TMA4195 Mathematical modelling 2005: Reading list

The important part of the reading list, or syllabus if you will, is the list of topics. This course is very example driven. There are some basic techniques that you are expected to learn, but equally important is a sort of attitude of the mathematical modeler that cannot possibly be taught: It has to be picked up by osmosis, as it were, from working with examples and seeing how experienced modelers approach the problem. I am hardly a very experienced modeler myself, but I have at least picked up some bits and pieces of the modelers' attitude. I hope I have conveyed some sense of that attitude, and that some of it has rubbed off.

But in preparing for an exam, techniques are important too. I list what we have been through, first by topic and then by reference.

Topics touched upon

• Dimensional analysis – Buckingham's pi theorem
• Nondimensionalisation and scaling
• Perturbation theory
  • Regular perturbations
  • Singular perturbations
  • Boundary layers and matching
• Conservation laws
  • Integral form and derivation from physical principles
  • Moving control volumes and the transport theorem
  • Shocks and the Rankine-Hugoniot condition
  • Characteristics
• Dynamical systems
  • Classification of equilibrium points
  • Bifurcations and bifurcation diagrams

Reading

First of all, there is The Book: Sam Howison: Practical Applied Mathematics – Modelling, Analysis, Approximation. Cambridge University Press, 2005. ISBN 0521603692.

Second, there are some of my notes. The list of notes is also conveniently collected in one place at the home page.

• Ch 1 – 2: These contain general background. I have not directly lectured on these, but you should at least look at them.
• Ch 3: Nondimensionalisation. I never touched on 3.1.3 or 3.2.1 (but we should have looked at 3.2.1: it's fun).
• Ch 7: PDEs. Bits of 7.1 and all of 7.3; just so we could do the next chapter.
  supplemented by a note on conservation laws and continuum mechanics (A5/A4)
• Ch 8: Traffic models. We didn't do 8.3.
• Ch 12: Asymptotic expansions
• Ch 13: Regular perturbation expansions
• Ch 16: Boundary layers. Not 16.5.3.
  supplemented by a note on enzyme kinetics (A5/A4)
• Ch 5 & 17: The thermistor
  supplemented by a note (A5/A4)
• Ch 18: Long thin domains – lubrication.
• Not in the book at all: But covered by two notes instead:
  Dynamical systems (A5/A4)
  Bifurcations (A5/A4)