Spring 2006
Math 312 Final Project
The final project for Math 312 can be either an extension of a model
that we have discussed in class, or a problem of your choice that you
analyze using the tools discussed in the class.
The three main types of models that we have discussed are
- Differential equations.
- Discrete maps.
- Markov chains (actually a special case of linear discrete maps).
We have also used dimensional analysis to study the relations
among the parameters in a problem, and we used nondimensionalization
to reduce the number of parameters.
Here are some of the models that we have discussed in class.
- Population growth: the logistic equation.
- The Solow growth model of macroeconomics.
- The pendulum (mainly as an example of dimensional analysis).
- Spread of a disease (SIR and SIQR models).
Even a brief search of the web will find many
variations and extensions of these models.
- Romance, including the simple "Romeo and Juliet" model, and the
more complicated "Laura and Petrarch" model.
- Lanchester's model of combat. Many interesting modifications
of this model are easily developed.
- Competing Species.
- Predator-Prey models.
- Periodic drug doses.
- Analysis of a simple game using Markov chains.
- and more...
The possibilities for modifying or extending these models
are endless. For example:
- Consider a higher dimensional problem--perhaps an arms
race between three countries, or a love triangle.
- Consider using different (maybe more realistic) assumptions
about the behavior of the system.
- If in class we used differential equations to develop a model,
consider how the problem would be modeled using a discrete map instead.
How would the analysis change?
- Analyze a familiar game. For example, on average, how many turns
will there be in a game of "Chutes and Ladders"? What are the most
common spots on which to land?
You may also have seen problems in other classes that could be
analyzed with the tools we have discussed in Math 312.
These could be problems in economics, political science,
sociology, biology, etc. For example, a model from another
class in which the mathematical details were glossed over
because the instructor wanted to avoid differential equations
would make a great project for Math 312.
Requirements
The project must include:
- A clear explanation of the problem that is to be modeled.
- A list of modeling assumptions.
- The mathematical formulation of the model.
- An analysis of the model, which may be some combination
of analytical analysis and computer simulation (see me for help
with computer simulation).
- A conclusion, and a brief discussion of enhancements to the
model that would be interesting to pursue.
I am not looking for a fifty page thesis. Think of it as designing
your own challenging Math 312 homework problem, and then writing a
detailed solution to the problem.
Numerical Computations
Several of the models that we discussed in class were simulated
on the computer using programs that I wrote and made available
on the web. If you have a problem that requires numerical
computations (for example, a set of nonlinear differential equations
with three or more dimensions), talk to me about setting up some
tools for you to use.
