Classes

An class for graduate students (e.g., master program in CES, SiSc, Math, Physics, etc) on mathematical models based on partial differential equations (PDEs). The course is recommended for master students. See also the RWTHonline entry of the class

Intention

The class presents a cohesive mathematical derivation and discussion of different partial differential equations as models for technical and physical processes. Basic framework will be the balance laws of mass, momentum and energy, as well as the Maxwell equations. Different constitutive material laws will yield different models.

We will consider among others: solid mechanics, fluid and gas dynamics, chemical reactions, magnetohydrodynamics. As application examples we study models for rubber, earthquakes, flames, shock waves, electric arcs, etc.

The aim is to view the connections of relevant PDEs in applied mathematics and master the process of modeling from the physical concept, the mathematical equation up to a concrete result.

Info

The formal structure of the class will be 3 hours of lectures and 1 hour of tutorial each week. The tutorials will be merged into 2 hours every other week, while the lectures will take turns with 2x 2 hours and 1x 2 hours per week.

The times for lectures and tutorials are given below.

• Lectures : Every Friday from 10:30 to 12:00 and every Tuesday from 14:30 to 16:00 1090|328. (First Lecture is Friday 12.04)
• Tutorials : First tutorial is Tuesday 16.04. and no class on 09.04

There are 6 ECTS points to earn with an oral exam of 30min. Currently the class is linked to several RWTH modules with different names - sorry... These are the module for CES and SiSc programs, and one only for the mathematics programs, and this here which includes the physics program.

See also online.rwth-aachen.de. Please register with the course to gain access to the Moodle-Page.

Content

1. Mathematical basics
2. Kinematics
3. Field equations
4. Solid state mechanics
5. Thermodynamics
6. Fluid mechanics
7. Kinetic gas theory
8. Electrodynamics
9. Magnetohydrodynamics

Some Background Literature

The course will be a combination of several topics and there is no single book which it can be based on. The list below is not meant to cover the course, most of these books actually go far beyond the course material. But the books certainly resonate with what is done in the course.

• R. Temam, A. Miranville, Mathematical Modeling in Continuum Mechanics, Cambridge University Press, 2000
• R. Greve, Kontinuumsmechanik, Springer, Berlin, 2003
• H. Schade & K. Neemann, Tensoranalysis, DeGruyter, Berlin, 2009
• I. Müller & P. Strehlow, Rubber and Rubber Balloons, Springer 2004
• C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, 3rd ed., Springer 2010
• S. R. De Groot & P. Mazur, Irreversible Themrodynamics, Dover, 1962
• A. Meister, Asymptotic single and multiple scale expansions in the low Mach number limit, SIAM J. Appl. Math 60/1, (1999)
• S. Chapman & T. G. Cowling, The Mathematical Theory of Non-uniform Gases, 3rd ed., Cambridge University Press, 1970
• F. Cap, Lehrbuch der Plasmaphysik und Magnetohydrodynamik, Springer, 1994

A very rough and non-authorized (possibly incomplete) collection of notes is given by the former CES students Simon Märtens et al. (Mathematical Modelling with PDEs (pdf))