__Course
plan. Mathematical modeling ____MMG510__

__spring 2008.__

Teaching will include lectures, exercises, 3 assignments to be
done at home, and a larger project done in working groups.

We will study during 7 weeks and have lectures and exercises on Mondays at 10.00 in MVF21 and on Thirsdays in MVF21 at 10.00.

On Wednesdays at 13:00 in MVF21 we will have lectures and exercises during first two weeks and later partially exercises and parially discussions on the projects.

One meeting per week with each working group for discussing the projects must take place at the Mathematical Center.

The week eight (VIII) will be for presentation of projects 30 minutes for every group. A working group can consist of maximum 3 people.

A short 2 hours written examination over the theoretical part of the course will take place at the end.

Well done home assignments will be counted as a bonus for the corresponding part of the written examination.

Total points for the course will be an average of points for the project (60%) and for the examination (40%).

Some changes in the order of chapters can be done.-------------------------------------------------------------------------------------------------------------------------------------------

**week 14(I)**
()

Monday 10.00:
Introduction. En example of a problem with modeling by ordinary
differential equations, PDE and stochastic processes.

Wednesday
13.00: Introduction. En example of a problem with
modeling by ordinary differential equations, PDE and stochastic
processes.

Thirsday
10:00: Three
views on diffusion: stochastic, kinetic, PDE. Modeling of different
types of transport processes by PDE and kinetic models.

**week 15 (II)**
()

Monday 10.00: Ordinary
differential equations. Phase portrait. Linear systems of ordinary
differential equations.

Wednesday 13.00: Stability of stationary points and solutions of differential equations. Ljapunov functions. Exercises on stability.

Thirsday 10:00 : First integrals. Periodic solutions to ODE. Poincaré-Bendixson theorem. Home assignment N1

OBS!!!
Important to have a plan for the project.

**week
16 (III)** ()

Monday 10:00: First
integrals. Periodic solutions to ODE. Poincaré-Bendixson
theorem. Examples of periodic solutions.

Wednesday 13.00 Exercises, projects

Thirsday 10:00:Hopf
bifurcation. Parameter
dependence of models. Bifurcations. Bifurcation diagrams.

**week 17 (IV)
**()

Monday Dimensions analysis and scaling. Equations in dimensionless variables. Asymptotic methods. Secular terms. Method with two time scales.

Wednesday 13.00:Exercises, projects

Thirsday 10:00:
Asymptotic
methods. Averaging method.Home
assignment N2

**week
18 (V)** ()

Monday Master
equation
for chemical kinetics by Monte Carlo method. The Gillespie
method.Home
assignment N3

**week
19 (VI)** ()

Monday - Monte
Carlo methods for a
stochastic
model for particles. General and direct modeling.

Wednesday - other activities

Thirsday
10:00: Monte
Carlo methods for a
stochastic
model for particles. General and direct modeling.

**week
20 (VII)** (14 – 16 May)

Monday 10:00: Non-linear
waves in reaction - diffusion processes.

Wednesday
13.00: Exercises,
projects

Thirsday Non-linear waves in reaction - diffusion processes.

**week 21 (VIII)
**()

Monday 10:00: Projects presentations
.

Wednesday 13.00: Exercises, projects

Thirsday
10:00: Projects presentations
.

**week 22 (IX)
**()

Wednesday
Examination.

Extra day: Consultation before examination.

Alexei Heintz <heintz@math.chalmers.se>