Accurate evaluation of layer potentials up to the boundary
Numerical solution of so-called "partial differential equations" is
crucial to science and engineering, e.g. in diffusion of heat and
chemicals, electricity and magnetism, and fluid flow. Computer
algorithms for this get only approximately the right answer, but
should be as accurate as possible. The project is to try out a new
way to evaluate smooth solutions to Laplace's equation (describing
e.g. steady heat flow) which are expressed as a sum of "point charges"
on the boundary, called a layer potential. You will need to write
(not too long) computer programs to do this, probably in MATLAB, and
produce plots showing how the accuracy improves. You will learn some
numerical math, such as interpolation and quadrature. You will also
use existing computer software (MPSpack) to get the point charge
strengths. The programming experience you get will be very useful for
future classes and your career.
You'll need Math 11 or 12 or 13, some computer programming experience,
and preferably Math 22 (linear algebra).