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Math 436

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Fall, 2000

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{\bf PROJECT, Due November  20}
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Your  project should be a well-written explanation of the topic 
and should convince me that you have a good understanding of it.
It should not be something copied from a book.
Include references and a list of any computer programs you used.  
Here are some possible topics.  If you want to do a topic not 
on this list, please discuss it with me ahead of time.

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1.  Learn something about numerical methods for solving pde.  Implement
a finite difference method on computer.  If you know how to use it, 
use the numerical solution to graph solutions, using a powerful 
graphics package (e.g., the one contained in MatLab.)  Discuss the 
accuracy of the numerical solutions.


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2.  Show how a computer can be used to calculate the first mode
of a \lq\lq drumhead" for some non-circular shapes of drums.  
Make a Maple, Mathematica or Matlab movie of the motion of the drum (with that mode).

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3.  Learn about the maximum principle and uniqueness results for the
heat equation.

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4.  Learn about weak solutions of the wave equation (1 space
variable) and learn more about how characteristics affect solutions.

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5.  Learn about Duhamel's principle.

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6.  Learn how to solve first order quasi-linear pde.

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7.  Learn the theory of Sturm-Liouville problems, including a proof
of completeness of the eigenfunctions under appropriate hypotheses.
Learn about the asymptotic behavior of the  $n\/$th eigenvalue and
eigenfunction as  $n\to \infty\/$.

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8.  Learn about the problem \lq\lq Can one hear the shape of a drum?"



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