Ordinary Differential Equations and Applied Dynamics Commons^™

Differential Equation Modeling For Sustainable Resource Management: A Steady-State Optimal Harvesting Approach, Iordanka N. Panayotova, Aleksei Talonov

CODEE Journal

Mathematical models based on differential equations provide a powerful framework for connecting real-world data to informed decision-making. In this work, we present a student-accessible project that uses an optimal-control framework to study the sustainable management of biological resources.

Motivated by fisheries management, we examine a predator--prey system in which harvesting decisions must balance ecological and economic considerations. The model is formulated as an optimal control problem that seeks to maximize the total discounted net revenue from harvesting. Rather than solving for the complete time-dependent harvesting trajectory, we restrict the analysis to positive controlled coexistence equilibria and characterize an interior stationary …

The General Solution Analysis Of Homogeneous Linear Equations, Jacob Schwamb, Edward Whipple

Discovery Day - Daytona Beach

The general solution analysis of homogeneous linear equations are any systems of equations in which all constant terms are equal to zero is classified as a homogeneous linear equation. Some key characteristics of homogeneous linear equations are that there are “zero” solutions, where every system has at least a single solution where all variables are zero, also all solutions to any homogenous linear equation is linearly independent, along with having preserved homogeneity, where if any variable (x) may be added to the system, then any scalar multiple of the variable is also a solution. The General solution of any homogeneous …

Bridging Discrete And Continuous Systems: Fibonacci Sequences And Exponential Growth From Odes, Martyna Wojcik

Discovery Day - Daytona Beach

Bridging Discrete and Continuous Systems: Fibonacci Sequences and Exponential Growth from Ordinary Differential Equations       It has been observed that nature often exhibits specific patterns of growth and structure in biological systems and spiral formations. The Fibonacci sequence, defined as a discrete recursive sequence where each term is generated as the sum of the two preceding terms, “has been applied extensively to understand some natural phenomena” (Pakdemirli, 2023). In contrast, exponential growth describes a continuous process in which the rate of change of a quantity is proportional to its current value. Such behavior is modeled using differential equations that “produce solutions …

A Modeling Scenario For Cooling A Hot Vehicle In Florida, Jared Bunn, Bernadette Mullins, Elizabeth Hale, Jaeyoun Oh

CODEE Journal

This paper presents a group project assigned in a Calculus 2 course that has students work to develop, analyze, and draw conclusions about a modeling scenario for cooling a hot car. Using a modeling-first approach, instructors supported the students in class throughout the beginning of the project, enabling the groups to complete the remainder of the project on their own. Students used parameter estimation to tune their models to provided data: one for windows being up, and one for windows being down. This project provides an example of how modeling can be introduced early in a calculus course, rather than …

Piracy, Terrorism, And The Law: Differential Equations In Hostage Situations, Gabriel Hallevy

Journal of Humanistic Mathematics

Pirates have taken the crew of an American ship hostage. They promise to release the hostages only if another pirate who is held in an American prison for commission of piracy crimes against American citizens, is released. Should the U.S. government enter into negotiations with them? Should they send armed forces and risk the hostages? Should they release the prisoner immediately and unconditionally? The article models and analyzes possible policies regarding sensitive situations involving hostages and other related risks using differential equations. The solutions are surprisingly simple, but not necessarily intuitive. Our analysis aims to demonstrate how powerful mathematics is …

A Differential Equation–Based Epidemiological Model Of Post-Operative Chronic Pain In Scoliosis Patients With Data-Driven Analysis, Paige Zhu, Padmanabhan Seshaiyer

CODEE Journal

Chronic post-surgical pain (CPSP) is a common and often overlooked complication following surgical correction of idiopathic scoliosis, impacting long-term patient wellbeing despite improvements in surgical outcomes. This project introduces a novel epidemiological framework to model the progression of CPSP using a compartmental structure. By applying a coupled system of nonlinear differential equations, we simulate pain trajectories over time and assess the effectiveness of surgical interventions. The model is implemented for a single-cohort population and extended to a two-cohort design to compare outcomes between Posterior Spinal Fusion (PSIF) and Vertebral Body Tethering (VBT) procedures. Further stratification by patient age enables us …

(R2147) Effect Of Mass Variation With Log-Logistic Distribution In Perturbed Interacting Cr3bp, Abdullah Abdullah, Majhar Ali, S. K. Sahdev

Applications and Applied Mathematics: An International Journal (AAM)

This paper investigates the motion of the infinitesimal body in the perturbed restricted three-body problem where the primary is heterogeneous in shape and secondary is with modified Newtonian potential. With the use of log-logistic distribution, space-time transformation and the above-said perturbations, we determine the equations of motion and quasi-Jacobian integral. Further, we numerically perform the locations of equilibrium points, their stability, regions of motion, periodic orbits and Poincaré surfaces of section.

(R2159) A Novel Extension Of Picard’S Method For Fractional Initial Value Problems With Convergence Analysis On Finite And Infinite Intervals, Jag Mohan, Anju Sood

Applications and Applied Mathematics: An International Journal (AAM)

In recent years, fractional differential equations have emerged as powerful tools for modeling phenomena with memory and hereditary effects, owing to their non-local characteristics. These equations excel in tackling intricate problems across physics, engineering, and other fields. As analytical solutions are often infeasible, numerical methods play a vital role in their practical application. In this study, we have generalized Picard’s method to address fractional differential initial value problems with Caputo derivative, establishing an existence and uniqueness theorem applicable to both finite and infinite intervals. To substantiate our findings, we provide an example with graphical evidence demonstrating the convergence of the …

From Cork To Coasting: A Multi-Stage Ode Model Of Water-Rocket Flight, Viktoria Savatorova, Patryk Kustra, Ethan Dyer, Connor Carlson, Aleksei Talonov

CODEE Journal

Water rockets provide an affordable and engaging context for exploring applications of differential equations. Motivated by outreach activities conducted with undergraduate students, we develop a four-stage mathematical model of vertical water-rocket flight that is suitable for use in an ODE or mathematical modeling course. The model includes the cork-release phase, water-thrust propulsion, air-thrust propulsion with compressible and potentially choked flow, and the final ballistic stage with quadratic drag. While retaining key physical features, the model can be formulated as a system of ordinary differential equations that can be integrated numerically using tools familiar to students. We compare model predictions with …

A Professional Development Course On Data-Driven Dynamical Systems At A Primarily Undergraduate Institution: Part A - Scientific Content, Alessandro M. Selvitella, Jeffrey R. Anderson

CODEE Journal

In the age of data-driven decision making, ordinary differential equations (ODEs) remain a powerful and interpretable framework for modeling dynamic processes, especially when integrated with modern tools from statistical learning and data-driven dynamical systems. Yet, general undergraduate and graduate curricula do not typically address key opportunities in data-driven dynamical systems.

This first paper in a series focuses on the mathematical and methodological core of a professional development course first developed in the academic year 2025-2026 at a Primarily Undergraduate Institution, Purdue University Fort Wayne. The curriculum developed in this course emphasized how regression, regularization, and sparse identification can be used …

Machine Learning For Modeling In An Elementary Differential Equations Class, Nathan Albin, Andrew G. Bennett, Abhinav Chand

CODEE Journal

Mixing machine learning with modeling is an area of increasing importance. This paper presents a lesson where students model a spring-mass system both using traditional analysis with linear damping and using machine learning to learn the damping from real data. The machine learning is implemented in a Jupyter notebook hosted on Google Colab, allowing students to train the neural network without requiring the students to carry out coding. Students get experience with how machine learning can fail, how it can work, and the time and data requirements for machine learning to succeed, and are asked to apply this knowledge to …

Dynamic Homeostasis In Relaxation And Bursting Oscillations, Christopher J. Ryzowicz

Biology and Medicine Through Mathematics Conference

No abstract provided.

Dynamical Systems Modeling To Determine The Role Of Crosstalk In Shaping Stat Signaling Profiles, Laura F. Strube, Anamarie Martinez, Neha Cheemalavagu, Karsen Shoger, James Faeder, Rachel Gottschalk

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 …

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 …

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 …

Stability Analysis Of Thermohaline Convection With A Time-Varying Shear Flow Using The Lyapunov Method, Kalin Kochnev

Honors Scholar Theses

This work applies the Lyapunov method to identify instabilities and compute the growth rate of a linear time-varying system. The linear system studied describes cold fresh water on top of hot salty water with a periodically time-varying background shear flow. A time-dependent weighting matrix is employed to construct a Lyapunov function candidate. The resulting linear matrix inequalities are discretized in time using the forward Euler method. As the number of temporal discretization points increases, the growth rate predicted by the Lyapunov method or Floquet theory, used for comparison, will converge to the same value obtained from numerical simulations. Furthermore, the …

Math Meets Climate: The Energy Balance Model, Maria I. Sanchez Muniz

Open Educational Resources

This assignment introduces students to the mathematics of Earth’s climate through the classical energy balance model. Students analyze how incoming solar radiation, outgoing thermal radiation, and temperature-dependent albedo interact to determine Earth’s equilibrium temperature. Using analytical calculations and computational tools, students identify equilibrium states, assess their stability, and interpret the results through the lens of dynamical systems and bifurcation theory. The activity builds conceptual understanding of climate feedbacks, greenhouse effects, and tipping behavior using a transparent, one-variable model. Designed for applied mathematics and interdisciplinary STEM courses, this assignment emphasizes computation, physical interpretation, and real-world relevance. It is released as a …

Understanding Enso Through Mathematical Models, Maria I. Sanchez Muniz

Open Educational Resources

This assignment introduces students to conceptual models of the El Niño–Southern Oscillation (ENSO) and guides them through a structured investigation of their physical and mathematical foundations. Students analyze the recharge–oscillator and delayed–oscillator frameworks, explore how differential equations capture ocean–atmosphere interactions, and evaluate parameter-driven changes in oscillatory behavior. A key component of the work is the guided use of generative AI as a research tool: students employ AI models to locate peer-reviewed literature, interrogate model extensions, and refine their understanding of complex mechanisms, while synthesizing all final explanations in their own words. By blending classical climate modeling with modern AI-supported inquiry, …

Modeling Synaptic Dysfunction As Neural Contagion: A Graph-Based Sedr Framework For Simulating Signal Spread, Michelle Marfo, Dr. Padmanabhan Seshaiyer, Alonso Ogueda-Oliva

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Modeling The Cancer Cell Growth Predictions Based On Classical Mathematical Models With Physics-Informed Neural Network, Widodo Samyono

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Expanding Ode Examples: Introducing Gene Regulation Dynamics Through Hill Functions, Maila Hallare, Jane M. Santamore

CODEE Journal

Gene regulation is a fundamental biological process that controls gene expression. It can explain phenomena such as cell differentiation, circadian rhythms, disease progression, and metabolic control, among many others. Despite their importance in mathematical biology, gene regulation models are rarely featured in traditional ODE textbooks, which focus mainly on examples from engineering, physics, chemistry, and population biology. This article introduces gene regulation dynamics as a valuable addition to ODE curricula, presenting the models from a mathematical perspective and building on properties of the Hill function. These models deepen the understanding of biology-inspired ODE applications, provide accessible research opportunities for students, …

Analytical And Numerical Approaches To Parameter Estimation In Damped Oscillatory Systems, Gracie Crooks, F. Ayça Çetinkaya

CODEE Journal

We investigate the inverse problem of identifying damping and stiffness parameters in one-dimensional damped oscillatory systems governed by second-order differential equations. Focusing on mass–spring–damper models, we analyze the qualitative behavior of solutions across underdamped, critically damped, and overdamped regimes, and derive explicit conditions for parameter recovery based on time-domain observations such as equilibrium crossings and turnaround points. Two numerical estimation methods are developed and compared: a finite-difference least-squares approach based on central difference approximations, and a finite element formulation derived from a variational framework using piecewise linear basis functions. Computational experiments using synthetic data assess the accuracy, stability, and noise …

Project-Based Learning With Odes: Modeling Straw Rocket Motion With Air Resistance, Viktoria Savatorova, Ethan Dyer, Aleksei Talonov

CODEE Journal

This paper presents a hands-on project that guides students through building and validating a mathematical model of projectile motion. The project starts with the idealized case of motion under gravity without air resistance and then introduces air drag : first as a linear force, and then as a nonlinear quadratic force, with the Reynolds number providing the justification for the quadratic model. Students perform experiments with vertical and angled launches, capturing and analyzing motion data using video analysis software. Vertical launch data allows parameter estimation via least squares fitting of the nonlinear drag model, yielding values for initial velocity and …

Derivation Of Adjoint Based Error Estimates For Nonlinear Ordinary Differential Equations With Application To Multistage Sir Models With Demographics, Daniel Alcala

Mathematics & Statistics ETDs

Ordinary Differential Equations (ODEs) are central to the mathematical modeling of various real-world phenomena, from mechanical systems governed by Newton’s laws to epidemic dynamics described by SIR-type ODEs. Since many ODEs do not admit closed-form analytic solutions, we approximate them numerically (e.g., with Euler’s, Runge–Kutta, or other such methods). This raises the key question: How accurate are these numerical solutions? In particular, reliably estimating the error in some quantity of interest (QoI) at time T without having an exact solution is of great scientific interest.

The first main contribution of this thesis is the development and analysis of adjoint-based error …

Stability Insights From Modeling Chronic Myelogenous Leukemia, Giovani Thai

Master's Theses

This thesis centers around a model for chronic myelogenous leukemia (CML) as it behaves under imatinib treatment, a common medication for CML patients, and the anti-leukemia immune response. The dynamics are represented with a system of nonlinear delay-differential equations first constructed by Kim et al. in 2008, capturing population changes of T-cells and various CML growth stages. We investigate stability in both the clinical and mathematical sense. Through numerical simulations, we computationally incorporate a supplementary treatment plan to determine its effectiveness in aiding immune response and medication in achieving remission and full elimination. The primary goal is to conduct a …

Stability Criteria For The Generalized El Borhamy-Rashad-Sobhy Equation, Mohamed El-Borhamy Assoc.Prof, Essam Eddin Rashad Prof., Fathi Mousa Dr., Mai Hamouda

Journal of Engineering Research

This article is concerned with the study of stability criteria for one of the generalization form of El Borhamy-Rashad Sobhy equation, which is a linear second-order ordinary differential equation with periodically time varying coefficients. Many engineering applications can be represented by this generalization, for instance, including the modeling of RLC circuit with time varying inductance, resistance and capacitance, and the vibration of a stretched string, whose mass per unit length is periodic, under a periodic motion. An approximate solution is derived by using the Wenhl-Kramers-Brillonin (WKB) approach. A method of constructing Liapunov function is employed to derive extra conditions for …

Modeling Covid-19 Spread And Effects Of Non-Pharmaceutical Interventions On A College Campus, Jakob Kotas, Priscilla Perey Ratonel

CODEE Journal

We consider an extension to the classical SIR compartmental model from mathematical epidemiology as applied to the spread of COVID-19 on a college campus. While the classical SIR model does not allow for recovered individuals to lose immunity, we alter the equations to allow for such (due to the rise of new variants), leading to a SIRS-type model. We study the system of ODEs analytically and through numerical simulation. Finally we discuss a quantitative approach for how college administrators can decide when to implement stricter non-pharmaceutical interventions (social distancing, mask mandates, quarantine, etc.) to eradicate the infection from the campus …

The Inverse Scattering Transform For The Nonlinear Schrödinger Equation, Ivan Casas-Rocha

Honors Undergraduate Theses

The Nonlinear Schrödinger (NLS) Equation, iψ_t + 1/2 ψ_xx ± |ψ|²ψ = 0, is a nonlinear partial differential equation which is used to model several physical phenomena including nonlinear effects inside optical fibers and the formation of rogue waves in shallow water. It is particu- larly difficult to study solutions to this equation due to the nonlinearity, and the nonlinearity leads to incredibly interesting solutions not found in linear PDEs such as solitons. In this thesis, we highlight two methods of obtaining solutions to the (NLS) equation: the Inverse Scattering Transform and the Dressing Method. Furthermore, …