**Research projects, Jonathan Goodman**

**Large time behavior of viscous planar shock profiles.**

Judith
Miller and I studied the large time behavior of perturbations of planar
shock fronts for scalar conservation laws with viscosity. We showed that
the large time behavior is given by an effective equation for the front
motion itself. There is a preprint in postscript
format.

**Anisotropic adaptive refinement in finite elements**

Klas Samelson, Anders
Szepessy, and I have been working on adaptive approximation and finite
elements with the goal of automatically creating very high aspect ratio
elements. There is a preprint in
postscript format.

### Large time behavior of linear and nonlinear waves in multidimensional
lattices

Peter Schultz and I (for Schultz' PhD thesis) studied the large time
behavior of solutions of spacially discretized versions of linear and nonlinear
wave equations.

**A nonlinear parabilic evolution equation.**

Matania Ben-Artzi (math dept, Hebrew U., Jerusalem), Arnon Levi, and
I studied a parabolic evolution equation with L1 initial data. We
proved uniqueness and regularization of solutions. There is a preprint
in postscript format.

**Some problems in finance
**

Dan Ostrov (math dept, Santa Clara University) and I studied the behavior
of the early exercise boundary in short time for American style options.
The method is to use an integral equation for the free boundary.
There is a
LaTeX file with the story. The figures for the paper are
figure 1,
figure 2, and
figure 3 .
You may also download a
C++ program used to calculate the free boundary.

Dan Ostrov (as above) and I the effect of small transaction costs on
the Merton problem of optimal dynamic asset allocation. We found a
simple derivation of the scalings that had been used and a duality
theory that makes it possible to treat other problems in a simple
intuitive way. The preprint
is in pdf format.

Back to Jonathan Goodman's home page.

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Department home page.