Publication Date

12-2012

Type of Culminating Activity

Thesis

Degree Title

Master of Science in Mathematics

Department

Mathematics

Supervisory Committee Chair

Barbara Zubik-Kowal, Ph.D.

Abstract

If used cautiously, numerical methods can be powerful tools to produce solutions to partial differential equations with or without known analytic solutions. The resulting numerical solutions may, with luck, produce stable and accurate solutions to the problem in question, or may produce solutions with no resemblance to the problem in question at all. More such numerical computations give no hope of solving this troublesome feature and one needs to resort to investing time in a theoretical approach. This thesis is devoted not solely to computations, but also to a theoretical analysis of the numerical methods used to generate computationally the approximate solutions. After deriving theoretical results for a wide class of problems, I use them to validate that my numerical computations produce reliable solutions. The fundamentals of this work are based on mathematical analysis with which the application of analysis to PDEs in a numerical and computational framework was possible.

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