Module Handbook

  • Dynamischer Default-Fachbereich geändert auf MAT

Module MAT-81-13-M-7

Nonlinear Partial Differential Equations (M, 9.0 LP)

Module Identification

Module Number Module Name CP (Effort)
MAT-81-13-M-7 Nonlinear Partial Differential Equations 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
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-81-13-K-7
Nonlinear Partial Differential Equations
P 84 h 186 h - - PL1 9.0 irreg.
  • About [MAT-81-13-K-7]: Title: "Nonlinear Partial Differential Equations"; 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: 86325 ("Nonlinear PDE")

Evaluation of grades

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


Contents

  • Fundamentals: Weak convergence and compactness,
  • Elliptic Partial Differential Equations: weak solution theory (Lax-Milgram Theorem, Fredholmalternative), regularity (in the interior, to the edge), eigenvalue problems, maximum principles,
  • Evolution equations: Duality theory, weak convergence in Hilbert spaces, Gelfand triple,
  • Spaces of time-dependent functions, parabolic equations (weak formulation, existence and uniqueness, regularity, maximum principles), hyperbolic partial differential equations,
  • Calculus of variations,
  • Theory of monotone operators and fixed point theorems,
  • Approximations: iterations, space discretisation (Galerkin), time discretisation (Rothe), regularisations.

Competencies / intended learning achievements

Upon successful completion of this module, the students have learned methods to treat non-linear partial differential equations. In particular, they are familiar with propositions on the existence, uniqueness and regularity, which are necessary to check the well-posedness of a PDE model. In addition, the students are able to gain important information about the qualitative behaviour of the solution of nonlinear partial differential equations by using these methods. They are able to name the essential statements of the lecture as well as to classify and to explain the connections.

In the exercise classes, the students have developed an experienced handling of the mathematical objects and methods that have been introduced and they have gained insights into interdisciplinary issues. They have learnt to apply these techniques to new problems, analyse them and to develop solution strategies.

Literature

  • R.A. Adams: Sobolev Spaces,
  • H. Brezis: Functional Analysis, Sobolev Spaces and Partial Differential Equations,
  • L.C. Evans: Partial Differential Equations,
  • G.M. Lieberman: Second Order Parabolic Partial Differential Equations,
  • M. Ruzicka: Nichtlineare Funktionalanalysis,
  • R. Showalter: Monotone operators in Banach space and nonlinear partial differential equations,
  • E. Zeidler: Nonlinear Functional Analysis and its Applications II/B: Nonlinear Mono-tone Operators.

Registration

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

References to Module / Module Number [MAT-81-13-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.)