Introduction to mathematical modelling

http://www.maths.ox.ac.uk/~fowler/courses/o11intro.html

Aims and objectives

The aim of the course is to develop and extend some of the theoretical ideas developed in the mainstream mathematics course and to show how they can be applied in simple modelling situations: the methods and models will be integrated so that techniques are illustrated or motivated by applications. Many of the mathematical techniques have been covered in papers a1 and b5 (both of which are prerequisites) but the emphasis in this course is on the interplay between the mathematics and the applications.

Prerequisites and connections

a1 and b5 are prerequisites; the course provides a platform for b6, b10 Mathematical Ecology and Biology, and for the fourth year courses Mathematics and the Environment, Mathematical Physiology, and Perturbation Methods.

Synopsis

Modelling, conservation and constitutive laws, nondimensionalization.

Basic ideas of graphical methods and perturbation methods.

Oscillations, nullclines, stability, bifurcations, hysteresis, waves, shocks, similarity solutions.

Nonlinear ordinary differential equations, first order partial differential equations, nonlinear diffusion, reaction-diffusion equations.

Applications may include traffic flow, combustion, spruce budworm infestations, ice sheet flow, snow melting.

Course materials

Problem sheets consist (more or less) of some of the exercises in the notes, and are also available in postscript or pdf format.

Specimen finals questions are available in postscript or pdf format. They are actually longer and harder than fhs questions.

Draft lecture notes are available and can be downloaded in postscript or pdf format.
Note: Make sure your version of the notes is the current one (see below).

Version numbers and date of issue

1.0, 9/10/02.
1.1, 14/10/02.
2.0, 14/10/03.

Reading

The principal source is the printed lecture notes.

Useful modelling books are:

N.D. Fowkes and J.J. Mahony 1994 An introduction to mathematical modelling. John Wiley, New York.
R. Haberman 1998 Mathematical models. SIAM, Philadelphia.
A.C. Fowler 1997 Mathematical Models in the Applied Sciences. CUP, Cambridge.
A.B. Tayler 1986 Mathematical Models in Applied Mechanics. OUP, Oxford.
Mathematical techniques are in:
J. Ockendon, S. Howison, A. Lacey and A. Movchan 1999 Applied Partial Differential Equations. OUP, Oxford.
E.J. Hinch 1991 Perturbation Methods. CUP, Cambridge.
D.W. Jordan and P. Smith 1999 Nonlinear Ordinary Differential Equations, 3rd ed. OUP, Oxford.

Lecture synopsis

Conservation and constitutive laws.
Non-dimensionalisation.
Approximations.
Graphical methods.
Stability.
Oscillations.
Hysteresis.
Resonance.
Waves and shocks.
Burgers' equation.
Fisher equation.
Snow melting.
Similarity solutions.
The viscous droplet.
Blow-up.
Reaction-diffusion.

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