Modern Methods of Solving Differential Equations
FSI-SDRCompulsoryMaster's (2nd cycle)Acad. year: 2016/2017Summer semester2. year of study5 credits
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 problems, existence of the solution. Application to the selected nonlinear equations of mathematical physics. Introduction to stochastic differential equations.
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 and construction of approximate solutions used for numerical computing. Students will obtain ideas of stochastic integral and stochastic differential equations.
Mode of delivery
90 % face-to-face, 10 % distance learning
Differential and integral calculus of one and more real variables,
ordinary and partial differential equations, functional analysis, function spaces,
Recommended optional programme components
Recommended or required reading
J. Franců: Moderní metody řešení diferenciálních rovnic, Akad. nakl. CERM, Brno 2006 (CS)
S. Fučík, A. Kufner: Nonlinear Differential Equations, Nort Holland, 1980.
K. Rektorys: Variational Methods in Mathematics, Science and Engineering, Dordrecht, D. Reidel Publ. Comp., 1980.
K. Rektorys: Přehled užité matematiky, Prometheus, Praha 1995. (CS)
J. Nečas: Direct Methods in the Theory of Elliptic Equations, Springer, Heidelberg 2012.
S. Fučík, A. Kufner: Nelineární diferenciální rovnice, SNTL, Praha 1978. (CS)
B. Oksendal: Stochastic Differential Equations, Springer, Berlin 2000.
J. Nečas: Direct Methods in the Theory of Elliptic Equations, Springer, Heidelberg 2012. (EN)
Planned learning activities and teaching methods
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.
Language of instruction
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.
Specification of controlled education, way of implementation and compensation for absences
Absence has to be made up by self-study.
Type of course unit
26 hours, optionally
Teacher / Lecturer
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.
26 hours, compulsory
Teacher / Lecturer
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.