Course detail

Equations of Mathematical Physics II

FSI-9RF2Acad. year: 2017/2018

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.

Language of instruction

Czech

Number of ECTS credits

2

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.

Prerequisites

Differential and integral calculus of one and more real variables,
ordinary and partial differential equations, functional analysis, function spaces.

Co-requisites

Not applicable.

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.

Course curriculum

Not applicable.

Work placements

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.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

L. C. Evans: Partial differential equations. AMS, Providence 1998
E. Zeidler: Nonlinear functional analysis and its applications. Springer, Berlin 1990
J. Nečas: Les méthodes en theorie des equations elliptiques. Academia, Praha 1967

Recommended reading

J. Franců: Moderní metody řešení diferenciálních rovnic. Akad.nakl.CERM, Brno, 2006

Type of course unit

 

Lecture

20 hours, optionally

Teacher / Lecturer

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.