This page describes projects I will direct. Please feel free to contact me directly if you have any questions about my projects.
Prerequisites: I want my students to have taken differential equations and linear algebra. It is preferable if the student has taken the upper division physics course in mechanics. All of these problems will require some programming, using either Mathematica, MATLAB, Java or C++.
We also study the Ordinary Differential Equation (ODE) for u(x), where the Laplacian is just the second derivative:
1. Use the GNGA to find stationary (time independent) solutions to the Swift-Hohenberg equation. This involves solving the PDE (2)
2. Use the polynomial speed-up for the PDE (1) or (2). This is is done on the square region in the original paper by Neuberger and Swift. Since that original paper, we have focused on algorithms that work for general nonlinearities. Our GNGA algorithm runs slowly on the cube and it could run much faster using this technique. Solve the PDE on the square or the interval with polynomial-speedup, and
3. Correctly handle the symmetry in solving the PDE (1) or (2) on the disk or circle. Modify our existing C++ code, or write a MATLAB program. The continuous symmetry of the circle gives some interesting challenges.
The folloing ideas are probably not going to be persued this summer.
3. Do the GNGA with using a Quasi-Newton's method in place of Newton's method. This has the possibility of greatly speeding up the algorithm, especially for PDEs on 3-dimensional regions.
5. Solve the PDE (1) on the surface defined by x4 + y4 + z4 = 1. This surface has the symmetry of the cube, but the region is 2-dimensional.