Date of Award

Fall 12-12-2020

Degree Type

Masters Thesis

Degree Name

Master of Science (MS)

School

Mathematics and Natural Sciences

Committee Chair

Dr. James Lambers

Committee Chair School

Mathematics and Natural Sciences

Committee Member 2

Dr. Haiyan Tian

Committee Member 2 School

Mathematics and Natural Sciences

Committee Member 3

Dr. Huiqing Zhu

Committee Member 3 School

Mathematics and Natural Sciences

Committee Member 4

Dr. Bernd Schroeder

Committee Member 4 School

Mathematics and Natural Sciences

Abstract

We propose to create a new numerical method for a class of time-dependent PDEs (second-order, one space dimension, Dirichlet boundary conditions) that can be used to obtain more accurate and reliable solutions than traditional methods. Previously, it was shown that conventional time-stepping methods could be avoided for time-dependent mathematical models featuring a finite number of homogeneous materials, thus assuming general piecewise constant coefficients. This proposed method will avoid the modeling shortcuts that are traditionally taken, and it will generalize the piecewise constant case of energy diffusion and wave propagation to work for an infinite number of smaller pieces, or a smoothly varying coefficient. We hypothesize that by treating a smoothly varying function as a piecewise constant function with infinitely many pieces, this potential method can be realized. Through the Uncertainty Principle, we will formulate highly accurate estimates of eigenvalues, and then through the QR algorithm, we will use these estimates to formulate highly accurate eigenvalues and eigenfunctions. Ultimately, we will produce a more efficient solution method that avoids traditional time-stepping.

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