Module Handbook

  • Dynamischer Default-Fachbereich geändert auf MAT

Module MAT-81-36-M-7

Mathematical Analysis of Linear PDEs (M, 9.0 LP)

Module Identification

Module Number Module Name CP (Effort)
MAT-81-36-M-7 Mathematical Analysis of Linear PDEs 9.0 CP (270 h)

Basedata

CP, Effort 9.0 CP = 270 h
Position of the semester 1 Sem. irreg.
Level [7] Master (Advanced)
Language [EN] English
Module Manager
Lecturers
+ further Lecturers of the department Mathematics
Area of study [MAT-TEMA] Industrial Mathematics
Reference course of study [MAT-88.105-SG] M.Sc. Mathematics
Livecycle-State [NORM] Active

Courses

Type/SWS Course Number Title Choice in
Module-Part
Presence-Time /
Self-Study
SL SL is
required for exa.
PL CP Sem.
4V+2U MAT-81-36-K-7
Mathematical Analysis of Linear PDEs
P 84 h 186 h - - PL1 9.0 irreg.
  • About [MAT-81-36-K-7]: Title: "Mathematical Analysis of Linear PDEs"; Presence-Time: 84 h; Self-Study: 186 h

Examination achievement PL1

  • Form of examination: oral examination (20-30 Min.)
  • Examination Frequency: each semester
  • Examination number: 86288 ("Mathematical Analysis of Linear PDE")

Evaluation of grades

The grade of the module examination is also the module grade.


Contents

  • revision of topics from functional analysis (linear spaces, dual spaces and weak convergence, compact operators, embedding, Fredholm theory, boundaries and regularity); crash course on Sobolev spaces (weak derivatives, definition of Sobolev spaces, approximation by smooth functions, traces, Sobolev inequalities and embedding, Poincaré inequalities),
  • elliptical linear PDE of second order: existence of solutions via the Lax-Milgram theorem, the Fredholm alternative, spectral theory, inner regularity, regularity up to the boundary, maximum principles (weak and strong),
  • parabolic linear PDE of second order: Sobolev spaces involving time, Bochner spaces, existence of weak solutions for parabolic PDE of second order via Galerkin approximations and via the Rothe method, uniqueness, regularity, maximum principles (weak and strong),
  • hyperbolic linear PDE of second order: existence of weak solutions via energy estimates and compactness, uniqueness, regularity.

Competencies / intended learning achievements

Upon successful completion of this module, the students have studied important methods required to deal with linear partial differential equations (PDE). The main focus here is on statements of existence, uniqueness and regularity, which can be used to check the well-posedness of a linear PDE. The students are able to use these methods to gain important information on the qualitative behaviour of the solution of linear PDE.

They are able to name and to prove the essential propositions of the lecture as well as to classify and to explain the connections.

By solving the given exercises, have gained a skilled and independent handling of the terms and methods of the lecture. They understand the proofs presented in the lecture and are able to reproduce and explain them and to apply them to concrete examples of (initial and) boundary value problems.

Literature

  • H. W. Alt: Lineare Funktionalanalysis,
  • L.C. Evans: Partial Differential Equations,
  • H. Brezis: Functional Analysis, Sobolev Spaces, and Partial Differential Equations.

Registration

Registration for the exercise classes via the online administration system URM (https://urm.mathematik.uni-kl.de).

Requirements for attendance (informal)

Knowledge from the modules [MAT-80-11B-M-4] and [MAT-70-12A-M-7] resp. [MAT-70-12-M-7] is useful but not necessarily required.

Modules:

Courses

Requirements for attendance (formal)

None

References to Module / Module Number [MAT-81-36-M-7]

Module-Pool Name
[MAT-81-MPOOL-7] Specialisation Partial Differential Equations (M.Sc.)
[MAT-8x-MPOOL-7] Specialisation Modelling and Scientific Computing (M.Sc.)
[MAT-AM-MPOOL-7] Applied Mathematics (Advanced Modules M.Sc.)
[MAT-RM-MPOOL-7] Pure Mathematics (Advanced Modules M.Sc.)