Date of Award

2025

Degree Name

Mathematics

College

College of Science

Type of Degree

M.A.

Document Type

Thesis

First Advisor

Dr. Tom Cuchta

Second Advisor

Dr. Stephen Deterding

Third Advisor

Dr. Chanaka Kottegoda

Abstract

This thesis explores the theory and application of discrete fractional Gompertz models—systems that integrate fractional difference operators into the classical Gompertz growth paradigm. By doing so, these models capture both discrete time steps and the long-range memory effects characteristic of fractional calculus. After outlining the fundamental notions of discrete calculus, discrete fractional sums and differences, and related special functions such as the discrete Mittag–Leffler function, we derive various fractional Gompertz-type equations. We prove the existence and uniqueness of solutions to these fractional difference equations, often employing discrete analogues of standard solution methods like variation of constants. We also investigate the long-term behavior of solutions, providing conditions under which they converge, blow up, or stabilize. Numerical examples illustrate how the interplay between the fractional order, the growth rate, and other parameters can substantially alter the trajectory of these systems. The results highlight the potential of discrete fractional modeling to capture complex dynamics in fields where data collection or intervention occurs in discrete units of time, and where past states continue to influence the present.

Subject(s)

Discrete mathematics.

Calculus.

Mathematics.

Mathematical models.

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