4 papers
Computation of Laplacian eigenvalues of two-dimensional shapes with dihedral symmetry
David Berghaus, Robert Stephen Jones, Hartmut Monien +1
We numerically compute the lowest Laplacian eigenvalues of several two-dimensional shapes with dihedral symmetry at arbitrary precision arithmetic. Our approach is based on the met…
The fundamental Laplacian eigenvalue of the ellipse with Dirichlet boundary conditions
Robert Stephen Jones
In this project, I examine the lowest Dirichlet eigenvalue of the Laplacian within the ellipse as a function of eccentricity. Two existing analytic expansions of the eigenvalue are…
The fundamental Laplacian eigenvalue of the regular polygon with Dirichlet boundary conditions
Robert Stephen Jones
The lowest eigenvalue of the Laplacian within the S-sided regular polygon with Dirichlet boundary conditions is the focus of this report. As suggested by others, this eigenvalue ma…
Computing ultra-precise eigenvalues of the Laplacian within polygons
Robert Jones
The main difficulty in solving the Helmholtz equation within polygons is due to non-analytic vertices. By using a method nearly identical to that used by Fox, Henrici, and Moler in…