Date of Award

Spring 1-1-2011

Document Type

Thesis

Degree Name

Master of Science (MS)

Department

Applied Mathematics

First Advisor

Bengt Fornberg

Second Advisor

James Curry

Third Advisor

Natasha Flyer

Abstract

A method for solving boundary value problems for linear partial differential equations in convex polygons developed by A.S. Fokas in the late 1990s is introduced. In order to solve well-posed boundary value problems using the novel Fokas approach, certain global relations must be derived. These global relations yield so-called Dirichlet to Neumann maps which not only allow us to solve Helmholtz-type PDEs using the Fokas method, but they are also of interest in their own right. Given a convex polygon and a prescribed set of boundary conditions associated with a PDE, the Dirichlet to Neumann map enables us to numerically recover unknown boundary conditions with relatively high accuracy without solving the PDE on the interior. The numerical implementation of the Dirichlet to Neumann map is shown to be an efficient and accurate method for resolving unknown boundary conditions. The map is also analyzed and certain parameters are optimized. With an accurate Dirichlet to Neumann map, solving the modified Helmholtz and the Helmholtz equations via the Fokas method becomes possible.

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