Date of Award

Spring 2020

Degree Type

Masters Thesis

Degree Name

Master of Science (MS)

School

Mathematics and Natural Sciences

Committee Chair

Dr. Haiyan Tian

Committee Chair School

Mathematics and Natural Sciences

Committee Member 2

Dr. James Lambers

Committee Member 2 School

Mathematics and Natural Sciences

Committee Member 3

Dr. Huiqing Zhu

Committee Member 3 School

Mathematics and Natural Sciences

Abstract

In this thesis, we solve nonlinear differential equations by the homotopy analysis method (HAM), which is a semi-analytic method first introduced by Shijun Liao in 1992. The modified HAM can be viewed as a more generalized method that encloses many perturbation and non-perturbation methods. It is different from perturbation or other analytical methods in that it allows considerable freedomformanyvariables. Using the modified HAM, especially zero-order and higher-order deformation equations, we solve a nonlinear initial value problem and a nonlinear eigenvalue problem. We adjust the convergence region of a solution by modifying auxiliary parameter values. The results converge in very few iterations under the proper choice of the values of auxiliary parameters. The approach is shown to be accurate and valid for differential equations with strong nonlinearity.

Keywords: Nonlinear initial value problem, Nonlineareigenvalueproblem, Homotopy analysis method, Duffing's equation, Perturbation theory.

ORCID ID

https://orcid.org/0000-0002-5616-1415

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