__Course
plan. Mathematical modeling ____MAI530,
TMV090____spring 2007.__

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 MV:H11 and on Fridays in MV:H11 at 10.00.

On Wednesdays at 13:00 in MV:H11 we will have lectures and exercises during first two weeks and later partially exercises and parially discussions ower the projects.

One meeting per week with each working group for discussing the projects will take place at the Mathematical Center at possibly additional useful time.

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 ower theoretical part of the cource will take place at the end.

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

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

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**week 12 (I)**
(19 – 23 Mars)

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

Sal: MV:H11

Wednesday 13.00: Introduction. En example of a problem with modeling by ordinary differential equations, PDE and stochastic processes. Sal: MV:H11

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

Sal: MV:H11

**week 13 (II)**
(26-30 Mars)

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

Sal:MV:H11

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

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

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

Sal:MV:H11

**week
14 - Easter vacation; week 15 - examination week **

**week
16 (III)** (16Apr – 20 April)

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

Sal:MV:H11

Wednesday 13.00 Exercises, projects

Sal:MV:H11

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

Sal:MV:H11

**week 17 (IV)
**(23-7pril)

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

Sal:MV:H11

Wednesday 13.00:Exercises, projects

Sal:MV:H11

·
Friday 10:00:
Asymptotic
methods. Averaging method.Home
assignment N2

Sal:MV:H11

**week
18 (V)** (24 - 30 April)

·
Monday Holiday

Sal:MV:H11

Wednesday 13.00: Master equation for chemical kinetics by Monte Carlo method. The Gillespie method.Home assignment N3

Sal:MV:H11

· Friday 10:00: Monte Carlo methods for integral equations.

Sal:MV:H11

**week
19 (VI)** (7 – 11 May)

·
Monday - HOLIDAY

Sal:MV:H11

Wednesday 13.00: Exercises, projects.

Sal:MV:H11

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

Sal:MV:H11

**week
20 (VII)** (14 – 16 May)

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

Sal:MV:H11

Wednesday 13.00: Exercises, projects

Sal:MV:H11

· Friday Holiday

**week 21 (VIII)
**(15-19 may)

·
Monday 10:00: Projects presentations

Sal: MV:H11

Wednesday 13.00: Exercises, projects

Sal:MV:H11

·
Friday 10:00: Projects presentations
.

Sal: MV:H11

· Extra day: Consultation before examination.

Alexei Heintz <heintz@math.chalmers.se>