Partial differential equations 1000-135RRC
Examples of partial differential equations, physical motivations. Equations of the first order and the method of characteristics (2 lectures).
Classical theory of elliptic equations (representation of solutions, maximum principles and their applications) (3 lectures).
Fourier Transform: definition, basic properties and applications to linear PDE (1 lecture).
Introduction to the theory of distributions and Sobolev spaces: notion and basic properties of weak derivatives, trace and imbedding theorems, the Rellich-Kondrashov Theorem (3 lectures).
Lax-Milgram Lemma and its applications in existence proofs for weak solutions of elliptic problems, Galerkin Method for linear problems (2 lectures).
Introduction to spectral theory. Fredholm alternative and its applications to linear elliptic problems (1 lecture).
Information on the regularity theory of elliptic equations. Examples of applications of fixed point theorems and Galerkin method to nonlinear problems.
(1 lecture).
Type of course
Prerequisites (description)
Course coordinators
Term 2024Z: | Term 2023Z: |
Learning outcomes
A student:
1. Knows basic properties of the Laplace operator and harmonic functions.
2. Knows the definition and basic properties of the Fourier transform and examples of its applications to linear PDE.
3. Knows basic properties of Sobolev spaces. Is aware of basic versions of trace and imbedding theorems
and knows how to apply them in the estimates for linear PDE.
4. Is able to apply the Lax-Milgram Theorem to prove the existence of weak solutions to elliptic problems.
5. Knows the Galerkin method and its basic applications in PDE.
6. Is familiar with classical fixed point theorems and their applications to simple nonlinear equations.
Bibliography
L.C.Evans, Partial differntial equations. AMS 1998
D.Gilbarg, N.S.Trudinger, Elliptic partial differential equations of second order. Springer-Verlag, Berlin 1983
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
- Bachelor's degree, first cycle programme, Mathematics
- Master's degree, second cycle programme, Mathematics
Additional information (registration calendar, class conductors, localization and schedules of classes), might be available in the USOSweb system: