Course detail

# Modern Methods of Solving Differential Equations

The course yields overview of modern methods for solving differential equations based on functional analysis. It deals with the following topics:

• Survey of spaces of functions with integrable derivatives.
• Linear elliptic equations: the weak and variational formulation of boundary value problems, existence and uniqueness of the solution, approximate solutions and their convergence.
• Characteristics of the nonlinear problems.
• Weak and variational formulation of the nonlinear coercive stationary problems, existence of the solution.
• Application to the selected nonlinear equations of mathematical physics.

Language of instruction

Czech

Number of ECTS credits

5

Mode of study

Not applicable.

Entry knowledge

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

Rules for evaluation and completion of the course

Course-unit credit is awarded on condition of having attended the seminars actively.
Examination has two parts: The practical part tests the ability of mutual conversion of the weak, variational and classical formulation of a particular nonlinear boundary value problem and analysis of its generalized solution. Theoretical part includes 4 questions related to the subject-matter presented at the lectures.
Absence has to be made up by self-study.

Aims

The aim of the course is to provide students an overview of modern methods applied for solving boundary value problems for differential equations based on function spaces and functional analysis including construction of the approximate solutions.
Students will be made familiar with the generalized formulations (weak and variational) of the boundary value problems for partial and ordinary differential equations and construction of approximate solutions used for numerical computing.
Students will obtain ideas of stochastic integral and stochastic differential equations.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

S. Fučík, A. Kufner: Nonlinear Differential Equations, Nort Holland, 1980. (EN)
K. Rektorys: Variational Methods in Mathematics, Science and Engineering, Dordrecht, D. Reidel Publ. Comp., 1980. (EN)
J. Nečas: Direct Methods in the Theory of Elliptic Equations, Springer, Heidelberg 2012. (EN)
B. Oksendal: Stochastic Differential Equations, Springer, Berlin 2000. (EN)

J. Franců: Moderní metody řešení diferenciálních rovnic, Akad. nakl. CERM, Brno 2006 (CS)
K. Rektorys: Přehled užité matematiky, Prometheus, Praha 1995. (CS)
S. Fučík, A. Kufner: Nelineární diferenciální rovnice, SNTL, Praha 1978. (CS)
S. Fučík, A. Kufner: Nonlinear Differential Equations, Nort Holland, 1980. (EN)
J. Nečas: Direct Methods in the Theory of Elliptic Equations, Springer, Heidelberg 2012. (EN)

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Classification of course in study plans

• Programme N-MAI-P Master's, 1. year of study, summer semester, compulsory
, 2. year of study, summer semester, compulsory

#### Type of course unit

Lecture

26 hours, compulsory

Teacher / Lecturer

Syllabus

1 Motivation. Overview of selected means of functional analysis.
2 Lebesgue spaces, generalized functions, description of the boundary.
3 Sobolev spaces, different approaches, properties. Imbedding and trace theorems, dual spaces.
4 Weak formulation of the linear elliptic equations.
5 Lax-Mildgam lemma, existence and uniqueness of the solutions.
6 Variational formulation, construction of approximate solutions.
7 Linear and nonlinear problems, various nonlinearities. Nemytskiy operators.
8 Weak and variational formulations of the nonlinear equations.
9 Monotonne operator theory and its applications.
10 Application of the methods to the selected equations of mathematical physics.
11 Introduction to Stochastic Differential Equations. Brown motion.
12 Ito integral and Ito formula. Solution of the Stochastic differential equations.
13 Reserve.

Exercise

26 hours, compulsory

Teacher / Lecturer

Syllabus

Illustration of the topics on the examples and application of theorems and theoretical results presented at the lectures to particular cases and in the selected equations of mathematical physics.

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