All ETDs from UAB

Advisory Committee Chair

Marius Nkashama

Advisory Committee Members

Ryoichi Kawai

Ian Knowles

Document Type

Thesis

Date of Award

2019

Degree Name by School

Master of Science (MS) College of Arts and Sciences

Abstract

Fractional calculus has existed since 1695, when L'Hôpital asked Leibniz about his notation for a derivative, $\dfrac{d^nf\left(x\right)}{dx^n}.$ L’Hôpital asked what the result would be if $n$ were $\dfrac{1}{2}$. Leibniz's response: ``An apparent paradox, from which one day useful consequences will be drawn.'' While fractional calculus has only been intensely explored in recent decades, this prediction seems to have been true, with fractional derivatives often seeming contradictory and difficult to work with. However, our exploration has found that these definitions are far more compatible after looking at their non-local properties, in particular their ability to account for causality. We then looked into applications of fractional calculus models to some typical problems involving differential equations such as oscillation and diffusion. Exploring different cases of fractional oscillation indicated that fractional calculus can better describe the evolution of oscillatory motion and damping over time, while exploring anomalous diffusion indicated that fractional calculus is more accurate when modeling the physical process of subdiffusion. The non-local property of fractional derivatives appears to be the same property that allows for these models to capture this additional information, though there remains much research to be done both in the theory of fractional calculus and in its applications.

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