__Course plan. Mathematical modeling
____MMG510, MVE160__

Teaching will include lectures, exercises, 2 assignments to be done at
home and giving two bonus points each, and a large project done in
working groups.

We will study during 8,5 weeks and have lectures and exercises on Mondays at 10.00 in MVF31 and on Thirsdays in MVF31 at 8.00.

On Tuesedays at 15:00 in MVF31 we will have lectures and exercises during first two weeks and later we will have discussions on the projects at this time.

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

The weeks 21 will be devoted to the presentation of projects 30 minutes for every group. A working group can consist of maximum 3 people.

A 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%).

The textbook for the course is: Arrowsmith D.K. , Place C.M.: Ordinary
Differential Equations.

A Qualitative Approach with
Applications. Chapmann and Hall. (1982).The reference to the
book below is A.P.

The order of chapters can be slightly shifted.

**week 12(I)** ()

Monday 15:15
Introduction to ODE. Phase portrate, trajectories. Particular examples.
A.P. Intorduction.

Exercises chapter 1: 3; 4c),f); 5;
19a),d); 25; 27iv); 29a)

Tuesday
15.15: Topics for projects.

En
example
of
a
problem
with
modeling
by
ordinary
differential
equations, PDE and stochastic processes. Notes to the
introduction lecture.

Modeling
of
different
types
of
transport
processes
by
PDE,
kinetic
models and stochastic processes.

Thirsday 8:00: Linear systems of ordinary
differential equations.
Classification of matrices.
A.P. Chapter 2.1-4.

Exercises chapter
2: 4; 6; 9b),c),d); 13b),c),e).

**week 13 (II)**
()

Monday 15.15:
Linear systems of ordinary differential equations. The
evolution operator. Affine systems.
A.P.Chapter
2.5,6,7.

Exercises chapter 2.
21 a),e); 3b);34c)

Tuesday 15.15: Exersises on
linear systems of ordinary
differential equations.
A.P.Chapter 2

Thirsday 8:00 : Stability of stationary points and solutions of non-linear
differential equations. Exercises on stability.
A.P.Chapter 3.1-4, 3.5-6.

Exercises chapter
3.1-3.3:1 b); 4
i),iii); 5 a),d),g);

Start working on Home assignment N1: 1) Find an example of non-linear ODE i
plane for each type of fixed points.

Calculate
eigenvalues
and separatrices in each case.

2)
Draw
phase protrait using Matlab (or by hand) for each of your
examples.

3) Try to find and
example of system with two different types of fixed points

OBS!!! Important to have a plan
for the project at the end of the second week.

**week
14
(III)**
()

Monday 15:15: Master equation for
chemical reactions. The Gillespie method. Paper by
Gillespie

Tuesday 15.15: Meetings on projects

**week
15
(IV)
**()

Start working on Home assignment N2 :

1) Choose a "chemical
system" with two types of particles such

that the corresponding ODE in plane
has a polynomial right hand side and a stable fixed point

2) Write a code in
Matlab that solves the ODE and draw some typical trajectories.

3) Model the same system by Gillespie method
and
compare random trajectories with those you got from ODE.

4) Scale up the system of
ODE ( 100 or 50 times ) intorducing fro example new X=100 x and new
Y=100 y.

One can see that coefficients for
linear terms will be the same. Coefficients at quadratic terms will
become 100 smaller and

coefficients at cubic terms will become
10000 smaller.

5) Solve the scaled up system with both methods again and
observe that random trajectories look very much like ODE
solutions.

Monday 15.15: Monte
Carlo methods for integral equations and particle systems. LectureMonteCarlo
.

Tuesday 15.15: Meetings on projects

Exercises chapter 3.5-7:14 a),b); 15;16.;17a),b); 20

Ready with Home assignment N1 at the end of the week 15

Holidays and examination weeks

**week 18 (V)**
()

Monday 15:15: Poincaré-Bendixsons theorem. Examples of periodic solutions A.P.Chapter 3.8,9

Exercises chapter 3.8,3.9: 23; 24 a),c);
27; 28

Tuesday 15:15: Meetings on projects

Thirsday 8:00:
Van Der Pol
oscillator. Lienard equation.
A.P. 4.4, 5.1

Ready with Home
assignment
N2

**week 19 (VI)
**()

Monday 15:15: Lyapunovs functions. Lyapunovs method. A.P. 5.4

Exercises chapter 5.4: 1 a),b); 3 a), b),c);7
a),b);8;
10.

Tuesday
15:15: Meetings on projects

Thirsday 8:00:. Exercises on
Lyapunovs method. A.P. 5.4

**week
20
(VII)**
(14 – 16 May)

Monday 15:15: Bifurcations. Hopf bifurcation. A.P. 5.5

Exercises chapter 5.5: 12 a),b),e),f); 13 a); 14 a); 15

Tuesday
15:15: Meetings on projects

Thirsday 8:00:
Dimensional
analysis.
Equations in non-dimensional form.

**week 21 (VIII) **()

Monday 15:15: Projects presentations .

Tuesday
15:15:
Projects presentations .

Thirsday
8:00:
Projects presentations .

**week 22 (IX) **()

Wednesday
1 june ; 8.30
-13.30 Examination

An additional examination on may 25-th
is ordered for Chalmers students travelling away before the ordinary
time