Course objectives:
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Students will be acquainted with basic variational approaches to solving nonlinear boundary value problems of mathematical analysis. They will focus mainly on finding the corresponding functional, on recognition of its geometry (functional bounded from below, saddle, mountain-pass, etc.) and on the conditions which guarantee the existence of the critical point (Palais-Smale condition). The particular methods will be illustrated by several examples and the students will try to apply them numerically.
The seminar assumes the knowledge of functional analysis within the range of lectures KMA/ODR and KMA/UFA.
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Requirements on student
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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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Week 1-3: variational formulation of nonlinear boundary value problem, comparison with topological methods Week 4-9: global minimum, mountain-pass, saddle point, general minimax, existence of critical point Week 10-13: examples and numerical approach
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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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26
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Graduate study programme term essay (40-50)
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40
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Total
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66
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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: |
Students should be familiar with the theory of functional analysis to the extent of the courses KMA/ UFA or KMA/FA. |
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Learning outcomes
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Knowledge - knowledge resulting from the course: |
By the end of the course, a successful student should be able to use basic variational methods and apply theoretical tools to real problems.
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Assessment methods
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Knowledge - knowledge achieved by taking this course are verified by the following means: |
Skills demonstration during practicum |
Seminar work |
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Teaching methods
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Knowledge - the following training methods are used to achieve the required knowledge: |
Seminar |
Task-based study method |
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