Here are four topics for the projects; I hope
to have two people working on each. Please tell me
if there are any preferences; otherwise, they will
be assigned randomly


1)=====================================================

The inverse problem for a wave equation.
Find a source of the wave by solving the wave equation
back in time. (PDE + finite differences).
This is a replication of the ongoing research
in the are of thermoacoustic tomography.

References:

Burgholzer P., Matt G., Haltmeier M. & Patlauf G. (2007)
Exact and approximate imaging methods for photoacoustic
tomography using an arbitrary detection surface.
Phys. Rev. E 75, 046706


Burgholzer P., Bauer-Marschallinger J., Grun H., Haltmeier M.
and G Paltauf (2007)
Temporal back-projection algorithms for
photoacoustic tomography with integrating line
detectors.
Inverse Problems 23, S65–S80

Hristova Yu., Kuchment P. and Nguyen L. (2008)
Reconstruction and time reversal in thermoacoustic
tomography in acoustically homogeneous and
inhomogeneous media.
Inverse Problems 24, 055006

2)=====================================================

A sensitivity analysis and a boundary value
problem (BVP) for a toy car
(to be solved by a shooting method)

Your algorithm has to find such values of parameters
for the promlem that the car starts at a given point A
and arrives into a given point B exactly at the given
time T.

References:

"Computer Methods for Ordinary differential equations
and Differential-Algebraic Equations"
Uri M Asher and Linda R Petzold

Section 4.6, example 4.7.

Section 6.4,

Chapter 7, problem 7.3





3)=====================================================

A shooting game "Helicopter war" (also a BVP).
(Obviosly, a shooting method)

Two helicopters whose coordinates are given take turns
trying to shoot each other by non-guided projectiles
with time fuses. The first helicopter to explode
a shell in the delta-neigborhood of the opponent
wins.

Model a realistic ballistic equation (wind, friction);
develop optimal winning strategies (i.e, fast converging
algorithms). May need the sensitivity analysis.


References:

"Computer Methods for Ordinary differential equations
and Differential-Algebraic Equations"
Uri M Asher and Linda R Petzold

Section 4.6, example 4.7.

Section 6.4,

Chapter 7, problem 7.3



4)======================================================

For a drum (or a membrane) of a given shape (a rectangle
or an L-shaped domain, for starters), find several lowest
frequencies (and modes) of vibration. Optionally, program
Matlab to generate the corresponding sound. (Use finite
differences to reduce the PDE into a system of linear
equations, and find the eigenvalues/eigenvectors).


References:

Some textbooks, TBA.