Project No.
|
Difficulty
|
Due Date
|
Description
|
| 1 |
2 |
10/20 |
Quaslinear 1st-order PDEs, traffic flow, compression waves, shock waves and rarefaction.
|
| 2 |
1 |
09/29 |
Vibrations of a string attached at x = 0 extending infinitely far in one direction.
|
| 3 |
0.5 |
09/29 |
A falling cable. This problem should be done using the Laplace transform. Along with that, it is largely a review of how to solve 2nd-order nonhomogeneous ODEs with constant coefficients.
|
| 4 |
1 |
10/13 |
The double (or two-dimensional) Fourier transform.
|
| 5 |
1 |
10/13 |
The Airy function and the linearized Korteweg-deVries equation.
|
| 6 |
1.5-2.0 |
10/20 |
Project 2.1 from pp. 82-84 in the text. This gets at numerical solutions of the 1D Poisson problem with homogeneous Dirichlet BCs in the case where the function f is only known from measurements. Whether you receive 1.5 or 2 points will be determined by how far you take part (g).
|
| 7 |
1 |
10/30 |
Solve Schroedinger's equation in 1 spatial dimension.
|
| 8 |
>=3 |
11/13 |
Solve heat and vibrational problems on various domains via the Fourier method. These problems will give a flavor of how the eigenfunctions can change with changes in BCs and/or shape of the domain.
|
| 9 |
>=1.5 |
12/1 |
Image processing is a relatively young field of research. In the last 10 years or so, many books and papers have appeared discussing the use of diffusion as an image processing tool. One use for it is in the denoising of images, which may have become fuzzy during transmission between locations. This project entails implementing such an algorithm for denoising. For more information, take a look, for instance, at this webpage, or do your own search using kewords such as “image denoising” and “diffusion”.
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| 10 |
>=1.5 |
11/17 |
Investigate the families of orthogonal functions known as Legendre and Tchebyshev polynomials.
|