Modern Methods of Solving Differential Equations (FSI-SDR)

Academic year 2020/2021
Supervisor: prof. RNDr. Jan Franců, CSc.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:
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.
Learning outcomes and competences:
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.
Prerequisites:
Differential and integral calculus of one and more real variables, ordinary and partial differential equations, functional analysis, function spaces,
probability theory.
Course contents:
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.
Introduction to stochastic differential equations.
Teaching methods and criteria:
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.
Assesment methods and criteria linked to learning outcomes:
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.
Controlled participation in lessons:
Absence has to be made up by self-study.
Type of course unit:
    Lecture  13 × 2 hrs. compulsory                  
    Exercise  13 × 2 hrs. compulsory                  
Course curriculum:
    Lecture 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 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.
Literature - fundamental:
1. S. Fučík, A. Kufner: Nonlinear Differential Equations, Nort Holland, 1980.
2. K. Rektorys: Variational Methods in Mathematics, Science and Engineering, Dordrecht, D. Reidel Publ. Comp., 1980.
3. J. Nečas: Direct Methods in the Theory of Elliptic Equations, Springer, Heidelberg 2012.
4. B. Oksendal: Stochastic Differential Equations, Springer, Berlin 2000.
Literature - recommended:
1. J. Franců: Moderní metody řešení diferenciálních rovnic, Akad. nakl. CERM, Brno 2006
2. K. Rektorys: Přehled užité matematiky, Prometheus, Praha 1995.
3. S. Fučík, A. Kufner: Nelineární diferenciální rovnice, SNTL, Praha 1978.
4. S. Fučík, A. Kufner: Nonlinear Differential Equations, Nort Holland, 1980.
5. J. Nečas: Direct Methods in the Theory of Elliptic Equations, Springer, Heidelberg 2012.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
M2A-P full-time study M-MAI Mathematical Engineering -- Cr,Ex 5 Compulsory 2 2 S