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.