Vorlesung: Partielle Differentialgleichungen (PDG1) (WiSe 2019/20)
Lecture (Vorlesung):
Tue 14-16 (in B 006) & Wed 10-12 (in B 004). LSF
Exercises (Übungen):
See
separate webpage (uni2work). LSF
Tutorials (Tutorien):
See separate webpage (uni2work).
Synopsis (Kurzbeschreibung):
This course gives an introduction to Partial Differential
Equations (PDEs), a vast area within Analysis. PDE's play an important rôle
in applications of Mathematics to other sciences (most prominently in
Physics and Engineering, but also in Biology and Financial
Sciences), as well as in Pure Mathematics (Analysis, Geometry,
Stochastics; Algebra less).
Among other things, we will study: the method of characteristics for (non-linear) first-order PDEs, the
classification of linear 2nd
order PDEs in elliptic, parabolic, and hyperbolic equations, explicit classical
solutions for the most prominent such equations
(Laplace and Poisson equations, heat equation, wave equation),
including boundary value problems and Cauchy problems.
(Die Vorlesung führt in die Theorie der partiellen
Differentialgleichungen ein. PDG'en spielen eine zentrale Rolle
sowohl in vielen Anwendungsgebieten der Mathematik, als auch in der
reinen Mathematik. Behandelt werden, unter anderem, die
Charakteristikenmethode, die Typeneinteilung in elliptische,
hyperbolische und parabolische Differentialgleichungen, explizite
Lösungsmethoden für die wichtigsten Typen linearer PDG'en zweiter
Ordnung (Laplacegleichung, Poissongleichung, Wellengleichung und
Wärmeleitungsgleichung), Randwert-Probleme, Cauchy-Probleme.)
Audience (Hörerkreis):
Bachelor students of Mathematics (WP16), Master students of
Mathematics (WP2), Master students of 'Finanz- und
Versicherungsmathematik' (WP49), TMP Master.
Credits:
9 (6+3) ECTS.
Prerequisites (Vorkenntnisse):
Analysis I-III, Lineare Algebra I-II.
You find a handout with the needed facts (without proofs, and to be updated!) in uni2work.
Language (Sprache):
English.
Exam (Prüfung):
See separate webpage (uni2work).
Content (Inhalt):
- Introduction and motivation
- Transport equations
- The Laplace Equation
- The Heat Equation
- The Wave Equation
- Method of Characteristics
- Fourier transform and PDE
Literature:
In uni2work
you will find a copy of the notes from the lecture (to be updated as we go
along).
Above you will find a short
description of the content of the lecture (to be updated as we go
along).
The lecture will mainly follow the books by Evans, and Arendt & Urban
mentioned below.
(Auf uni2work
wird es ein Mitschrift aus der Vorlesung geben. Hier wird laufend eine Kurzübersicht der
Vorlesung erstellt.
Die Vorlesung wird sich
größtenteils auf folgenden zwei Büchern (von denen mehrere Exemplare
in der Bibliothek vorhanden sind) basieren:)
- [E] L. C. Evans, Partial Differential Equations: Second Edition, AMS, Providence, RI, 2010.
- [A-U] W. Arendt, K. Urban, Partielle Differenzialgleichungen, Springer Spektrum, 2018. (Login with your Campus-account.)
Supplementary literatur (Ergänzende Literatur):
- E. DiBenedetto, Partial Differential Equations (2nd edition), Birkhäuser Cornerstones, 2010.
- M. Renardy, R.C. Rogers, An Introduction to Partial Differential Equations, Springer, 2004.
- J. Jost, Partial Differential Equations, Springer, 2013.
- B. Schweizer, Partielle Differentialgleichungen, 2. Auflage, Springer Spektrum, 2018.
- F. John, Partial Differential Equations,Springer, 1982.
- J. Rauch, Partial Differential Equations, Springer, 1991.
- G.B. Folland, Introduction to Partial Differential Equations, Second Edition, Princeton University Press, 1995.
- P.J. Olver, Introduction to Partial Differential Equations, Springer, Cham, 2014.
- Q. Han, A Basic Course in Partial Differential Equations, AMS, 2011.
Here
a longer liste.
Office hours (Sprechstunde):
Thursday 10:15-11:00 (Room B 408) or by appointment via email.
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Letzte Änderung: 03 March 2021 (
No more updates
).
Thomas Østergaard Sørensen