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

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