The MATLAB function inpolygon determines the points of a rectangular grid that are in a specified region. I want to investigate simple finite difference methods for this problem. The two regions are specified by the xy-coordinates of their vertices. I will look at the simplest known isospectral pair. It is possible that for convex regions the answer to Kac's question is "yes". This affects the accuracy and the rate of convergence of finite difference methods. These corners lead to singularities in most of the eigenfunctions - the gradients are unbounded. Toby developed methods, not involving finite differences, for computing the eigenvalues very accurately. A reprint of his 1997 paper in SIAM Review is also available from his Web site. A summary of Toby's work is available at his Web site. I was introduced to isospectral drums by Toby Driscoll, a professor at the University of Delaware. I want to show that the finite difference operators on these regions have the same sets of eigenvalues, so they are also isospectral. I am interested in finite difference methods for membrane eigenvalue problems. Wikipedia has a good background article on Kac's problem. The regions are known as "isospectral drums". In fact, they produced several different pairs of such regions. They demonstrated a pair of regions that had different shapes but exactly the same infinite set of eigenvalues. In 1991, Gordon, Webb and Wolpert showed that the answer to Kac's question is "no". If one specifies all of the eigenvalues, does that determine the region? Kac asked about the opposite implication. ![]() ![]() The MathWorks logo comes from this partial differential equation, on an L-shaped domain, , but this article is not about our logo.Ī region determines its eigenvalues. The values of $\lambda$ that allow nonzero solutions, the eigenvalues, are the squares of the frequencies of vibration, and the corresponding functions $u(x,y)$, the eigenfunctions, are the modes of vibration. Requiring $u(x,y) = 0$ on the boundary of a region in the plane corresponds to holding the membrane fixed on that boundary. The vibrations are modeled by the partial differential equation Kac's article is not actually about a drum, which is three-dimensional, but rather about the two-dimensional drum head, which is more like a tambourine or membrane.
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