Trinity University
University of Nebraska-Lincoln
Virginia Commonwealth University
Rhodes College
Western Kentucky University
Central Connecticut State University
William & Mary
Analytical And Numerical Solutions For The Hydrogen Atom,
2026
Embry-Riddle Aeronautical University
Analytical And Numerical Solutions For The Hydrogen Atom, Kassidy Myers
Discovery Day - Daytona Beach
The Schrödinger equation is the foundational equation of non-relativistic quantum mechanics. The hydrogen atom is the simplest system for solving this equation, as it consists of only one proton and one electron. In this project, we work on the Schrödinger equation that models the spherically symmetric states of the hydrogen atom that depend only on the radial coordinate. We simplified and nondimensionalized the radial equation and solved the resulting equation using a power series (Frobenius) method. This approach revealed the physically meaningful solutions and led to quantized energy levels. In addition to finding the analytical solution, we numerically solve the …
The Motion Of A Falling Object Under Linear Drag And How Differential Equations Can Be Used To Find It,
2026
Embry-Riddle Aeronautical University
The Motion Of A Falling Object Under Linear Drag And How Differential Equations Can Be Used To Find It, Lukas Estrella
Discovery Day - Daytona Beach
This project, "The Motion of a Falling Object Under Linear Drag and How Differential Equations Can Be Used To Find It," investigates the motion of a falling object subject to air resistance through a combination of mathematical modeling and fundamental physical principles. The analysis is grounded in Newton’s second law, which yields a differential equation describing the forces acting on the object. Assuming a linear drag model, in which the resistive force is proportional to velocity, the governing equation reduces to a first-order ordinary differential equation for velocity. This equation is solved using the integrating factor method, yielding an explicit …
The General Solution Analysis Of Homogeneous Linear Equations,
2026
Embry-Riddle Aeronautical University
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 …
Modeling Population Growth With Logistic And Modified Logistic Equations,
2026
Embry-Riddle Aeronautical University
Modeling Population Growth With Logistic And Modified Logistic Equations, Mihil Dimpal Patel
Discovery Day - Daytona Beach
Population growth models are essential tools for understanding how biological populations change over time under environmental constraints. This study examines population dynamics by comparing the classical exponential growth model with the logistic growth model. While exponential growth assumes unlimited resources and results in unbounded population increase, the logistic model incorporates a carrying capacity that limits growth as resources become scarce. To better represent real-world conditions, the logistic model is extended by introducing modifications such as harvesting terms and time-varying carrying capacities, which account for external removal of individuals and changing environmental limits. The equilibria of these models are determined, and …
Modeling Seiche Oscillations Using Damped Vibration Differential Equations,
2026
Embry-Riddle Aeronautical University
Modeling Seiche Oscillations Using Damped Vibration Differential Equations, Bianca Gerity, Arineh Shahbazi
Discovery Day - Daytona Beach
A seiche oscillation is a standing wave that oscillates in an enclosed body of water, like a lake or pool. Seiches are caused by strong winds, earthquakes, and rapid atmospheric changes. Seiches are an excellent real-world example of damped harmonic motion. The physics of these unique vibrations can actually be modeled using a second-order differential equation for damped oscillators of the general form mx''+cx'+kx=0, where m represents the mass of the vibrating water column, c represents the energy dissipation due to friction and viscosity, and k represents the force governed by gravity and the basin's geometry. The objective of this …
Bridging Discrete And Continuous Systems: Fibonacci Sequences And Exponential Growth From Odes,
2026
Embry-Riddle Aeronautical University
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 …
Mechanical Vibrations And Damping,
2026
Embry-Riddle Aeronautical University
Mechanical Vibrations And Damping, Axon Deadrick, Will Standish, Tanay Agarwal
Discovery Day - Daytona Beach
Mechanical vibrations occur in many engineering systems and can be described using second-order differential equations. In this project, the motion of vibrating systems is studied using the mass–spring model. The focus is on three types of oscillations: free vibrations, dampened vibrations, and forced oscillations. Free vibration describes how a system moves when it is displaced and then released without any external force. Damped vibration includes effects such as friction or resistance that cause the motion to gradually decrease over time. Forced oscillations occur when an external force acts on the system and continuously drives the motion. This project also examines …
Numerical Modeling Of Badminton Shuttlecock Trajectories,
2026
Embry-Riddle Aeronautical University
Numerical Modeling Of Badminton Shuttlecock Trajectories, Lola G. Torres, Cassandra Pumphrey, Jadyn Peterson, Domenic Barsotti
Discovery Day - Daytona Beach
The Trajectory of a badminton Shuttlecock can vary significantly when compared to a classic projectile motion, primarily due to aerodynamic drag. This project aims to model the flight of the shuttlecock using Newton's second law for gravitational and drag related forces, resulting in a nonlinear system of a first order differential equation. The given parameters include the shuttlecock mass, cross-sectional area, air density, as well as the drag coefficient, determining the overall magnitude of the drag force. The resulting initial value problem is solved numerically using a multitude of Runge_Kutta methods to compare the accuracy and stability across different computational …
Motion With Air Resistance,
2026
Embry-Riddle Aeronautical University
Motion With Air Resistance, Gauge Mccain, Jacob Bealefeld, Francesca Wise
Discovery Day - Daytona Beach
The motion of objects moving through air is influenced not only by gravity but also by air resistance, which affects the speed and acceleration of the object over time. This project examines the motion of a falling object by modeling it with an ordinary differential equation that accounts for both gravitational force and a resistive drag force proportional to velocity. Using Newton’s Second Law, a first-order differential equation is derived to describe how the velocity of the object changes as it falls. The solution of this equation demonstrates how the velocity increases initially and gradually approaches a constant value known …
A Differential Equation Approach To Heat Flow In A Thin Rod,
2026
Embry-Riddle Aeronautical University
A Differential Equation Approach To Heat Flow In A Thin Rod, Alexandria Krol, David Cardona, Collin Petrie
Discovery Day - Daytona Beach
A Differential Equation Approach to Heat Flow in a Thin Rod examines how differential equations can be used to model and understand heat conduction in a fundamental physical system. Heat transfer in solids is a key concept in physics and engineering, particularly in systems where temperature changes over time. A thin rod provides a useful one-dimensional model for studying how heat moves through a material and how temperature varies along the rod as time passes. The primary objective is to develop a mathematical description of this process using differential equations. The analysis begins with physical principles such as conservation of …
Numerical Analysis Of The Sir Model For Predicting Disease Spread,
2026
Embry-Riddle Aeronautical University
Numerical Analysis Of The Sir Model For Predicting Disease Spread, Victoria Gaibor, Isabel Tejada, Kate Moore
Discovery Day - Daytona Beach
This project, Numerical Solutions of the SIR Model for Predicting Disease Spread, investigates the application of numerical methods to analyze the dynamics of infectious diseases using the classical Susceptible–Infected–Recovered (SIR) model. The SIR model, a system of nonlinear ordinary differential equations, is widely used to describe how diseases such as COVID-19 propagate through a population. The primary objective of this study is to solve the SIR initial value problem using multiple numerical techniques, including Euler’s method, Runge–Kutta methods, and multistep methods, and to compare their accuracy and efficiency. The model is implemented using given initial conditions and parameters, and additional …
Mechanical Vibrations And Damping – Structural Analysis,
2026
Embry-Riddle Aeronautical University
Mechanical Vibrations And Damping – Structural Analysis, Kelsey Hunsicker, Sarah Kraus, Sophia Muller Martinelli De Souza
Discovery Day - Daytona Beach
Mechanical Vibrations are crucial in understanding and structural analysis of engineering systems such as bridges and airplane wings. If not considered, these vibrations can lead to structural fatigue or failure. By using differential equations, structural vibrations will be examined. Researching the different kinds of vibrations and damping will help to find the vibration behavior of the system. For example, a mass-spring damper system will use second-order linear differential equations. The systems model can be shown to be underdamped, overdamped, or critically damped. These will compare the amplitudes and oscillation differences between the systems by using computational code. Analyzing these differences …
Numerical Investigation Of The Nonlinear Simple Pendulum And The Dependence Of Oscillation Period On Initial Angle,
2026
Embry-Riddle Aeronautical University
Numerical Investigation Of The Nonlinear Simple Pendulum And The Dependence Of Oscillation Period On Initial Angle, Kelly Wold, Aidan Hart, Patrick Gilliam
Discovery Day - Daytona Beach
Numerical Investigation of the Nonlinear Simple Pendulum and the Dependence of Oscillation Period on Initial Angle examines how the oscillation period of a simple pendulum varies with initial angular displacement and evaluates the accuracy of numerical methods in capturing this behavior. In classical treatments, the small-angle approximation simplifies the governing differential equation and predicts a constant period independent of amplitude; however, this assumption breaks down for larger angles, where the system exhibits nonlinear dynamics. The objective of this project is to model the full nonlinear equation of motion and quantify how the period depends on initial conditions. To achieve this, …
Simulating Pacemakers And Heartbeat Recovery Through Mathematical Modeling,
2026
Embry-Riddle Aeronautical University
Simulating Pacemakers And Heartbeat Recovery Through Mathematical Modeling, Thomas Estrada, Jayla Edwards
Discovery Day - Daytona Beach
Title: Simulating Pacemakers and Heartbeat Recovery Through Mathematical Modeling This study utilizes the Fitzhugh-Nagumo model to simulate cardiac electrical activity and the regulatory role of pacemakers through ordinary differential equations (ODEs). By defining the rate of change for membrane voltage, 𝑑𝑣/dt, and a recovery variable, 𝑑𝑤/dt, the model captures the heart's excitability and resting states. Central to the analysis is the stimulus current parameter, which represents the "kick" provided by a pacemaker to correct flatline conditions or weak heartbeats. Using Euler’s method for numerical integration, the research compares unstable cardiac rhythms against corrected periodic oscillations. Additionally, the project implements vector …
A Modeling Scenario For Cooling A Hot Vehicle In Florida,
2026
Florida Polytechnic University
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,
2026
Faculty of Law, Ono Academic College
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,
2026
Phillips Academy
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 …
Dynamical Behavior, Extinction And Persistence In A Stochastic Delayed Two-Strain Epidemic Model With A Generalized Crowley-Martin Incidence Rate,
2026
Hassan II University of Casablanca, Morocco
Dynamical Behavior, Extinction And Persistence In A Stochastic Delayed Two-Strain Epidemic Model With A Generalized Crowley-Martin Incidence Rate, Amina Allali, Dounia Bentaleb, Saida Amine
Mathematical Modelling and Numerical Simulation with Applications
This paper develops and analyzes a novel delayed stochastic SIR epidemic model with two interacting strains and general incidence functions. By establishing the existence and uniqueness of a positive global solution, the well-posedness of the model under stochastic perturbations is ensured. Extinction occurs when the stochastic reproduction number falls below unity, as demonstrated through Itô calculus and martingale convergence theorems, while persistence is guaranteed under the Crowley–Martin incidence when it exceeds one. Numerical experiments based on the Positive Preserving Truncated Euler–Maruyama (PPTEM) scheme confirm the analytical predictions and highlight the influence of time delay and noise intensity on the long-term …
(R2147) Effect Of Mass Variation With Log-Logistic Distribution In Perturbed Interacting Cr3bp,
2026
Dyal Singh College, University of Delhi, India
(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.
(R2155) Mathematical Insights Into Cancer Cells Growth Stability And Chaos,
2026
University of Delhi
(R2155) Mathematical Insights Into Cancer Cells Growth Stability And Chaos, Pardeep Kumar, Kashish Agarwal, Tripti Anand, Sarita Jha
Applications and Applied Mathematics: An International Journal (AAM)
In this research paper, we examined a recently proposed three-dimensional dynamical model of cancer cell growth that explicitly couples the populations of tumour cells, healthy host cells, and immune cells (Pardeep et al.). We aim to elucidate the model’s novel biological features relative to classical tumour–immune interaction models and to characterize its dynamical regimes. This model is governed by nonlinear differential equations featuring a quadratic tumour proliferation term and bilinear coupling terms for tumour–immune and tumour–host interactions. Then, we performed the rigorous analysis by solving the fixed-point equations and from the Jacobian matrices at the resulting equilibria to identify the …
