## Abstract

This dissertation examines the power series solutions---and their analytic continuation via gauge transformations---of three classical nonlinear ordinary differential equations arising in fluid mechanics that are mathematically related by their large-distance asymptotic behaviors in semi-infinite domains. The first problem examines the influence of surface tension and gravitational forces to form a static air--liquid interface that intersects a flat wall at a given contact angle and limits to a flat pool away from the wall. The second problem considers the related configuration of an axisymmetric air--liquid interface formed when the flat wall is replaced with a right circular cylinder. In both problems, we show that although power series solutions for the interface shapes are readily obtainable, the series diverge due to the influence of convergence-limiting singularities. In both cases, these singularities are mapped to new locations through transformations that are motivated by long-distance asymptotic behaviors of the solution. Convergent power series solutions to both problems are thus obtained. The third problem considers the Sakiadis boundary layer induced by a rapidly moving wall in a semi-infinite fluid domain, for which no exact analytic solution has been put forward in prior literature. For this problem, we follow the approach of the previous problems, and derive a convergent power series solution for both Newtonian and Ostwald-de Waele power law non-Newtonian fluids. Overall, this dissertation demonstrates that asymptotic behaviors, in conjunction with gauge function choices, can lead to exact convergent power series solutions to problems in mathematical physics.

## Library of Congress Subject Headings

Gauge invariance; Power series; Differential equations, Nonlinear--Asymptotic theory

## Publication Date

2-27-2024

## Document Type

Dissertation

## Student Type

Graduate

## Degree Name

Mathematical Modeling (Ph.D)

## Department, Program, or Center

Mathematics and Statistics, School of

## College

College of Science

## Advisor

Steven J. Weinstein

## Advisor/Committee Member

Mohamed A. Samaha

## Advisor/Committee Member

Nathaniel Barlow

## Recommended Citation

Naghshineh, Nastaran, "On the use of gauge functions to obtain convergent power series solutions to nonlinear ODEs" (2024). Thesis. Rochester Institute of Technology. Accessed from

https://repository.rit.edu/theses/11718

## Campus

RIT – Main Campus

## Plan Codes

MATHML-PHD