Course detail

# Equations of Mathematical Physics II

FSI-9RF2Acad. year: 2016/2017Winter semesterNot applicable.. year of study1  credit

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.

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.

Recommended optional programme components

Not applicable.

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.

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

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.