commences to finish it successfully: |
formulovat a vysvětlit základní pojmy teorie obyčejných diferenciálních rovnic (v rozsahu předmětu KMA/ODR) |
formulovat a vysvětlit základní pojmy teorie parciálních diferenciálních rovnic (v rozsahu předmětu KMA/PDR) |
Skills - students are expected to possess the following skills before the course commences to finish it successfully: |
vyřešit ODR 1. řádu se separovatelnými proměnnými |
vyřešit počáteční a okrajové úlohy pro lineární ODR 1. a 2. řádu |
řešit počátečně-okrajové úlohy pomocí Fourierovy metody a metody integrálních transformací
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řešit počáteční úlohy pro transportní, vlnovou a difuzní rovnici pomocí základních metod |
aplikovat obyčejné a parciální diferenciální rovnice a jejich řešení na úlohy z praxe
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Competences - students are expected to possess the following competences before the course commences to finish it successfully: |
N/A |
N/A |
N/A |
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Learning outcomes
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Knowledge - knowledge resulting from the course: |
definovat a vysvětlit pojem klasického a slabého řešení okrajové úlohy pro ODR |
definovat a vysvětlit pojem klasického a slabého řešení okrajové úlohy pro PDR |
popsat a vysvětlit základní pojmy teorie bifurkací |
popsat a vysvětlit základní variační metody (věty typu Mountain pass a Sedlový bod) |
Skills - skills resulting from the course: |
najít slabé řešení základních okrajových úloh pro ODR |
najít slabé řešení základních okrajových úloh pro PDR |
zformulovat větu o implicitní funkci a aplikovat ji v teorii bifurkací |
aplikovat stupeň zobrazení v teorii bifurkací |
formulovat potenciální bifurkační věty |
aplikovat variační metody na úlohy z praxe |
Competences - competences resulting from the course: |
N/A |
N/A |
N/A |
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Assessment methods
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Knowledge - knowledge achieved by taking this course are verified by the following means: |
Combined exam |
Skills - skills achieved by taking this course are verified by the following means: |
Skills demonstration during practicum |
Competences - competence achieved by taking this course are verified by the following means: |
Combined exam |
Skills demonstration during practicum |
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Teaching methods
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Knowledge - the following training methods are used to achieve the required knowledge: |
Lecture supplemented with a discussion |
Interactive lecture |
Self-study of literature |
Skills - the following training methods are used to achieve the required skills: |
Practicum |
Task-based study method |
Self-study of literature |
Individual study |
Competences - the following training methods are used to achieve the required competences: |
Task-based study method |
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