Date of Award

5-2016

Document Type

Thesis

Degree Name

Master of Science (MS)

College/School

College of Science and Mathematics

Department/Program

Mathematical Sciences

Thesis Sponsor/Dissertation Chair/Project Chair

A. David Trubatch

Committee Member

Arup Mukjerjee

Committee Member

Bogdan G. Nita

Abstract

Numerical analysis is a powerful resource in all mathematical sciences especially in the study of partial differential equations (PDEs). It allows evaluating and demonstrating derived solutions for PDEs and whenever the solution can’t be derived analytically it provides us with ability to calculate the solution function numerically and predict its behavior over time. This work presents a numerical method to evaluate an analytically known solution, demonstrates the needed parameters to achieve the desired accuracy, extends the methodology into the sphere of the mathematical unknown to be able to predict the results by using the same numerical methodology. The equation in question which we are going to analyze is a short pulse equation (SPE) which is an alternative model for the nonlinear Schrodinger equation. The SPE finds applications, for example, as a model for ultrashort pulses in optical fibers and has a form: uxt — u + | ( u3)xx. SPE is an integrable nonlinear partial differential equation. The soliton solutions of the NLSE have played an important role in the development of fiber-optic communications. But when the pulse becomes short, results produced by the NLSE worsen but the SPE generates good output. For this reason, it is very interesting to find the exact or numerical way to solve the SPE which represents the ultra-short light pulses.

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