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| V 11 (1) |
Literature |
Context |
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| Monday 15:15 | A.P.Introduction Borr. 444-458 |
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| Tuesday 15.15: | Hab. pp. 224-255 Borr. 444-458 Borr. 86-93 |
Examples
with modeling
by
ordinary differential
equations. Predator-Pray, and Competing species
models. Topics for projects. Notes to the second lecture |
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| Thirsday 8:00: | Hab. pp.257-322 |
Problems
with
modeling by PDE. Wave phenomena. Topics for projects. Notes to the third lecture |
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Important to have a plan
for the project at the end of the second week. |
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| Monday 15:15 | C.C.B.) Chapter 7 |
Problems
with
modeling by PDE. Diffusion and wave phenomena. Topics for projects. Material to the fourth lecture. |
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Notes
to the fifth lecture. Luo's paper |
Examples
of
problems
with
modeling
by
mesoscopic
models. Lattice Boltzmann equation. Topics for projects. Notes to the fifth lecture. |
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| Thirsday 8:00: |
Paper
by
Gillespie |
Examples
of problems
with
modeling by stochastic processes. The Gillespie method. Topics for projects. |
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| V 13 (3) | ||||||
| Monday 15:15 | A.P. Chapter 2.1-4. | Linear systems of ordinary
differential equations.
Classification of matrices. Exercises chapter 2: 4; 6; 9b),c),d); 13b),c),e). |
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| Tuesday 15.15: | Meetings on projects | |||||
| Thirsday 8:00: | A.P.Chapter 2.5,6,7. | Linear systems of ordinary differential
equations. The
evolution operator. Affine systems. Exercises chapter 2. 21 a),e); 3b);34c) Start working on Home assignment N1 (do the assignmen with your project group and leave after vacations) |
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| V 14 -15 |
Examination week and Easter
vacations week |
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| V 16 (4) | ||||||
| Monday 15:15 | A.P.Chapter 3.1-4, 3.5-6. | Stability of stationary points and solutions
of
non-linear
differential equations. Global phase
portraits. Exercises chapter 3.1-3.3:1 b); 4 i),iii); 5 a),d),g); |
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| Tuesday 15.15: |
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| Thirsday 8:00: | A.P. Chapter 5.4 Borrelli |
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| 17 (5) | ||||||
| Monday 15:15 |
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Lyapunovs
functions. Lyapunovs method Exercises on Lyapunovs method. Exercises A.P. chapter 5.4: 1 a),b); 3 a), b),c);7 a),b);8; 10. Borrelli. Ch. 8: Ex. 3-10; Ex. 17-19 |
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| Tuesday 15.15: |
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Periodic solutions to ODE. Van der Pol equation. Poincaré-Bendixsons theorems Examples of periodic solutions Exercises chapter 3.5-7:14 a),b); 15;16.;17a),b); 20 Start working on Home assignment N2. (Two weeks) |
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| V 18 (6) | ||||||
| Thirsday 8:00: |
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Poincaré-Bendixsons theorems Examples of periodic solutions A.P., Exercises chapter 3.8,3.9: 23; 24 a),c); 27; 28 Borrelli, Chapter 9.1-2, Exercises 6-16 |
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| V 19 (7) | ||||||
| Monday 15:15 | Borrelli Chapter. 2.9, Chapter. 9.3 A.P. Chapter 5.5 |
Bifurcations,
Hopf
bifurcation. Borelli Chapter 2.9; Borelli Chapter 9.3, Exercises 1-9, 20, 21 A.P. Chapter 5.5, Exercises: 12 a),b),e),f); 13 a); 14 a); 15 |
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| Tuesday 15.15: |
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| Thirsday 8:00: | Notes on non-
dimensional variables |
Non-dimensional
variables
in
differential
equations.
Ready with Home assignment 2 |
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| V 20 (8) | ||||||
| Monday 15:15 | Notes
on
travelling
waves Haberman |
Travelling
waves
in
reaction
diffusion
equations Models with hyperbolic equations. Characteristics. shock waves |
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| Tuesday 15.15: | Repetition
before
examination Project presentations. ( 1 or 2) |
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| V 21 (9) | ||||||
| Monday 15:15 | Projects
presenations |
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| Tuesday 15.15: | Projects presenations | |||||
| Thirsday 8:00: | Projects presenations | |||||
| V 22 |
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| Wednesday 8:30-12:39 |
Examination |
1) Consider an
non-linear ODE in the plane: x'=V(x) with
V(1)=
-(x(1)-100)+ 5*(x(2)-200) + 0.01*(x(1)-100)*(x(2)-200);
V(2)=
-(x(1)-100)- (x(2)-200) - 0.02*(x(1)-100)*(x(2)-200);
or another nonlinear ODE with a stable fixed point (100,200).
2) Write a
code in
Matlab that solves the ODE and draw some typical trajectories for
initial data with large (2000-3000) coordinates.
3) Interpret your ODE system as a "chemical
system" with two types of particles.
4) Model
the
system
of
particles
by
Gillespie
method.
5) Compare
ODE
trajectories
and
random
trajectories
from
Gillespies
method.
| Topic: |
References: |
| Development of an epidemy by Populations Mass Action Law (malaria, AIDS, influensa, etc). | Borr. |
| Modeling ecosystems by Populations Mass Action Law (fishes, plankton, birds, people, parties, etc.). | Borr. |
| Modeling genetics of
evolution by ODE |
HofbSigm) FHopp) Murray v. I |
| Traffic flow depending on individual reactions of drivers on the road. | |
| Inverted pendelum with oscillating base. |
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| Autocatalization
and bifurcations |
Borr. 545 |
| Vine fermentation process. | Velt. |
| Oscillating processes in cells. | C.C.B. |
| Dynamics of a satellite in the system Earth-Moon or a planet in in a double star system. |
| Topic: |
References |
| Development of an epidemy by Gillespies method (malaria, AIDS, influensa, etc). | Gillespie paper |
| Modeling ecosystems by Gillespies method (fishes, plankton, birds, people). | Gillespie paper |
| Stochastic processes in cell membranes. | C.C.B. |
| Transport of neutrons through protection of a nuclear reactor. | Sobol, chapter 7 |
| Topic: |
References: |
| Waves and jems traffic flow by hyperbolic equations. | Hab. Chapter 3 |
| Neuron pulse propagation. | Sch. p. 167,
p.181, C.C.B. |
| Process of freezing and melting a lake. | Matt. p.592 |
| Reaction-diffusion process in pellets |
Matt. p.555 |
| Modeling spacial effects in genetics of evolution by PDE | Murray v. I and v. II |
| Bifurcation of a
viscous flow in a channel by the Lattice Boltzmann model. |
Luo's
paper 1 and paper 2 |
| Viscous flow in a channel with obstacles by
the Lattice Boltzmann model. |
Luo's
paper 1 |
| Patterns
formation by
reaction-diffusion equations. |
Murray J.D. v. II |
| Patterns formation in activation - inhibition reactions in diffusive media. | Murray J.D. v. II |
| Modeling stripe patterns of
angelfishs Pomakanthus semicirculatus
by reaction-diffusion
equation Modeling chemotaxis by reaction-diffusion equations. Cancer modelling and simulation. |
Science 261,
(1993) pp.189-192 Nature 376, (1995), pp. 765-768 Murray J.D. v II. |
| Spreading of waves in a channel by the shallow water equation. |
A written examination over the theoretical part of the course will take place at the end.
Well done home assignments will give two bonus points each for the written examination.
Total points for the course will be an average of points for the project (60%) and for the written examination (40%).
