In a few words, mathematical modeling is the
process of taking a problem of practical interest, casting it in a
mathematical
form and arriving at a meaningful or useful solution. This is a vast
enterprise
which really describes all of applied mathematics. This course will
consider
problems whose mathematical formulation consists of differential or
difference
equations. The emphasis will be on analytical (pencil and paper) methods
for solving and analyzing differential equations; however, numerical
(computer)
methods will be mentioned and used for demonstrations. Given the
developments
of the past twenty years, it is impossible to study differential
equations
at this level without an encounter with dynamical systems and chaos.
During
this semester we will explore both the modeling process as it applies to
problems in engineering, biology, ecology, physics and physiology as
well
as the exciting new developments in nonlinear dynamics.
The final grade in the course will be
determined
by regular (nearly weekly) problem sets (70%) and a project (30%).
Graduate
students will have extra probelms on every problem set.
It is not too soon to begin thinking about a topic for a project. Your project must deal with a specific example of modeling that involves ODEs and/or PDEs. The model may be analytical or numerical in nature. Otherwise, there are no other conditions. You should begin talking with me as soon as you get ideas so we can refine and focus your topic. A detailed outline with an annotated bibliography is due no later than October 15. The project is due no later than the last day of classes. The actual paper should be on the order of 15-20 pages for undergraduates and 25-30 pages for graduate students. It should be carefully written and well-organized, "typed" preferably in LaTeX, and should contain figures, tables, numerical output as needed. The paper must include references (five minimum) and show evidence of library research. The paper itself needn't contain original work. Once again, start your thinking and planning today!
You have until the tenth week of classes to
drop the course with only the instructor's (but not a Dean's) signature.
The incomplete policy of the Department and College is strictly
enforced:
incompletes are given only in situations in which a student who has been
in good standing all semester, is prevented from completing a course
assignment (for
example, the final exam) by circumstances beyond his/her control (for
example,
hospitalization, death in the family).
The course will follow the book fairly closely since it contains so many relevant and exciting topics. However there will be a few instances in which we will stray from the book to stress the modeling aspects of the course. Here is a list of anticipated topics.
0. Read Chapter 1 of the text.
1. ODE review - Models that involve first order ODEs
2. Stirred tank problems - Reality models
3. Population models
4. One-dimensional flows - Chapter 2
5. Bifurcations - Chapter 3
6. Pharmacokinetics and compartment models - Notes
7. Flows on the circle; fireflies - Chapter 4
8. Linear systems in 2-d - Chapter 5
9. Phase plane analysis and geometry - Chapter 6
10. A quick look at limit cycles - Chapter 7
This is an ambitious list of topics; at the same time it is a regrettably brief survey of modeling and dynamical systems. As you read, look for omitted topics in the book for possible project topics.
There is a wealth of reading to be done on
nonlinear
dynamics and ODE/modeling at all levels. Please visit the library and do
some exploring. You should definitely have one or more ODE books nearby.
Here are a few relevant sources (in no particular order).