Date of Award

Spring 2021

Document Type

Open Access Thesis

Department

Mathematics

First Advisor

Hong Wang

Abstract

Materials exhibiting both elastic and viscous properties have been termed the name viscoelastic materials and have been modeled using a combination of integer order derivatives affixed in varying ways called viscoelastic models. This results in highly complicated numerical procedures necessitating highly expensive computational time which we will show. To that end the use of fractional derivatives were researched and determined to be the ideal solution for modeling these materials, of which this paper is focused on exploring. Such research began as a theoretical study, however over time the applied benefits were discovered and utilized and have since been expanded on, allowing the complicated numerical procedures mentioned above to be replaced with succinct numerical schemes. From these schemes we wished to focus on how to hone the numerical respect of material property contributions to the results, to which we focused on three different types of definitions for how to numerically model them, specifically the Caputo fractional derivative in this paper.

In this thesis we will first give an introduction of viscoelastic materials and the integer order models that were conceived, along with an exploration of how complicated these models can become. From there we will introduce fractional derivatives and the different definitions that exist, focusing on the Caputo fractional derivative. We will then explore the benefits of fractional viscoelastic models compared to integer order models, and from there we will expand, introduce, and compare three different definitions of variable order versions of such fractional models.

Included in

Mathematics Commons

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