Math 5334 & 6333: PDEs Extra!!

Instructor: Thomas W. Carr

ODEs Extra!!

See ODEs Extra for "Extra!!" dealing with ordinary differential equations:

Gibb's Phenomena and Laplace's Equation

Here is a short paper that describes how to remove Gibb's phenomena from the separation of variables series solution of Laplace's equation. It is assume there are jump discontinuities only at the corners of a rectangular boundary. When the series solution is truncated there will be "Gibb's Phenomena" at the corners.

The method presented adds a function b(x,y) so that the the boundary condition at the corners of the new function v is always 0. Notice that b is "bi"-linear so that the Laplacian(b) = 0.

While there are some new terms the paper is not too difficult to read. Note that the solution method where u is solved for in four different parts appears in section 2.5 of our book.

"Can one hear the shape of a drum?" M. Kac, 1966

microwave cavities Mark Kac's question is a translation of the question, knowing the spectrum, i.e., the eigenvalues, of the Laplacian with Dirichlet boundary conditions, can we determine the shape of the boundary?

$\displaystyle \nabla^2 \phi(x,y) +\lambda \phi(x,y)$ = $\displaystyle 0 \mbox{ in } R,$  
$\displaystyle \phi$ = $\displaystyle 0 \mbox{ on }\partial R.$  


(The figures to the right are from Dr. Srinivas Sridhar at http://sagar.physics.neu.edu/isospec.html )
Given the wave equation with fixed boundaries we obtained the eigenvalue problem for the Laplacian as shown above. The eigenvalues determine the frequency of vibration of the membrane. Hearing the sound of the drum translates to knowing the eigenvalues. Does this uniquely determine the shape of of the drum? The answer is no. Two drums with different shapes can produce the same eigenvalues; they are said to be "isospectral."

In the figures above Dr. Sridhar shows the first three eigenfunctions for two different microwave cavities. They experimentally confirmed an earlier theoretical result by Gordon, Webb and Wolpert (1992) that the eigenvalues for each cavity (drum) were the same.

An easy "popular press" article on this subject appear in Science News. Dr. Sridhar's web page has more information on the microwave cavity experiments. Dr. Tony Driscal has recently improved techniques to compute the eigenvalues of isospectral drums.

Jaws! = Monster Waves!

Search online for photots and videos os "Jaws waves". Here is some cool physics . jaws wave

Solar-wind shock

In Feb 2000, NASA recorded data on a plasma shock wave emitted by the sun: shock.

Dispersion

Maple file