A second difference matrix for the Laplacian in Elliptical Coordinates

Horath, Jimie (2018) A second difference matrix for the Laplacian in Elliptical Coordinates. Masters thesis, Northern Arizona University.

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Abstract

We use finite differences and a matrix approach to approximate the Laplacian in Elliptical coordinates which is known to be ∆Φ = 1 (∂2Φ + ∂2Φ). c2(sinh2 μ + sin2 ν) ∂μ ∂ν Using Matlab, we code our construction of the Laplacian in Elliptical coordinates and use it to compute eigenvalues for the Elliptical Laplacian with an error of order O(h2). We compare our eigenvalues against published eigenvalues on the ellipse with zero-Dirichlet boundary conditions, and eigenvalues on the disk with zero-Neumann boundary conditions. We also compute eigenvalues with an error of order O(h2) for the ellipse with zero-Neumann boundary conditions for several cases, and we use our Elliptical Laplacian to solve the heat and wave equations as well as a semi-linear application.

Item Type:	Thesis (Masters)
Publisher’s Statement:	© Copyright is held by the author. Digital access to this material is made possible by the Cline Library, Northern Arizona University. Further transmission, reproduction or presentation of protected items is prohibited except with permission of the author.
Keywords:	Dirichlet; Eigenvalues; Ellipse; Elliptic Coordinates; Laplacian; Neumann
Subjects:	Q Science > QA Mathematics
NAU Depositing Author Academic Status:	Student
Department/Unit:	Graduate College > Theses and Dissertations College of the Environment, Forestry, and Natural Sciences > Mathematics and Statistics
Date Deposited:	05 Jul 2022 21:24
Last Modified:	05 Jul 2022 21:24
URI:	https://openknowledge.nau.edu/id/eprint/5282

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