The Mathematica Journal
Volume 9, Issue 3

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T R O T T ' S C O R N E R
Michael Trott

Introduction

For fans and friends of partial differential equations (PDEs), Mathematica 5 is a great step forward. The solution capabilities for parabolic and hyperbolic equations have been greatly expanded compared with Version 4. Now it is possible to choose various derivative approximation schemes and solve problems. This means that now we can calculate, for example, Schrödinger equation solutions, Wigner functions, nonlinear reaction diffusion equations, and others in a variety of situations. Elliptic PDEs, on the other hand, cannot (yet) be solved using a direct call to NDSolve. But Version 5 has another excellent new toolset--sparse arrays and various linear algebra functions that deal with them. For simple linear elliptic PDEs, this is enough to program a solution "manually." In this Corner, I will discuss a vibrating drum with an unusual shape--one with many corners.

For physicists, the solutions of the Helmholtz eigenvalue problem, , have a magic attraction; seeing this equation makes their eyes start to glow and unavoidably thoughts about "Sturm-Liouville," "Weyl deficiency indices," and "spectral density" come to mind. This equation describes many physical phenomena ranging from vibrating drums to quantum mechanics. Depending on the dimension, the boundary shape and boundary conditions, and the form of , an astonishing number of interesting problems and phenomena arise. Typical investigations deal with the distribution of the eigenvalues, as well as quantitative and qualitative features of the eigenfunctions. One recent area of interest is nontrivial shapes. The eigenvalue problem for fractal Koch squares and Koch triangles was investigated in great detail in [1]. In this Corner, we will use a two-dimensional domain with another unusual boundary, with set to zero and simple homogeneous Dirichlet conditions. We will calculate some of the eigenvalues and the eigenfunctions of a highly irregular drum.



     
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