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Course detail

Equations of Mathematical Physics II

Course unit code: FSI-9RF2
Academic year: 2016/2017
Year of study: Not applicable.
Semester: winter
Number of ECTS credits:
Learning outcomes of the course unit:
Students will be made familiar with the generalized formulations (weak and variational) of the boundary value problems for partial and ordinary differential equations, construction of approximate solutions.
Mode of delivery:
Not applicable.
Prerequisites:
Differential and integral calculus of one and more real variables, ordinary and partial differential equations, functional analysis, function spaces.
Co-requisites:
Not applicable.
Recommended optional programme components:
Not applicable.
Course contents (annotation):
The course is a free continuation of subject Equations of Mathematical Physics I.
It focuses on modern methods of solving linear and nonlinear differential equations.
By means of functional analysis generalized formulation of stationary boundary value problems is introduced and existence of their solution is studied.
Finite dimensional approximations of solutions being base for numerical solving are introduced, too.
Recommended or required reading:
L. C. Evans: Partial differential equations. AMS, Providence 1998
J. Franců: Moderní metody řešení diferenciálních rovnic. Akad.nakl.CERM, Brno, 2006
E. Zeidler: Nonlinear functional analysis and its applications. Springer, Berlin 1990
J. Nečas: Les méthodes en theorie des equations elliptiques. Academia, Praha 1967
Planned learning activities and teaching methods:
The course is taught through lectures explaining the basic principles and theory of the discipline.
Assesment methods and criteria linked to learning outcomes:
Practical part of the examination tests ability of mutual conversion of the weak, variational and classical formulation of the particular nonlinear boundary value problem and analysis of its generalized solution. The theoretical part consists of 3 questions related to the subject-matter presented at the lectures.
Language of instruction:
Czech, English
Work placements:
Not applicable.
Course curriculum:
Not applicable.
Aims:
The aim of the course is to provide students an overview of modern methods applied
for solving boundary value problems for differential equations by means of function
spaces and functional analysis including construction of the approximate solutions.
Specification of controlled education, way of implementation and compensation for absences:
Absence has to be made up by self-study.

Type of course unit:

Lecture: 20 hours, optionally
Teacher / Lecturer: prof. RNDr. Jan Franců, CSc.
Syllabus: 1 Spaces of integrable functions.
2 Spaces of functions with integrable derivatives.
3 Imbedding theorems, theorem on traces, dual spaces.
4 Weak formulation of linear elliptic equations and their solvability.
5 Variational formulation, finite dimension approximate solutions.
6 Linear and nonlinear problems, various nonlinearities, Nemytskiy operators.
7 Variational problems and its solvability, convexity problems.
8 Applications to selected problems.
9 Solvability of abstract operator equations.
10 Applications to the selected equations of mathematical physics.

The study programmes with the given course