Applied Mathematics Commons^™

Toward Mapping Multiphase Multicomponent Mixtures With Neural Networks, Kristen L. Hallas, Melissa De Jesus, Christine J. Wu, Jianzhi Li, Jason Bernstein, Philip C. Myint

School of Mathematical & Statistical Sciences Faculty Publications

Equation of state (EOS) tables are commonly used in hydrodynamic simulations of high-pressure, high-temperature phenomena in fields like planetary science, astrophysics, and high-energy-density science. However, generating and storing EOS tables for multiphase, multicomponent mixtures over a wide range of pressures and temperatures is computationally infeasible due to their memory-intensive nature. To address this issue, we have developed a neural network-based machine learning model to predict new EOS tables for binary mixtures. In particular, a deep feedforward neural network trained on a set of ten EOS tables at particular mixture compositions is able to predict nine new (hold-out) EOS tables at …

Study Of A Nonlinear Delayed Parabolic Model For Prion Disease Dynamics With The Unfolded Protein Response, Gangadhara Boregowda, Laurent Pujo-Menjouet, Zhaosheng Feng, Michael R. Lindstrom

School of Mathematical & Statistical Sciences Faculty Publications

Prion diseases are neurodegenerative disorders characterized by the dynamic spread of misfolded toxic proteins in the brain. In this process, the normal cellular prion protein (PrPC) produced by neurons misfolds into a toxic form known as scrapie prion protein (PrPSc). These misfolded proteins propagate through the brain by converting healthy prions into their toxic form. This biological mechanism can be modeled by a system of nonlinear parabolic partial differential equations, accompanied by a nonlinear delayed integral boundary condition. Our primary objective is to establish the existence of nonnegative classical solutions to this system. Furthermore, we derive a priori estimates for …

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 …

Fast Computation And Model Order Reduction Of The Friction Stir Welding Process With Pod-Deim, Joshua Kay, Zilong Song

Mathematics and Statistics Student Research and Class Projects

Friction stir welding (FSW) is a solid-state manufacturing process widely used in joining aluminum and other metal workpieces. The FSW process can be modeled by a coupled system of non-Newtonian Navier–Stokes and heat-transfer equations. However, solving this non-linear system with high accuracy requires significant computational power. This work refines the system by introducing corrected coefficients and new treatments for boundary conditions near the tool. Then, model order reduction, including the Proper Orthogonal Decomposition (POD) and Discrete Empirical Interpolation Method (DEIM), is applied to efficiently solve the FSW system in a low-dimensional space. To enhance accuracy and effectiveness, two novel treatments …

The Inverse Elasto-Acoustic Problem, Patrick Grice

Dissertations

A stable and numerically efficient boundary integral method formulation of the elasto-acoustic problem is presented, based on Fourier analysis. The method generalizes well to multiple scattering. The Frechet derivative of the elasto-acoustic problem with respect to shape perturbations is derived, and geometric flow theory is used to design stable numerical methods for the simulation of moving boundaries. The shape derivative is used to define a regularized Gauss-Newton algorithm for shape fitting of elasto-acoustic scatterers.

Geometric Convergence And State-Space Decompositions For Stochastic Gradient Descent Markov Chains, Philip Zaleski

Dissertations

No abstract provided.

Electronic Structure Discretization And Compression Using Diagonal Basis Sets, Casey Lee Dowdle

Dartmouth College Ph.D Dissertations

Numerically solving the electronic structure problem is a fundamentally difficult problem due to the exponential growth in the dimension of the Hilbert space as the system size increases. In order to solve problems at a chemically relevant accuracy, both the choice of basis set and numerical method are important factors that are intrinsically connected.

In this thesis, we study the discretization and resulting compression of electronic Hamiltonians using diagonal basis sets. A diagonal basis set approximately diagonalizes the matrix and tensor representations of the one- and two-body potentials. This can reduce storage, simplify matrix-vector products, and lower the complexity of …

Modeling The Clonal Rosette Composition Of A Bromeliaceae Genet: A Combinatorial Approach, Layla K. Lammers, Erin N. Bodine

Spora: A Journal of Biomathematics

Bromeliaceae, a neo-tropical plant family encompassing over 3,000 species, exhibit two modes of reproduction: sexual reproduction via flowers and seeds, and asexual reproduction via genetically identical clonal rosettes. The vegetative bodies of bromeliads form rosettes with new leaves emerging from the center and clonal rosettes emerging above a single leaf in the rosette, resulting in a genetic individual consisting of a seed-grown rosette and multiple iterations of clonal rosettes. This research develops a combinatorial model of probability that a single genetic individual will include at least n clonal rosettes when a single rosette can produce at most 1 or 2 …

Human-Centered Modeling Of Traffic As A Complex System, Poorendra P. Ramlall

Discovery Day - Daytona Beach

Traffic systems are driven not only by motion, but by interaction: vehicles influence one another, drivers continuously adapt to surrounding behaviour, and cognitive processes shape decisions that can propagate through the flow of traffic. Understanding these layered interactions is essential for improving traffic safety and for designing the next generation of intelligent, connected, and automated transportation systems. This PhD research develops a multiscale, data-driven framework for identifying, modelling, and ultimately interpreting interaction structure in traffic systems. The work first established an information-theoretic basis for this problem, demonstrating how information flow can uncover directional relationships in traffic dynamics and help infer …

Low-Rank Spectral Analysis For The Reddening Of The Seven Sisters Star Cluster, Eric Rodarte, Angelina Scalice, Madison Warner, Kevin Numbe

Discovery Day - Daytona Beach

The Pleiades, also known as the Seven Sisters, is a stunning star cluster located approximately 440 light-years from Earth. This vibrant assemblage of hot blue stars in the Taurus constellation can be admired with the naked eye or through binoculars during early autumn. In this presentation, we utilize spectral theory to measure the reddening in the Pleiades star cluster. To evaluate the impact of interstellar dust on reddening, we employ principal component analysis (PCA) on a matrix representing color indices from various photometric bands linked to the cluster’s photometric data. This dataset was obtained from VIZIER. Our PCA analysis of …

Analyzing Fungal Growth Dynamics Under Different Environmental Conditions Using A Lotka–Volterra Competition System, Maria Ordonez, Fabrio Araujo

Discovery Day - Daytona Beach

Fungi play a critical role in ecosystems as decomposers that recycle nutrients and maintain environmental balance. Their populations are influenced by multiple environmental factors such as temperature, humidity, nutrient availability, and interactions with other organisms. In this project, the Lotka–Volterra model is used to analyze how competing fungal species interact and how these interactions influence population dynamics over time. By modeling two fungal populations competing for the same limited resources, the equations illustrate how environmental conditions and competition coefficients determine whether one species dominates; both species coexist, or one species becomes extinct. The model provides insight into how changes in …

Supply Chain Analysis: The Oregonator Autocatalytic Case Study, Abigail Butcher

Discovery Day - Daytona Beach

Understanding stability in complex supply chains remains a critical challenge due to nonlinear feedback, delayed responses, and sensitivity to parameter changes. This project presents a novel framework that applies bifurcation analysis to evaluate system stability, using the Oregonator autocatalytic chemical reaction model as an analog for supply chain dynamics. A parameter sweep of key model variables, particularly the stoichiometric factor f and the reaction rate constants k, is used to identify transitions between stable and oscillatory regimes. These transitions provide insight into how variations in feedback strength can drive instability in real-world systems. The framework will then be extended to …

Modeling Stellar Structure: Comparing Numerical Solutions Of The Lane-Emden Equation, Jasman Jasmanjot, Bailey Dale, Ryan Dickey

Discovery Day - Daytona Beach

The Lane-Emden equation is a differential equation that is often used in astrophysics to describe the distribution of the density inside a star, and by extension, its pressure distribution. Analytic solutions of the Lane-Emden equation can only be found at polytropic indices n = 0,1,5, matching certain physical conditions. For all other polytropic values, a numerical solution is needed. In this work, a comparison between two numerical schemes for solving the Lane-Emden equation is presented, namely the Euler method and the classical 4th order Runge-Kutta method. The accuracy of these models is first compared to the cases with known analytical …

Numerical Modeling Of Thermo-Poroelasticity Using Finite Element Method, Maya Mckean

Discovery Day - Daytona Beach

We propose a numerical method for solving and modeling thermo-poroelasticity problems using a finite element formulation. Thermo-poroelasticity models describe the coupled interaction between mechanical deformation, fluid flow, and heat transfer in specific materials or environments over time. These models demonstrate the evolution of displacement, pressure, and temperature; we compute these fields in this work using backward Euler time discretization and Enriched Galerkin finite element spatial discretization. For these computations we used FreeFEM, a partial differential equation solver that uses the finite element method, which produced our numerical results. We then compared these values with the expected analytical solution. This was …

Numerical Methods For Nonlinear Problems Using The Finite Element Method, Logan Price

Discovery Day - Daytona Beach

Numerical Methods for Nonlinear Problems Using the Finite Element Method is a computational mathematics capstone that builds and tests finite element method (FEM) workflows for nonlinear partial differential equations in FreeFEM++, with ParaView used for visualization. Two nonlinear model problems are used to demonstrate the approach. The first is a semilinear reaction-diffusion equation with a cubic nonlinearity. A manufactured solution is used so accuracy can be checked at a fixed final time, and refinement studies in both time step and mesh size are run while nonlinear iteration counts are tracked to show solver effort. The second problem is the steady …

Classification Of Sequential Factors In Aviation Accident Cause Prediction, Sophia Nasca, Addyson Wolfe

Discovery Day - Daytona Beach

Uncovering the root causes of aviation accidents is a critical component of improving aviation safety. Traditional approaches are largely reactive, relying on post-incident analysis rather than proactively identifying risk factors. This project addresses the need for proactive safety by using a multi-source dataset that integrates aviation accident records, weather conditions, and maintenance data extracted from investigative reports. The objective of this work is to move beyond predicting broad probable causes and instead model the sequence of contributing factors that lead to aviation incidents. Using the Swiss Cheese Model, the study will capture layered failures across operational, environmental, and maintenance domains. …

Quantifying The Effect Of Metallicity On Stellar Properties And Evolutionary Timescales, Jacob Becker

Discovery Day - Daytona Beach

Stellar age estimations derived from asteroseismology depend on stellar models that are sensitive to metallicity (Z). Variations in this parameter could alter the agreement between gyrochronological and asteroseismic ages, as well as main sequence lifetimes and observational properties such as effective temperature and luminosity. We test how much typical metallicity differences (±0.2 dex) affect main-sequence models of solar-type stars. Using MESA, evolutionary tracks are created for 1 M☉ stars at three metallicities (Z = 0.009, 0.014, 0.022), and it is measured how these changes shift the positions of the zero-age main sequence and the corresponding main sequence lifetimes. This work …

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 …

Ai-Driven Scheduling Algorithms For Private Aviation, Tayan Benson, Jessica Buskey, Gabriel Camacho, Caitlyn A. Gabrinowitz

Discovery Day - Daytona Beach

Private aviation scheduling is complex and dynamic, requiring frequent aircraft repositioning based on demand and operational constraints, unlike fixed commercial airline schedules. As fleets grow beyond 300 aircraft, traditional deterministic methods become too slow, leading to the use of approaches such as genetic algorithms, but neural network-based methods have not seen in-depth exploration. This project models aircraft scheduling as a network, where airports and flights form a graph. It explores advanced AI methods, including graph neural networks and spatio-temporal graph neural networks (STGNNs), to capture both network structure and time constraints. The goal is to generate efficient daily schedules from …

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 …