MA302 Final Project

Due Thursday, 12/2, or 11/24 if you prefer not to have anything due in the final week

You may work in a team up to three persons. All the team members receive the same score. Each team should  submit only one report. If you don't have a partner, e-mail me at zhilin@math.ncsu.edu and I will try to match you with somebody.

The project is about the Oregonator from Section 14.7.  Additional reference is available from me upon request. Alternatively, you can choose a problem from your own favorite field . The problem of your choice has to have two or more equations and two or more parameters. You have to follow the following Project Format.

Preamble:  (10 points)
 
  • Title of your project.
  • Names and student ID of your team member(s).
  • Abstract
    • Note: Abstract is a brief summary of your project.  It contains the problem that you intend to solve, the method(s) you used in the project and main results and conclusions of your analysis or numerical results. It should be about ten sentences long.

    Introduction (15 points)

  • Describe the problem , its importance, and history if possible.
  • Define and explain all of your variables, parameters and their units, and notations.
    • Theoretical Discussions (10 points)
  • Non-dimensionalize the problem if possible.
  • Find and/or verify any steady state solutions (equilibriums).
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    Numerical Computation and Analysis (55 points)
     
  • Write a Matlab code to define the system of differential equations.

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  • Write a Matlab drive code to define the parameters, input the initial condition, solve the problem using Matlab ode23s (recommended), or ode15s, or ode45, and plot the solutions versus time and get the phase plot. You need to mark the initial point on the phase plot and the direction of the trajectories.

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  • Try to re-produce Figure 14.4  and do the experiments as suggested in the text book, page 362, if you choose the problem in Section 14.7.

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  • Briefly describe the numerical method that you used, for example, ode23s. How is the method related to Runge Kutta methods? It is an explicit  or implicit method? How accurate is the method? Does it use fixed or variable time steps? What is the data structure of its output? When do we want to use ode45 or ode23s?

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  • What kind of steady state solution (equilibrium) does the system have? Is it stable or unstable limit cycle? (The answer may depends on the choice of parameters)

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  • Explain the meanings of your solutions and the relation with the parameters, for example, if k5=0, or w=0, then y tends to zero and there are no oscillations.

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  • Present results from selected , clearly labeled plots (title, xlabel, ylabel, zlabel).  I expect eight plots from you. Hint: There are four paragraphs in the text book discuss about the numerical experiments, you should produce two plots from each paragraph, the solution plots and the phase plots.

    • Hint:  Type help semilogy;  help subplot; and help plot3 in Matlab to see their usages.

                y1=y(:,1); y2=y(:,2); y3=y(:,3);
                subplot(311); semilogy(t,y1); xlabel(' ....
                subplot(312); semilogy(t,y2); xlable('....
                subplot(313); semilogy(t,y3); xlable(' ...
                figure(2); plot3(y1,y2,y3); title('...
     

    Appendix  (10 points)
     
  • Lists of the Matlab  codes you used, the function and the driver.
  • References  (text book or any other related materials you have used).

    • Extra Credits (10 points):

  • (5 points) Use some advanced options of ode23s. Hint: type help odeset in Matlab.

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  • (5 points) Find the cputime and the number of the floating point operations of your Matlab code. Hint: type help cputime and help flops in Matlab.