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):

0. Introduction and motivation

1. Transport equations

2. The Laplace Equation

3. The Heat Equation

4. The Wave Equation

5. Method of Characteristics

6. 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