Partial Differential Equations Commons^™

Inhomogeneous Branching Random Walks: Incorporating Genealogy And Density Effects, Lauren Ajax, Beatrice Durham, Pratima Hebbar, Cade Johnston, Jiayi Zhang

Spora: A Journal of Biomathematics

We introduce a novel framework using inhomogeneous branching random walks (BRWs) to model biological processes, specifically by introducing genealogy-dependence in branching rates and displacement distributions to model bacterial colony growth. Current stochastic models often either assume independent and identical behavior of individual agents or incorporate only spatiotemporal inhomogeneity, ignoring the effect of genealogy-based inhomogeneity on the long-time behavior of these processes. Such asymptotics are of independent mathematical interest and are crucial in understanding the emergence of patterns. We propose several inhomogeneous BRW models in 2D space where displacement distributions and branching rates vary with time, space, and genealogy. A combined …

Entropic Foundation Of Finance And Physics: Securities Price Dynamics And Quantum Theory, Mohammad Abedi

Electronic Theses & Dissertations (2024 - present)

In many scientific and financial contexts, we must reason and make predictions under conditions of incomplete information. This dissertation develops Entropic Dynamics (ED) as a unified framework for deriving dynamical laws directly from principles of inference. Within this approach, probability distributions represent states of knowledge, and their evolution is determined through entropy maximization subject to relevant constraints. This leads to a novel concept of entropic time and a formulation of dynamics as an inferential process. In this talk, I will present how ED provides a common foundation across multiple domains. In physics, quantum dynamics for particles and scalar fields in …

Performance Of Numerical Methods Applied To The Black–Scholes Model, Scott Cameron Williams

UNF Graduate Theses and Dissertations

We compare five numerical approaches for approximating solutions to the Black–Scholes partial differential equation for pricing European call options: FTCS, BTCS, Crank– Nicolson, Monte Carlo simulation, and a physics–informed neural network (PINN). These methods span finite difference techniques, probabilistic simulation, and machine learning. Performance is evaluated based on computational efficiency and accuracy relative to the analytical Black–Scholes solution.

Among the methods, Crank–Nicolson and the PINN demonstrated the strongest overall performance. Crank–Nicolson achieved the highest accuracy but exhibited increased runtime as the number of underlying stock price grid points grew. In contrast, the PINN produced slightly less accurate results but with …

Mathematical Contributions To The Study Of Chemotaxis And Cell Signaling, Hajr Zam

Graduate Theses, Dissertations, and Problem Reports (ETD)

This dissertation presents results from two mathematical projects concerned with the biology of cells. Chapter 1 provides biological background and places the two mathematical problems in the context of cell signaling. The larger project, with Prof. H. Hattori on a chemotaxis model is presented in Chapters 3 and 4. Work with Prof. \'{A}. Hal\'{a}sz on a chemical reaction network system with linear multimers and two types of labels is presented in Chapter 2. The chemotaxis system describes the one-dimensional dynamics of a species of cells with two chemical species, a chemo-attractant and chemo-repellent. The goal is to analyze the behavior …

Investigation Of Space Charge Effects On Co2 Electrocatalytic Reduction On Gd-Doped Ceria Via Scanning Kelvin Probe And Model-Based Bayesian Analysis, Alejandro Mejia

Graduate Theses, Dissertations, and Problem Reports (ETD)

In studying novel energy conversion and storage systems, such as high-temperature electrolysis, numerous underlying fundamental physical processes remain unclear or inadequately understood. Among these, the modeling and comprehension of surface reaction mechanisms, coupled with the intricate effects of space‑charge interfaces, remains an unclear and challenging area of research.

The work of this dissertation involves the development of a 2D finite element analysis model, leveraging the robust MOOSE framework from INL. This model, featuring inhomogeneous defect thermodynamics for near-surface chemistry, formulated through Poisson‑Cahn variational theory, has been exploited for studying the electrocatalytic reduction of CO₂ on gadolinia doped ceria. The …

Simulation Of Wave Propagation In Granular Particles Using A Discrete Element Model, Syed Tahmid Hussan

College of Graduate Studies: Theses & Dissertations

The understanding of Bender Element mechanism and utilization of Particle Flow Code (PFC) to simulate the seismic wave behavior is important to test the dynamic behavior of soil particles. Both discrete and finite element methods can be used to simulate wave behavior. However, Discrete Element Method (DEM) is mostly suitable, as the micro scaled soil particle cannot be fully considered as continuous specimen like a piece of rod or aluminum. Recently DEM has been widely used to study mechanical properties of soils at particle level considering the particles as balls. This study represents a comparative analysis of Voigt and Best …

Last Passage Time And Excursion Theory For Solvable Diffusions With Applications In Mathematical Finance, Yaode Sui

Theses and Dissertations (Comprehensive)

In this dissertation, we investigate the properties of last passage times and excursion theory in one-dimensional solvable diffusions, emphasizing their applications in financial modeling, particularly in option pricing. We derive closed-form formulas for the marginal distribution of last passage times and their joint distribution with process values, including the maximum and minimum of the process value. The focus is on time-homogeneous diffusions with various boundaries and imposed killing. Employing spectral expansion theory, we derive explicit formulas for distributions of last passage times in common processes such as Drifted Brownian Motion (BM), Squared Bessel (SQB), Ornstein-Uhlenbeck (OU), and Cox-Ingersoll-Ross (CIR) models. …

Multiscale Modelling Of Brain Networks And The Analysis Of Dynamic Processes In Neurodegenerative Disorders, Hina Shaheen

Theses and Dissertations (Comprehensive)

The complex nature of the human brain, with its intricate organic structure and multiscale spatio-temporal characteristics ranging from synapses to the entire brain, presents a major obstacle in brain modelling. Capturing this complexity poses a significant challenge for researchers. The complex interplay of coupled multiphysics and biochemical activities within this intricate system shapes the brain's capacity, functioning within a structure-function relationship that necessitates a specific mathematical framework. Advanced mathematical modelling approaches that incorporate the coupling of brain networks and the analysis of dynamic processes are essential for advancing therapeutic strategies aimed at treating neurodegenerative diseases (NDDs), which afflict millions of …

Beginner's Analysis Of Financial Stochastic Process Models, David Garcia

HMC Senior Theses

This thesis explores the use of geometric Brownian motion (GBM) as a financial model for predicting stock prices. The model is first introduced and its assumptions and limitations are discussed. Then, it is shown how to simulate GBM in order to predict stock price values. The performance of the GBM model is then evaluated in two different periods of time to determine whether it's accuracy has changed before and after March 23, 2020.

Mathematical Models Yield Insights Into Cnns: Applications In Natural Image Restoration And Population Genetics, Ryan Cecil

Electronic Theses and Dissertations

Due to a rise in computational power, machine learning (ML) methods have become the state-of-the-art in a variety of fields. Known to be black-box approaches, however, these methods are oftentimes not well understood. In this work, we utilize our understanding of model-based approaches to derive insights into Convolutional Neural Networks (CNNs). In the field of Natural Image Restoration, we focus on the image denoising problem. Recent work have demonstrated the potential of mathematically motivated CNN architectures that learn both `geometric' and nonlinear higher order features and corresponding regularizers. We extend this work by showing that not only can geometric features …

Sparse Spectral-Tau Method For The Two-Dimensional Helmholtz Problem Posed On A Rectangular Domain, Gabriella M. Dalton

Mathematics & Statistics ETDs

Within recent decades, spectral methods have become an important technique in numerical computing for solving partial differential equations. This is due to their superior accuracy when compared to finite difference and finite element methods. For such spectral approximations, the convergence rate is solely dependent on the smoothness of the solution yielding the potential to achieve spectral accuracy. We present an iterative approach for solving the two-dimensional Helmholtz problem posed on a rectangular domain subject to Dirichlet boundary conditions that is well-conditioned, low in memory, and of sub-quadratic complexity. The proposed approach spectrally approximates the partial differential equation by means of …

Stochastic Navier-Stokes Equations With Markov Switching, Po-Han Hsu

LSU Doctoral Dissertations

This dissertation is devoted to the study of three-dimensional (regularized) stochastic Navier-Stokes equations with Markov switching. A Markov chain is introduced into the noise term to capture the transitions from laminar to turbulent flow, and vice versa. The existence of the weak solution (in the sense of stochastic analysis) is shown by studying the martingale problem posed by it. This together with the pathwise uniqueness yields existence of the unique strong solution (in the sense of stochastic analysis). The existence and uniqueness of a stationary measure is established when the noise terms are additive and autonomous. Certain exit time estimates …

Multi-Level Small Area Estimation Based On Calibrated Hierarchical Likelihood Approach Through Bias Correction With Applications To Covid-19 Data, Nirosha Rathnayake

Theses & Dissertations

Small area estimation (SAE) has been widely used in a variety of applications to draw estimates in geographic domains represented as a metropolitan area, district, county, or state. The direct estimation methods provide accurate estimates when the sample size of study participants within each area unit is sufficiently large, but it might not always be realistic to have large sample sizes of study participants when considering small geographical regions. Meanwhile, high dimensional socio-ecological data exist at the community level, providing an opportunity for model-based estimation by incorporating rich auxiliary information at the individual and area levels. Thus, it is critical …

From Wave Propagation To Spin Dynamics: Mathematical And Computational Aspects, Oleksii Beznosov

Mathematics & Statistics ETDs

In this work we concentrate on two separate topics which pose certain numerical challenges. The first topic is the spin dynamics of electrons in high-energy circular accelerators. We introduce a stochastic differential equation framework to study spin depolarization and spin equilibrium. This framework allows the mathematical study of known equations and new equations modelling the spin distribution of an electron bunch. A spin distribution is governed by a so-called Bloch equation, which is a linear Fokker-Planck type PDE, in general posed in six dimensions. We propose three approaches to approximate solutions, using analytical and modern numerical techniques. We also present …

483— Effectiveness Of Mmr Vaccination In Orthodox Jewish Neighborhoods, Meenu Mundackal

GREAT Day Posters

Measles is a highly contagious disease, where large outbreaks arise by direct contact between susceptible (unvaccinated) and infectious individuals. Many Orthodox Jewish neighborhoods were affected by measles from 2018-2019. To quantify the vaccination effort on this susceptible population, a retrospective analysis was used to study the NYC and Rockland County populations using a differential equations model. A subsequent model, known as a realistically-structured network model, studied only the NYC population, in relation to typical household size. Vaccination strategies were applied to three cohorts: unvaccinated family members, members with 1 prior MMR dose, and members with 2 prior MMR doses. The …

The Analysis Of Neural Heterogeneity Through Mathematical And Statistical Methods, Kyle Wendling

Theses and Dissertations

Diversity of intrinsic neural attributes and network connections is known to exist in many areas of the brain and is thought to significantly affect neural coding. Recent theoretical and experimental work has argued that in uncoupled networks, coding is most accurate at intermediate levels of heterogeneity. I explore this phenomenon through two distinct approaches: a theoretical mathematical modeling approach and a data-driven statistical modeling approach.

Through the mathematical approach, I examine firing rate heterogeneity in a feedforward network of stochastic neural oscillators utilizing a high-dimensional model. The firing rate heterogeneity stems from two sources: intrinsic (different individual cells) and network …

Determinism Of Stochastic Processes Through The Relationship Between The Heat Equation And Random Walks, Gurmehar Singh Makker

Publications and Research

We study the deterministic characteristics of stochastic processes through investigation of random walks and the heat equation. The relationship is confirmed by discretizing the heat equation in time and space and determining the probability distribution function for random walks in dimension d = 1, 2. The existence of the relationship is presented both through theoretical analysis and numerical computation.

Study Of Specially And Temporally Dependent Adsorption Coefficient In Heterogeneous Porous Medium, Dilip K. Jaiswal, Gulrana _

Applications and Applied Mathematics: An International Journal (AAM)

One-dimensional advection-dispersion equation (ADE) is studied along unsteady longitudinal flow through a semi-infinite heterogeneous medium. Adsorption coefficient is considered temporally and spatially–dependent function i.e., expressed in degenerate form. The dispersion parameter is considered as inversely proportional to adsorption coefficient. The input source is of pulse type. The Laplace Transformation Technique (LTT) is used to obtain the analytical solution by introducing certain new independent variables through separate transformations. The effects of adsorption, heterogeneity and unsteadiness are investigated and discussed with the help of various graphs.

The Martingale Approach To Financial Mathematics, Jordan M. Rowley

Master's Theses

In this thesis, we will develop the fundamental properties of financial mathematics, with a focus on establishing meaningful connections between martingale theory, stochastic calculus, and measure-theoretic probability. We first consider a simple binomial model in discrete time, and assume the impossibility of earning a riskless profit, known as arbitrage. Under this no-arbitrage assumption alone, we stumble upon a strange new probability measure Q, according to which every risky asset is expected to grow as though it were a bond. As it turns out, this measure Q also gives the arbitrage-free pricing formula for every asset on our market. In …

Ergodicity For The 3d Stochastic Navier-Stokes Equations Perturbed By Lévy Noise, Manil T. Mohan, K. Sakthivel, Sivaguru S. Sritharan

Faculty Publications

In this work we construct a Markov family of martingale solutions for 3D stochastic Navier–Stokes equations (SNSE) perturbed by Lévy noise with periodic boundary conditions. Using the Kolmogorov equations of integrodifferential type associated with the SNSE perturbed by Lévy noise, we construct a transition semigroup and establish the existence of a unique invariant measure. We also show that it is ergodic and strongly mixing.
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Masked Instability: Within-Sector Financial Risk In The Presence Of Wealth Inequality, Youngna Choi

Department of Applied Mathematics and Statistics Faculty Scholarship and Creative Works

We investigate masked financial instability caused by wealth inequality. When an economic sector is decomposed into two subsectors that possess a severe wealth inequality, the sector in entirety can look financially stable while the two subsectors possess extreme financially instabilities of opposite nature, one from excessive equity, the other from lack thereof. The unstable subsector can result in further financial distress and even trigger a financial crisis. The market instability indicator, an early warning system derived from dynamical systems applied to agent-based models, is used to analyze the subsectoral financial instabilities. Detailed mathematical analysis is provided to explain what financial …

Physical Applications Of The Geometric Minimum Action Method, George L. Poppe Jr.

Dissertations, Theses, and Capstone Projects

This thesis extends the landscape of rare events problems solved on stochastic systems by means of the \textit{geometric minimum action method} (gMAM). These include partial differential equations (PDEs) such as the real Ginzburg-Landau equation (RGLE), the linear Schroedinger equation, along with various forms of the nonlinear Schroedinger equation (NLSE) including an application towards an ultra-short pulse mode-locked laser system (MLL).

Additionally we develop analytical tools that can be used alongside numerics to validate those solutions. This includes the use of instanton methods in deriving state transitions for the linear Schroedinger equation and the cubic diffusive NLSE.

These analytical solutions are …

Flow Anisotropy Due To Thread-Like Nanoparticle Agglomerations In Dilute Ferrofluids, Alexander Cali, Wah-Keat Lee, A. David Trubatch, Philip Yecko

Department of Applied Mathematics and Statistics Faculty Scholarship and Creative Works

Improved knowledge of the magnetic field dependent flow properties of nanoparticle-based magnetic fluids is critical to the design of biomedical applications, including drug delivery and cell sorting. To probe the rheology of ferrofluid on a sub-millimeter scale, we examine the paths of 550 μm diameter glass spheres falling due to gravity in dilute ferrofluid, imposing a uniform magnetic field at an angle with respect to the vertical. Visualization of the spheres’ trajectories is achieved using high resolution X-ray phase-contrast imaging, allowing measurement of a terminal velocity while simultaneously revealing the formation of an array of long thread-like accumulations of magnetic …

Thermodynamics Of Coherent Structures Near Phase Transitions, Julia M. Meyer, Ivan Christov

The Summer Undergraduate Research Fellowship (SURF) Symposium

Phase transitions within large-scale systems may be modeled by nonlinear stochastic partial differential equations in which system dynamics are captured by appropriate potentials. Coherent structures in these systems evolve randomly through time; thus, statistical behavior of these fields is of greater interest than particular system realizations. The ability to simulate and predict phase transition behavior has many applications, from material behaviors (e.g., crystallographic phase transformations and coherent movement of granular materials) to traffic congestion. Past research focused on deriving solutions to the system probability density function (PDF), which is the ground-state wave function squared. Until recently, the extent to which …

Kinetic Monte Carlo Methods For Computing First Capture Time Distributions In Models Of Diffusive Absorption, Daniel Schmidt

HMC Senior Theses

In this paper, we consider the capture dynamics of a particle undergoing a random walk above a sheet of absorbing traps. In particular, we seek to characterize the distribution in time from when the particle is released to when it is absorbed. This problem is motivated by the study of lymphocytes in the human blood stream; for a particle near the surface of a lymphocyte, how long will it take for the particle to be captured? We model this problem as a diffusive process with a mixture of reflecting and absorbing boundary conditions. The model is analyzed from two approaches. …

A Semiparametric Estimation For Regression Functions In The Partially Linear Autoregressive Time Series Model, R. Farnoosh, M. Hajebi, S. J. Mortazavi

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, a semiparametric method is proposed for estimating regression function in the partially linear autoregressive time series model . Here, we consider a combination of parametric forms and nonlinear functions, in which the errors are independent. Semiparametric and nonparametric curve estimation provides a useful tool for exploring and understanding the structure of a nonlinear time series data set to make for a more efficient study in the partially linear autoregressive model. The unknown parameters are estimated using the conditional nonlinear least squares method, and the nonparametric adjustment is also estimated by defining and minimizing the local L2 -fitting …

An Approximate Analytical Solution Of The Fractional Diffusion Equation With External Force And Different Type Of Absorbent Term - Revisited, S. Das, R. Kumar, P. K. Gupta

Applications and Applied Mathematics: An International Journal (AAM)

In this article Homotopy Perturbation Method (HPM) is applied to obtain an approximate analytical solution of a fractional diffusion equation with an external force and a reaction term different from the reaction term used by Das and Gupta (2010). The anomalous behavior of diffusivity in presence or absence of linear external force due to the presence of this force of reaction term are obtained and presented graphically.

The Substitution Theorem For Semilinear Stochastic Partial Differential Equations, Salah-Eldin A. Mohammed, Tusheng Zhang

Articles and Preprints

In this article we establish a substitution theorem for semilinear stochastic evolution equations (see's) depending on the initial condition as an infinite-dimensional parameter. Due to the infinitedimensionality of the initial conditions and of the stochastic dynamics, existing finite-dimensional results do not apply. The substitution theorem is proved using Malliavin calculus techniques together with new estimates on the underlying stochastic semiflow. Applications of the theorem include dynamic characterizations of solutions of stochastic partial differential equations (spde's) with anticipating initial conditions and non-ergodic stationary solutions. In particular, our result gives a new existence theorem for solutions of semilinear Stratonovich spde's with anticipating …

The Stable Manifold Theorem For Semilinear Stochastic Evolution Equations And Stochastic Partial Differential Equations, Salah-Eldin A. Mohammed, Tusheng Zhang, Huaizhong Zhao

Articles and Preprints

The main objective of this paper is to characterize the pathwise local structure of solutions of semilinear stochastic evolution equations (see’s) and stochastic partial differential equations (spde’s) near stationary solutions. Such characterization is realized through the long-term behavior of the solution field near stationary points. The analysis falls in two parts 1, 2.

In Part 1, we prove general existence and compactness theorems for C^k-cocycles of semilinear see’s and spde’s. Our results cover a large class of semilinear see’s as well as certain semilinear spde’s with Lipschitz and non-Lipschitz terms such as stochastic reaction diffusion equations and the …

The Truncated Cauchy Distribution: Estimation Of Parameters And Application To Stock Returns, Paul G. Staneski

Mathematics & Statistics Theses & Dissertations

The problem addressed in this dissertation is the existence and estimation of the parameters of a truncated Cauchy distribution. It is known that when a number of distributions with infinite support are truncated to a finite interval that the maximum likelihood estimator of the scale parameter fails to exist with positive probability. In particular, necessary and sufficient conditions which give rise to instances of non-existence have been found for the exponential (Deemer and Votaw (1955)), gamma (Broeder (1955), Hegde and Dahiya (1989)), Weibull (Mittal and Dahiya (1989)) and normal distribution (Barndorff-Nielsen (1978), Mittal and Dahiya (1987), Hegde and Dahiya (1989)). …