Content

• Weak derivatives and partial integration, mollifiers, properties.
• Hölder spaces, boundaries and regularity.
• Sobolev spaces, approximations by smooth functions.
• Extensions and traces.
• Sobolev inequalities and embeddings.
• Poincare's inequality.
• Some applications to elliptic PDEs.

We recommend this lecture for everyone who wants to work with partial differential equations

Contact time

2 SWS lectures
1 SWS exercises

Lectures:

Tuesday 10:15 - 11:45 (Raum 48-538)

Exercises:

Friday, 12:15 - 13:45 (Raum 48-582)

Material

OLAT

Kis

Inhalt

Partial Differential Equations play an important role in natural sciences and engineering. Stationary processes can be modelled by differential equations vthat involve more than one spatial variable, e.g. equations for membrans or electrostatic and gravitational potentials. The equations can also include time derivatives to describe transient processes like growth processes, wave propagation, heat transfer or fluid flow. In this introductory lecture the three most important types of second order PDEs are presented: elliptic, parabolic and hyperbolic equations. Explicit solution techniques and the qualitative baviour of solutions is discussed. Special knowledge of results from functional analysis is not required.

Kontaktzeit

2 SWS lectures
1 SWS exercises

Vorlesungen:

Montag 10:15 - 11:45 (Raum 48-538)

Donnerstag 08:15 - 09:45 (Raum 48-582)

2. Semesterhälfte

Übungen:

Donnerstag, 14:00 - 15:30 (Raum 48-582)

Material

OLAT

Kis

Content

Recap on the basics of ODEs and PDEs from the viewpoint of functional theory

- Recap on numerical methods for ODEs (deterministic methods: Euler, RK, finite difference, finite elements)

- Neural networks (NNs) and approximation theorems

- NNs as a method for solving differential functional equations

- NNs as a method for solving functional state estimation problem

- NNs as a method for solving functional optimization problem

- Different types of NN design for PDEs and ODEs

- Physics informed neural networks, neural differential equations and diffusion models

- Applications in process automation, automated robot and vehicle control

In this course students shall learn the fundamental principles governing complex systems via the theory of dynamical systems. We shall mainly rely on PDEs and ODEs as prototypical models to capture the dynamics of various systems from vehicle motion to the spreading of diseases. In the first quarter of the course, we shall make a brief recapitulation of the basic topics such as wellposedness, phase space diagrams, stability theory and bifurcation theory, classes of PDEs and also look at standard numerical methods for solving ODEs and PDEs. With the emergence of neural networks as universal function approximators they offer compelling advantages for dealing with complex and high-dimensional PDEs and ODEs. The course shall focus on understanding how neural networks and the concepts of machine learning can be used to obtain solutions to different types of dynamical systems especially the ones specified using PDEs and ODEs. Going further, we shall also focus on the problem of estimating unobserved states and the problem of controlling the system states based on partial observations with the help of neural networks. Finally, we shall see how the three core blocks of prediction, estimation and control can be integrated seamlessly in a network to obtain a generic unified algorithm for solving dynamical problems. By the course's end, students will have gained a solid foundational understanding of dynamical systems, ODEs, PDEs and neural networks, equipped with analytical and computational tools to address interdisciplinary problems effectively.

Contact time

2 SW lectures

Lectures

To be announced

Material

OLAT

Kis

Ziele

Modellierung von Problemen aus den Lebenswissenschaften mithilfe von Differenzengleichungen: x_{t+1}= f(x_t ).

• Mathematische Werkzeuge für Differenzengleichungen kennenlernen.
• Untersuchung der Modelle hinsichtlich (Stabilitäts)Analyse, Simulationen.

Literatur

• E.S. Allman, J.A. Rhodes: Mathematical Models in Biology.
  An Introduction, Cambridge Univ. Press, 2004.
•  L. Edelstein-Keshet: Mathematical Models in Biology, SIAM 2005.
• S. Elaydi: An Introduction to Difference Equations, 3rd edition, Springer, 2005.
  Weitere (themenspezifische) Literatur wird noch bekannt gegeben.

Inhalt

• Mathematisches Werkzeug:
   ▶Dynamik von Differenzengleichungen 1. Ordnung:
      Gleichgewichtspunkte, Einspinndiagramme, Stabilitä
• tskriterien,
      Verzweigungsdiagramme.
  ▶ Systeme von Differenzengleichungen, Stabilitätstheorie.

•  Anwendungen (z.B.):

           ▶ Wirt-Parasit-Interaktionen
           ▶ Kannibalische Mehlkäfer
           ▶ Infektionsausbreitungen (SIR, Ebola, Zombies)

Das Seminar (siehe Folien) findet in der ersten Semesterhälfte des WS 2026/27 statt. Weitere Informationen werden noch bekanntgegeben.

Bei Interesse bitte melden bei  surulescu@mathematik.uni-kl.de.

Inhalt

Single species models in continuous time:

• Population growth models: Malthus, habitat variability, Verhulst, contest and scramble, generalist predation.
• Elementary bifurcations and catastrophes (example: insect outbreak), harvesting.
• Spatial spread of a single population, traveling waves, and similarity solutions.

Multispecies models in continuous time:

• Lotka-Volterra models (predator-prey, competition, symbiosis).
• Enzyme kinetics, inner and outer solutions, asymptotic methods.
• Turing pattern formation, animal coat patterns.
• Invariant sets.
• Spatial spread of populations: reaction-diffusion equations, traveling waves, taxis, models for tumor growth and invasion.

Kinetic transport equations: multiscale modeling, upscaling, method of moments.

• Examples: Brain tumors and their spread in anisotropic tissues,  animal movement on path networks.

Kontaktzeit

4 SWS lectures
2 SWS exercises

Lectures:

Monday 10:15 - 11:45 (room 48-582)
Thursday, 12:15 - 13:45 (room 48-582)

Exercises:

Friday, 14:00 - 15:30 (room 48-582)

Material

OLAT

Kis

Inhalt

• Differentialgleichungen erster Ordnung: Autonome Differentialgleichungen erster Ordnung, Variation der Konstanten, explizit lösbare Fälle, Anfangswertprobleme,
• Existenz und Eindeutigkeit: Funktionalanalytische Grundlagen, Banachscher Fixpunktsatz, Satz von Picard-Lindelöf, Fortsetzbarkeit von Lösungen, Existenzsatz von Peano,
• Qualitatives Verhalten: Lemma von Gronwall, Stetige Abhängigleit von den Daten, Ober- und Unterfunktionen,
• Lineare Differentialgleichungen: Homogene lineare Systeme, Matrix-Exponentialfunktion, Variation der Konstanten, Differentialgleichungen n-ter Ordnung,
• Stabilität: Dynamische Systeme, Phasenraum, Hamiltonsche Systeme, Asymptotisches Verhalten, Stabilitätstheorie nach Lyapunov

Kontaktzeit

2 SWS lectures
1 SWS exercises

Lectures:

Friday 12:15 - 13:45 (room 48-208)

Exercises:

Wednesday, 12:15 - 13:45 (room 11-205)
Wednesday, 15:45 - 17:15 (room 46-268)
every 2 weeks

Material

OLAT

Kis

Inhalt

Die Vorlesung führt ein in die Integration von skalaren und vektoriellen Funktionen über Kurven und Flächen. Grundlegende Sätze, wie die Green'schen Formeln und die Sätze von Gauss und Stokes werden gezeigt. Inhalte der Vorlesung sind

• Parametrisierung von Kurven und (skalare und vektorielle) Kurvenintegrale
• Parametrisierung von Flächen und (skalare und vektorielle) Oberglächenintegrale
• Mannigfaltigkeiten und Tangentialräume, Differentiale differenzierbarer Abbildungen
• klassische Differentialoperatoren: div, grad, rot, Laplace
• Green'sche Formeln, Satz von Gauss, Satz von Stokes
• Anwendungen in der Physik

Kontaktzeit

2 SWS lectures

1 SWS exercises

Lectures:

Monday 12:15 - 13:45 (room 48-210)

Exercises:

Wednesday 12:15 - 13:45 (room 11-205)

Wednesday 15:45 - 17:15 (room 46-268)

Material

OLAT

Kis

Inhalt

Kontaktzeit

4 SWS lectures
2 SWS exercises

Lectures:

Tuesday 10:15 - 11:45 (room 48-538)
Thursday, 10:15 - 11:45 (room 48-538)

Exercises:

Friday 12:15 - 13:45 (room 48-582)

Material

OLAT

Kis

Inhalt

Kontaktzeit

2 SWS lectures
1 SWS exercises

Lectures:

Wednesday 12:15 - 13:45 (room 48-208)
 

Exercises:

Monday 10:15h - 11:45h (every 2 weeks.), 46-268
Thursday 10:15h - 11:45h (every 2 weeks), 48-582

Material

OLAT

Kis

Instructor: Prof. Dr. Christina Surulescu

Dates: To be announced

Inhalt

Single species models in continuous time:

• Population growth models: Malthus, habitat variability, Verhulst, contest and scramble, generalist predation.
• Elementary bifurcations and catastrophes (example: insect outbreak), harvesting.
• Spatial spread of a single population, traveling waves, and similarity solutions.

Multispecies models in continuous time:

• Lotka-Volterra models (predator-prey, competition, symbiosis).
• Enzyme kinetics, inner and outer solutions, asymptotic methods.
• Turing pattern formation, animal coat patterns.
• Invariant sets.
• Spatial spread of populations: reaction-diffusion equations, traveling waves, taxis, models for tumor growth and invasion.

Kinetic transport equations: multiscale modeling, upscaling, method of moments.

• Examples: Brain tumors and their spread in anisotropic tissues,  animal movement on path networks.

Kontaktzeit

4 SWS lectures
2 SWS exercises

Lectures:

Monday 10:15 - 11:45 (room 48-582)
Thursday, 12:15 - 13:45 (room 48-582)

Exercises:

Friday, 14:00 - 15:30 (room 48-582)

Material

OLAT

Kis

Inhalt

Kontaktzeit

4 SWS lectures
2 SWS exercises

Lectures:

Tuesday 12:15 - 13:45 (room 48-582)
Thursday, 08:15 - 09:45 (room 48-582)

Exercises:

to be announced

Material

OLAT

Kis