Outer Approximation of the Laplacians of Random Carpets and Julia Sets

This page displays the work done in summer 2007 at Cornell University on the Outer Approximation of the Laplacians of Random Carpets and Julia Sets using the Matlab r2006a PDE Toolbox. Our ultimate goal is to understand Laplacians on fractal domains. We hope to gain insights (i.e. conjectures) from these computational experiments on random carpets and Julia sets. We surround the set in question with a larger set, considered as a subset of R^2, and compute the eigenvalue problem Δ(u)=λu on the approximating domain. We look for convergence and interesting characterstics of the list of eigenvalues, also known as the spectrum.

• Download Julia Set Files (tar.gz | zip) (program, source code, readme, and companion scripts for Matlab)
• Download Random Carpet Files (tar.gz | zip) (program, source code, readme, and companion scripts for Matlab)
• Detailed explanation of the algorithm and approaches to the problem
• Related resources including computation of Julia set approximating domains, and quantum graphs (an area where the outer approximating method has been proven to work)
• Data from selected trials

Additions, Corrections or Feedback on this page to:

smh82@cornell.edu

Additional acknowledgements:

Mandi Fix, Steve Gaarder, Erin Pearse, Doug Rizzolo, Luke Rogers, ,Bob Strichartz, and Russ Thompson

Last Update: 31 July 2008