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.
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.
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