Project for Math 615
Overview:
- The project is in two parts, the preliminary part due
Tuesday, April 2 and the final part due at noon on Thursday
May 9th.
- The preliminary part is worth 50 points and the final part 100 points.
But the whole project will be normalized to 40% of the final grade.
- You are required to talk to me in person about your project
before turning in the preliminary part. You will likely discover that
I want to simplify everything.
- The preliminary part should be between 5 and 10 pages of genuine content
.
- The final part should be longer but I will regard it as a bad thing
if there is more than 25 pages of text and numbers. That is, you might
have a number of figures but I think that good technical writing will allow
you to do the main part of the job in well under 25 pages.
What to include in the preliminary part:
- Start with a relatively detailed description of the problem you propose
to address. This will typically mean a description of an equation, first
of all by explaining the source of the equation and then by explaining the
particular form of the problem you are interested in. For instance,
in some cases the goal is to understand the effect of a parameter on the solutions.
In some cases you are doing a pure boundary problem, in other cases
an initial and boundary value problem. You don't have to do a PDE but
it should involve a continuum model. That is, to specify an exact solution
must require infinitely many numbers. Pure ODE problems are not allowed.
Inverse problems, delay differential equations, and differential algebraic
equations are interesting non-PDE classes and are encouraged; you may
discover that your scientific/engineering situation requires such mathematical
frameworks.
- Please try to do a really simple case of your problem in the preliminary
part.
- Then propose what numerical approach you intend to follow for the
remainder of the project. This is the most important thing in the preliminary
part! For instance, spend a page on describing existing literature.
Spend a page on a search for existing software. A first place
to look is in the Matlab documentation. For instance, linear
algebra routines of varius types, ODE routines, splines, Fourier transforms,
etc. can all be done well without leaving the world of Matlab. After
that you may want to look at Matlab toolboxes, but you are in charge
of getting access to these. Some of these toolboxes are too general
and too complete--I encourage you not to just run one of them on you problem
though that may be possible in some cases. For further existing codes,
see Netlib at
netlib.bell-labs.com/netlib
.
What to include in the final part:
- Everything else. I hope that your emphasis will be on understanding
and evaluating numerical quality. That is, some of your project must
be used to explain the mathematical model in which you are interested, but
mostly you want to compare numerical approaches and/or evaluate the quality
of the numerical approach(es). For instance, examine in
detail a simple case where you can get an analytical solution or analytical
approximation. More realistically, a comparison of the result of certain
calculations for different parameter values and different methods may make
up the bulk of your project.
- If it is at all possible, do a local truncation error calculation
and a stability analysis if they are appropriate.
- Include a bibliography that shows you have looked into existing ways
of solving your kind of problem and also shows where your problem comes from.
Ed Bueler’s office: Chapman 301C. Phone: 474-7693.
Email: ffelb@uaf.edu
. Web: http://www.math.uaf.edu/~bueler/
.