Course detail
Modern Methods of Solving Differential Equations
FSI-SDR-A Acad. year: 2023/2024 Summer semester
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.
Language of instruction
English
Number of ECTS credits
5
Supervisor
Department
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.
The study programmes with the given course
Programme N-MAI-A: Mathematical Engineering, Master's, compulsory
Programme N-AIM-A: Applied and Interdisciplinary Mathematics, Master's, 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.