Ordinary Differential Equations and Applied Dynamics Commons^™

Mathematical Modeling Of The Combined Effects Of Thermal Burn And Local Irradiation, Quintessa Hay, Rachel Jennings, Amy Creel, Kyle Gaffney, Christina Wagner, Kidist Maxwell, Ginu Unnikrishnan, Tyler Dant

Biology and Medicine Through Mathematics Conference

No abstract provided.

Mitigating Parameter Identifiability Issues Through Model Calibration On The Data-Informed Active Subspace: An Example In Tumor Growth, Allison L. Lewis, Rebecca A. Everett

Biology and Medicine Through Mathematics Conference

No abstract provided.

Incorporating Thermal Performance Curves Into Population Dynamic Models For The West Nile Vector, Culex Pipiens, Benjamin Bruncati, Helle Aronson, Chloé Lahondère, Michael A. Robert

Biology and Medicine Through Mathematics Conference

No abstract provided.

A Two-Phase Perspective On A Related-Rates Paradox, Charlie Vazquez Acosta

Honors Capstones

Related-rates problems are a standard topic in first-year calculus and have appeared in textbooks for over 150 years. These problems are used to teach implicit differentiation and the relationship between changing quantities. Common examples include the falling ladder, the fishing bobber, the melting snowball, and the leaking conical tank. In each of these problems, a quantity is changing at a constant rate, and students are asked to find the rate of change of another related quantity. While the computations themselves are usually straightforward, the standard models lead to unrealistic results near the end of the motion. For example, the falling …

Phenological Overlap In Obligate Plant-Pollinator Mutualism, Austin J. Carlson

2026 Spring Honors Capstones Projects

Plant-pollinator mutualisms require temporal overlap between flowering and pollinator activity, so climate-driven timing shifts can weaken the interaction and, in severe cases, destabilize the system. This work investigates how reduced overlap affects persistence in an obligate plant-pollinator pair using a coupled differential equation model in which a phenological overlap factor scales the saturating mutualistic benefit. Simplification with a constant overlap enables closed-form equilibrium and stability analysis, revealing that below a critical overlap threshold, coexistence is no longer maintained. Rescaling reduces the parameter space from ten quantities to seven dimensionless groups, and sensitivity analysis identifies the degree of species dependence and …

Stochastic Universal Differential Equations For Epidemiological Modeling: Uncertainty Quantification In Disease Transmission Dynamics, Alice Menaya Armah-Bonney

Electronic Theses and Dissertations

Epidemic forecasting requires not only predictions of expected case counts, but also quantification of uncertainty, although existing surrogate modeling frameworks for agent-based models remain fundamentally deterministic. In this thesis a Stochastic Universal Differential Equation framework is presented that extends the deterministic Universal Differential Equation approach by incorporating a learnable stochastic diffusion term, enabling calibrated probabilistic forecasts while preserving the mechanistic interpretability and computational efficiency of the deterministic baseline. In doing so, a two-phase training algorithm is introduced to ensure stable convergence and the framework is validated against the ensemble output from ExaEpi, an exascale agent-based model of a COVID-19 outbreak …

Safe Control Design For Quadruped Locomotion In Unstructured Environments Using Linear Transfer Operators, Sriram Sundar Krishnamoorthy Shankara Narayanan

All Dissertations

Deploying quadruped robots in unstructured, obstacle-rich environments requires control and planning methods that remain safe and reliable despite complex terrain geometry, limited sensing, and inevitable modeling errors. This thesis develops operator-theoretic tools for safe control design of robotic systems using linear transfer operators, with a focus on quadruped locomotion in unstructured environments. The central goal is to develop a unified operator-theoretic framework for safe control design based on the Perron–Frobenius (P–F) and Koopman operators. In particular, the thesis leverages \emph{density functions} to develop safe navigation frameworks in the dual space of densities. In the operator-theoretic perspective, the P–F operator governs …

Modeling Bitcoin Dynamics Using Differential Equations, Boone M. Fleenor

Departmental Honors & Graduate Capstone Projects

In this thesis, we develop and analyze two nonlinear systems of ordinary differential equations to model Bitcoin price dynamics. Analytical techniques are used to obtain exact or approximate solutions where possible. Then, numerical simulations using a fourth-order Runge–Kutta method are employed to explore system behavior beyond analytically tractable regimes. Finally, model outputs are compared to historical Bitcoin price data using normalized and resampled time series. These results suggest that deterministic models can provide meaningful insight into the structural behavior of Bitcoin markets, while highlighting the need for stochastic or time-dependent extensions for more realistic modeling.

Employing Effective Solution Methods For Caputo-Based Sequential Fractional Models, Eman A. A. Ziada, Mohamed F. Abouelenein, Hijaz Ahmad, Monica Botros

Mathematical Modelling and Numerical Simulation with Applications

This paper investigates a class of nonlinear sequential singular fractional differential equations (FDEs) involving Caputo derivatives. This type of equation has several key advantages that enhance its value, such as capturing memory and hereditary effects. Viscoelastic materials and anomalous diffusion, as well as biological systems, can take advantage of this feature. In addition, fractional derivatives possess a sequential structure that enables the implementation of multiscale processes and hierarchical memory responses. Moreover, it provides an effective and flexible framework for solving differential equations compared to classical differential equations. It can therefore be used to model complex systems in physics, biology, and …

Numerical Simulations And Hyers-Ulam Stability Of A Novel Nonlocal Anthropogenic Cutaneous Leishmaniasis Mathematical Model, Khalid Fanoukh Al Oweidi, Zakirullah -, Kamal Shah, Thabet Abdeljawad

Mathematical Modelling and Numerical Simulation with Applications

In this work, the fractal-fractional Atangana-Baleanu derivative with the Mittag-Leffler kernel is employed to capture the memory and hereditary effects inherent to anthropogenic cutaneous leishmaniasis transmission dynamics. The Banach fixed-point theorem and contraction mapping principle are used to prove the existence and uniqueness of solutions, while Hyers-Ulam stability of the system is analyzed to demonstrate the robustness of solutions with respect to small perturbations. Using a nonlinear least-squares approach, model parameters and fractional order are estimated using epidemiological data from the World Health Organization. The basic reproduction number $R_0 = 0.53$ indicates that the disease is under control after adding …

Exploring The Dynamics Of Romantic Relationships Through The Lens Of Prem Rog, Umang Jain, Dheeraj Sharma, Pranay Goswami, Kuldeep Malik

Journal of Humanistic Mathematics

Romantic relationships are dynamic events that begin, grow, and frequently remain for a long time in a stagnant or fluctuating state until possibly dissipating. Although they are unquestionably the most significant dynamic events in our lives, dynamic systems theory has only recently included them in its formal framework. Without a mathematical model, it would be impossible to analyze and comprehend the dynamics because, in general, love stories are too brief to allow things to stabilize and are affected by the ups and downs of the surrounding community. In this paper, we set up models made up of four ordinary differential …

Modeling French Language Preservation: An Optimal Control Problem, Charlotte Blasi

Scripps Senior Theses

The French language is one of the most globally spoken languages, with over 300 million speakers and designated the official language of 29 nations. French continues to prosper as a result of its titular nations' imperialist history that emphasized linguistic diffusion as well as the support of international organizations dedicated to promoting both the language and the culture of French-speaking nations. In this thesis, we explore the nuanced history of French-speaking countries and one institution dedicated  to promoting the French language, the OIF. We further examine the media mechanism of the OIF and construct a system of Ordinary Differential Equations …

A Stability Analysis Of The Phase-Lock Equations, Brian M. Sunguza

UNF Graduate Theses and Dissertations

Ginzburg and Landau have provided a set of equations that relate superconductivity to magnetic fields. Through a transformation process, Zhan has derived what are now called the phase-lock equations. A stability analysis of the spatially-independent phase-lock equations is the purpose of this presentation. This simplification is significant since it allowed for the analytical determination of equilibria, their stability, and the influence of a periodic forcing function. Through the use of an original code, numerical simulations are shown to corroborate the analytical results described above.

This analysis includes novel Lyapunov functions that allowed for the analytical determination of the instability region. …

Mathematical Model Of Graphene, Douglas M. Sanor

Williams Honors College, Honors Research Projects

Graphene, a single-atom-thick layer of carbon arranged in a hexagonal lattice, exhibits exceptional mechanical, electrical, and thermal properties that make it a promising material for a wide range of engineering applications. This paper presents a mathematical framework for modeling the mechanical behavior of graphene, with a focus on atomistic-to-continuum approaches. We begin with a onedimensional Frenkel-Kontorova model that represents graphene as a discrete chain of particles interacting with both their nearest neighbors through harmonic spring potentials and an underlying substrate through van der Waals forces. Numerical simulations of this discrete model demonstrate the commensurate-toincommensurate phase transition, revealing how geometric mismatch …

Managing Multi-Drug Resistance: An Evolutionary Game Theory And Optimal Control Approach, Shukhrat Nasrulloev

Theses and Dissertations

Multi-drug resistance is an evolutionary process in which treatment eliminates sensitive cells, allowing resistant clones to dominate. This thesis investigates this process using a framework integrating population dynamics, evolutionary game theory, and optimal control theory. We develop a two-population logistic growth model describing competition between drug-sensitive and drug-resistant cells under treatment, construct dose-dependent payoff matrices and replicator dynamics to characterize evolutionary competition, and derive a critical drug level D_crit = (r_S - rR)/(dS - d_R) at which resistant cells gain a fitness advantage. An optimal control problem is formulated via Pontryagin's Maximum Principle to identify schedules …

Type Ii Diabetes Treatment Comparison Via Compartment Modeling, Abigail M. Collins

Theses and Dissertations

Type II diabetes mellitus affects one in ten adults worldwide, yet the effects of treatment type and adherence level on developing complications and quality of life have not been well characterized at the population level, and mathematical modeling offers a structured way to examine these dynamics. This thesis adapts the Boutayeb et al. (2004) model to incorporate dynamic treatment types and levels of adherence, producing nine scenarios in which complication development rate and complication recovery rate differed, to compare peak complications and quality of life across treatment and adherence conditions. Using a system of ordinary differential equations and compartment modeling, …

A Novel Mathematical Model Of Hiv Transmission Incorporating The Effects Of Treatment And Pre-Exposure Prophylaxis: Sensitivity Analysis And Numerical Simulations, Erick Manuel Delgado Moya

Mathematical Modelling and Numerical Simulation with Applications

Human immunodeficiency virus (HIV) continues to be a public health problem in many countries of the world, and Pre-exposure prophylaxis (PrEP) is a preventive method for HIV, which has shown great efficacy and is in use worldwide. This work presents a new mathematical model for HIV transmission incorporating PrEP use and evaluates the impact of PrEP along with its increasing use in a population. The construction of the model takes into account three forms of diagnosis: diagnosis of individuals in risky sexual contact, diagnosis after risky contact (diagnosis in the undiagnosed infected compartment), and diagnosis associated with attempting to enter …

Construction And Data-Driven Analysis Of A Stochastic, Individual-Based Opioid Epidemiology Network Model, Leigh Bennett Pearcy, Owen Queen, Vincent Jodoin, Suzanne Lenhart, Christopher Strickland

Mathematical Modelling and Numerical Simulation with Applications

While substance use epidemiology has been an active area of mathematical research in recent years, the social and mental processes that are involved in the development of substance use disorders have presented challenges to advancing the epidemiological theory and how they differ from the contraction of pathogenic disease. Such distinction is especially pertinent in the context of the current United States opioid epidemic and its intersection with the recent COVID-19 pandemic, as both prescription drugs and social influence play major roles in the development of opioid use disorder. In this paper, we construct a stochastic network model capturing how individual …

Assessing The Electrical Efficiency Of The Thompson Coil, Christopher Wengert, Dominick Dingus, Madouna Barsoum, Jacqueline Ceballos, Michael Ent, Jackson Stephens

Student Scholar Symposium

The Thompson Coil is a well-known system used in undergraduate electromagnetism courses to demonstrate electromagnetic induction. Traditionally, the system employs a “jumping ring” to visualize induced currents. This project seeks to repurpose this phenomenon for practical energy transfer by replacing the jumping ring with a wire coil, allowing the measurement of the power delivered to an electric load. Using Faraday’s Law of Induction, 𝜖 = −𝑑Φ𝐵 /𝑑𝑡 , where Φ𝐵 represents the magnetic flux, the induced electromotive force (EMF) in the receiving coil is determined. Finally, the system’s power transfer efficiency is evaluated using the formula 𝜂 =𝑃𝑜𝑢𝑡/ 𝑃𝑖𝑛 ×100% …

Complex Numbers And Phone Signals, Ava Barrett, Rebecca Brazeal, Javier Gutierrez

Student Scholar Symposium

Though widely used, complex numbers are often introduced in coursework without real-world context. Our team examines their application in electromagnetic waves and communication systems, focusing on how they help model and analyze phone signals. By connecting concepts from physics and math, we demonstrate their role in signal modulation and transmission. Through this project, we aim to highlight the practical importance of complex numbers in modern technology and provide a clearer understanding of their use in communication systems that power everyday devices like smartphones.