Physical Sciences and Mathematics Commons^™

An Application Of M-Matrices To Preserve Bounded Positive Solutions To The Evolution Equations Of Biofilm Models, Richard S. Landry Jr.

University of New Orleans Theses and Dissertations

In this work, we design a linear, two step implicit finite difference method to approximate the solutions of a biological system that describes the interaction between a microbial colony and a surrounding substrate. Three separate models are analyzed, all of which can be described as systems of partial differential equations (PDE)s with nonlinear diffusion and reaction, where the biological colony grows and decays based on the substrate bioavailability. The systems under investigation are all complex models describing the dynamics of biological films. In view of the difficulties to calculate analytical solutions of the models, we design here a numerical technique …

Homogenization Techniques For Population Dynamics In Strongly Heterogeneous Landscapes, Brian P. Yurk, Christina A. Cobbold

Faculty Publications

An important problem in spatial ecology is to understand how population-scale patterns emerge from individual-level birth, death, and movement processes. These processes, which depend on local landscape characteristics, vary spatially and may exhibit sharp transitions through behavioural responses to habitat edges, leading to discontinuous population densities. Such systems can be modelled using reaction–diffusion equations with interface conditions that capture local behaviour at patch boundaries. In this work we develop a novel homogenization technique to approximate the large-scale dynamics of the system. We illustrate our approach, which also generalizes to multiple species, with an example of logistic growth within a periodic …

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 …

Radial Basis Function Differential Quadrature Method For The Numerical Solution Of Partial Differential Equations, Daniel Watson

Dissertations

In the numerical solution of partial differential equations (PDEs), there is a need for solving large scale problems. The Radial Basis Function Differential Quadrature (RBFDQ) method and local RBF-DQ method are applied for the solutions of boundary value problems in annular domains governed by the Poisson equation, inhomogeneous biharmonic equation, and the inhomogeneous Cauchy-Navier equations of elasticity. By choosing the collocation points properly, linear systems can be obtained so that the coefficient matrices have block circulant structures. The resulting systems can be efficiently solved using matrix decomposition algorithms (MDAs) and fast Fourier transforms (FFTs). For the local RBFDQ method, the …

Introduction To The Usu Library Of Solutions To The Einstein Field Equations, Ian M. Anderson, Charles G. Torre

Tutorials on... in 1 hour or less

This is a Maple worksheet providing an introduction to the USU Library of Solutions to the Einstein Field Equations. The library is part of the DifferentialGeometry software project and is a collection of symbolic data and metadata describing solutions to the Einstein equations.

Analytical Solutions For The Black-Scholes Equation, Jalil Manafian, Mahnaz Paknezhad

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, the Black-Sholes equation (BS) has been applied successfully with the Cauchy-Euler method and the method of separation of variables and new analytical solutions have been found. The linear partial differential equation (PDE) transformed to linear ordinary differential equation (ODE) as well. We acquired three types of solutions including hyperbolic, trigonometric and rational solutions. Descriptions of these methods are given and the obtained results reveal that three methods are tools for exploring partial differential models.

Fractional Order Thermoelastic Deflection In A Thin Circular Plate, J. J. Tripathi, S. D. Warbhe, K. C. Deshmukh, J. Verma

Applications and Applied Mathematics: An International Journal (AAM)

In this work, a quasi-static uncoupled theory of thermoelasticity based on time fractional heat conduction equation is used to model a thin circular plate, whose lower surface is maintained at zero temperature whereas the upper surface is insulated. The edge of the circular plate is fixed and clamped. Integral transform technique is used to derive the analytical solutions in the physi-cal domain. The numerical results for temperature distributions and thermal deflection are com-puted and represented graphically for Copper material.

On The Lp-Spaces Techniques In The Existence And Uniqueness Of The Fuzzy Fractional Korteweg-De Vries Equation’S Solution, F. Farahrooz, A. Ebadian, S. Najafzadeh

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, is proposed the existence and uniqueness of the solution of all fuzzy fractional differential equations, which are equivalent to the fuzzy integral equation. The techniques on L^P-spaces are used, defining the L^p_F F ([0; 1]) for 1≤P≤∞, its properties, and using the functional analysis methods. Also the convergence of the method of successive approximations used to approximate the solution of fuzzy integral equation be proved and an iterative procedure to solve such equations is presented.

Mathematical Modeling Of Mixtures And Numerical Solution With Applications To Polymer Physics, John Timothy Cummings

Doctoral Dissertations

We consider in this dissertation the mathematical modeling and simulation of a general diffuse interface mixture model based on the principles of energy dissipation. The model developed allows for a thermodynamically consistent description of systems with an arbitrary number of different components, each of which having perhaps differing densities. We also provide a mathematical description of processes which may allow components to source or sink into other components in a mass conserving, energy dissipating way, with the motivation of applying this model to phase transformation. Also included in the modeling is a unique set of thermodynamically consistent boundary conditions which …

Examining The Electrical Excitation, Calcium Signaling, And Mechanical Contraction Cycle In A Heart Cell, Kristen Deetz, Nygel Foster, Darius Leftwich, Chad Meyer, Shalin Patel, Carlos Barajas, Matthias K. Gobbert, Zana Coulibaly

Spora: A Journal of Biomathematics

As the leading cause of death in the United States, heart disease has become a principal concern in modern society. Cardiac arrhythmias can be caused by a dysregulation of calcium dynamics in cardiomyocytes. Calcium dysregulation, however, is not yet fully understood and is not easily predicted; this provides motivation for the subsequent research. Excitation-contraction coupling (ECC) is the process through which cardiomyocytes undergo contraction from an action potential. Calcium induced calcium release (CICR) is the mechanism through which electrical excitation is coupled with mechanical contraction through calcium signaling. The study of the interplay between electrical excitation, calcium signaling, and mechanical …

Height Transitions, Shape Evolution, And Coarsening Of Equilibrating Quantum Nanoislands, Mikhail Khenner

Mathematics Faculty Publications

No abstract provided.

Heads And Tails, Julie Simons

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

A Method For Sensitivity Analysis And Parameter Estimation Applied To A Large Reaction-Diffusion Model Of Cell Polarization, Marissa Renardy, Tau-Mu Yi, Dongbin Xiu, Ching-Shan Chou

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Modeling Microtubule-Based Transport In The Frog Egg Cell, Maria-Veronica Ciocanel

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Analysis And Implementation Of Numerical Methods For Solving Ordinary Differential Equations, Muhammad Sohel Rana

Masters Theses & Specialist Projects

Numerical methods to solve initial value problems of differential equations progressed quite a bit in the last century. We give a brief summary of how useful numerical methods are for ordinary differential equations of first and higher order. In this thesis both computational and theoretical discussion of the application of numerical methods on differential equations takes place. The thesis consists of an investigation of various categories of numerical methods for the solution of ordinary differential equations including the numerical solution of ordinary differential equations from a number of practical fields such as equations arising in population dynamics and astrophysics. It …

Evolution Of Superoscillations For Schrödinger Equation In A Uniform Magnetic Field, Fabrizio Colombo, Jonathan Gantner, Daniele C. Struppa

Mathematics, Physics, and Computer Science Faculty Articles and Research

Aharonov-Berry superoscillations are band-limited functions that oscillate faster than their fastest Fourier component. Superoscillations appear in several fields of science and technology, such as Aharonov’s weak measurement in quantum mechanics, in optics, and in signal processing. An important issue is the study of the evolution of superoscillations using the Schrödinger equation when the initial datum is a weak value. Some superoscillatory functions are not square integrable, but they are real analytic functions that can be extended to entire holomorphic functions. This fact leads to the study of the continuity of a class of convolution operators acting on suitable spaces of …

Stochastic Quasilinear Evolution Equations In Umd Banach Spaces, Manil T. Mohan, Sivaguru S. Sritharan

Faculty Publications

In this work we prove the existence and uniqueness up to a stopping time for the stochastic counterpart of Tosio Kato's quasilinear evolutions in UMD Banach spaces. These class of evolutions are known to cover a large class of physically important nonlinear partial differential equations. Existence of a unique maximal solution as well as an estimate on the probability of positivity of stopping time is obtained. An example of stochastic Euler and Navier–Stokes equation is also given as an application of abstract theory to concrete models.

A Regression Model To Predict Stock Market Mega Movements And/Or Volatility Using Both Macroeconomic Indicators & Fed Bank Variables, Timothy A. Smith, Alcuin Rajan

Publications

In finance, regression models or time series moving averages can be used to determine the value of an asset based on its underlying traits. In prior work we built a regression model to predict the value of the S&P 500 based on macroeconomic indicators such as gross domestic product, money supply, produce price and consumer price indices. In this present work this model is updated both with more data and an adjustment in the input variables to improve the coefficient of determination. A scheme is also laid out to alternately define volatility rather than using common tools such as the …

Interplay Of Quantum Size Effect, Anisotropy And Surface Stress Shapes The Instability Of Thin Metal Films, Mikhail Khenner

Mathematics Faculty Publications

Morphological instability of a planar surface ([111], [011], or [001]) of an ultra-thin metal film is studied in a parameter space formed by three major effects (the quantum size effect, the surface energy anisotropy and the surface stress) that influence a film dewetting. The analysis is based on the extended Mullins equation, where the effects are cast as functions of the film thickness. The formulation of the quantum size effect (Z. Zhang et al., PRL 80, 5381 (1998)) includes the oscillation of the surface energy with thickness caused by electrons confinement. By systematically comparing the effects, their contributions into the …

On Honey Bee Colony Dynamics And Disease Transmission, Matthew I. Betti

Electronic Thesis and Dissertation Repository

The work herein falls under the umbrella of mathematical modeling of disease transmission. The majority of this document focuses on the extent to which infection undermines the strength of a honey bee colony. These studies extend from simple mass-action ordinary differential equations models, to continuous age-structured partial differential equation models and finally a detailed agent-based model which accounts for vector transmission of infection between bees as well as a host of other influences and stressors on honey bee colony dynamics. These models offer a series of predictions relevant to the fate of honey bee colonies in the presence of disease …

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 …

Solution Of Pdes For First-Order Photobleaching Kinetics Using Krylov Subspace Spectral Methods, Somayyeh Sheikholeslami

Dissertations

We solve the first order reaction-diffusion equations which describe binding-diffusion kinetics using a photobleaching scanning profile of a confocal laser scanning microscope approximated by a Gaussian laser profile. We show how to solve these equations with prebleach steady-state initial conditions using a time-domain method known as a Krylov Subspace Spectral (KSS) method. KSS methods are explicit methods for solving time- dependent variable-coefficient partial differential equations (PDEs). KSS methods are advantageous compared to other methods because of their stability and their superior scalability. These advantages are obtained by applying Gaussian quadrature rules in the spectral domain developed by Golub and Meurant. …

Numerical Solution Of Partial Differential Equations Using Polynomial Particular Solutions, Thir R. Dangal

Dissertations

Polynomial particular solutions have been obtained for certain types of partial differential operators without convection terms. In this dissertation, a closed-form particular solution for more general partial differential operators with constant coefficients has been derived for polynomial basis functions. The newly derived particular solutions are further coupled with the method of particular solutions (MPS) for numerically solving a large class of elliptic partial differential equations. In contrast to the use of Chebyshev polynomial basis functions, the proposed approach is more flexible in selecting the collocation points inside the domain. Polynomial basis functions are well-known for yielding ill-conditioned systems when their …

Eignefunctions For Partial Differential Equations On Two-Dimensional Domains With Piecewise Constant Coefficients, Abdullah Muheel Momit Aurko

Master's Theses

In this thesis, we develop a highly accurate and efficient algorithm for computing the solution of a partial differential equation defined on a two-dimensional domain with discontinuous coefficients. An example of such a problem is for modeling the diffusion of heat energy in two space dimensions, in the case where the spatial domain represents a medium consisting of two different but homogeneous materials, with periodic boundary conditions.

Since diffusivity changes based on the material, it will be represented using a piecewise constant function, and this results in the formation of a complicated mathematical model. Such a model is impossible to …

Numerical Methods For Non-Divergence Form Second Order Linear Elliptic Partial Differential Equations And Discontinuous Ritz Methods For Problems From The Calculus Of Variations, Stefan Raymond Schnake

Doctoral Dissertations

This dissertation consists of three integral parts. Part one studies discontinuous Galerkin approximations of a class of non-divergence form second order linear elliptic PDEs whose coefficients are only continuous. An interior penalty discontinuous Galerkin (IP-DG) method is developed for this class of PDEs. A complete analysis of the proposed IP-DG method is carried out, which includes proving the stability and error estimate in a discrete W^2;p-norm [W^2,p-norm]. Part one also studies the convergence of the vanishing moment method for this class of PDEs. The vanishing moment method refers to a PDE technique for approximating these PDEs by a …

Regularized Solutions For Terminal Problems Of Parabolic Equations., Sujeewa Indika Hapuarachchi

Electronic Theses and Dissertations

The heat equation with a terminal condition problem is not well-posed in the sense of Hadamard so regularization is needed. In general, partial differential equations (PDE) with terminal conditions are those in which the solution depends uniquely but not continuously on the given condition. In this dissertation, we explore how to find an approximation problem for a nonlinear heat equation which is well-posed. By using a small parameter, we construct an approximation problem and use a modified quasi-boundary value method to regularize a time dependent thermal conductivity heat equation and a quasi-boundary value method to regularize a space dependent thermal …

Some Problems Arising From Mathematical Model Of Ductal Carcinoma In Situ., Heng Li

Electronic Theses and Dissertations

Ductal carcinoma in situ (DCIS) is the earliest form of breast cancer. Three mathematical models in the one dimensional case arising from DCIS are proposed. The first two models are in the form of parabolic equation with initial and known moving boundaries. Direct and inverse problems are considered in model 1, existence and uniqueness are proved by using tool from heat potential theory and Volterra integral equations. Also, we discuss the direct problem and nonlocal problem of model 2, existence and uniqueness are proved. And approximation solution of these problems are implemented by Ritz-Galerkin method, which is the first attempt …

Numerically Solving A System Of Pdes Modeling Chronic Wounds Treated With Oxygen Therapy, Stefan Stryker

Mahurin Honors College Capstone Experience/Thesis Projects

Chronic wounds such as diabetic foot ulcers are the leading cause of non-traumatic amputations in developed countries. For researchers to better understand the physiology of these wounds, a mathematical model describing oxygen levels at the wound site can be used to help predict healing responses. The model utilizes equations that are modified from work by Guffey (2015) that consists of four variables – oxygen, bacteria, neutrophils, and chemoattractant within a system of partial differential equations. Our research focuses on numerically solving these partial differential equations using a finite volume approach. This numerical solver will be important for future research in …

Elimination For Systems Of Algebraic Differential Equations, Richard Gustavson

Dissertations, Theses, and Capstone Projects

We develop new upper bounds for several effective differential elimination techniques for systems of algebraic ordinary and partial differential equations. Differential elimination, also known as decoupling, is the process of eliminating a fixed subset of unknown functions from a system of differential equations in order to obtain differential algebraic consequences of the original system that do not depend on that fixed subset of unknowns. A special case of differential elimination, which we study extensively, is the question of consistency, that is, if the given system of differential equations has a solution. We first look solely at the ``algebraic data" of …

Application Of Kudryashov Method For The Ito Equations, Mozhgan Akbari

Applications and Applied Mathematics: An International Journal (AAM)

In this present work, the Kudryashov method is used to construct exact solutions of the (1+1)- dimensional and the (1+2)-dimensional form of the generalized Ito integro-differential equation. The Kudryashov method is a powerful method for obtaining exact solutions of nonlinear evolution equations. This method can be applied to non-integrable equations as well as integrable ones.