Partial Differential Equations Commons^™

Godunov-Type Upwind Flux Schemes Of The Two-Dimensional Finite Volume Discrete Boltzmann Method, Leitao Chen, Laura Schaefer

Publications

A simple unified Godunov-type upwind approach that does not need Riemann solvers for the flux calculation is developed for the finite volume discrete Boltzmann method (FVDBM) on an unstructured cell-centered triangular mesh. With piecewise-constant (PC), piecewise-linear (PL) and piecewise-parabolic (PP) reconstructions, three Godunov-type upwind flux schemes with different orders of accuracy are subsequently derived. After developing both a semi-implicit time marching scheme tailored for the developed flux schemes, and a versatile boundary treatment that is compatible with all of the flux schemes presented in this paper, numerical tests are conducted on spatial accuracy for several single-phase flow problems. Four major …

The Pope's Rhinoceros And Quantum Mechanics, Michael Gulas

Honors Projects

In this project, I unravel various mathematical milestones and relate them to the human experience. I show and explain the solution to the Tautochrone, a popular variation on the Brachistochrone, which details a major battle between Leibniz and Newton for the title of inventor of Calculus. One way to solve the Tautochrone is using Laplace Transforms; in this project I expound on common functions that get transformed and how those can be used to solve the Tautochrone. I then connect the solution of the Tautochrone to clock making. From this understanding of clocks, I examine mankind’s understanding of time and …

Swelling As A Stabilizing Mechanism During Ion Bombardment Of Thin Films: An Analytical And Numerical Study, Jennifer M. Swenson

Mathematics Theses and Dissertations

Irradiation of semiconductor surfaces often leads to the spontaneous formation of rippled structures at certain irradiation angles. However, at high enough energies, these structures are observed to vanish for all angles, despite the absence of any identified, universally-stabilizing physical mechanisms in operation. Here, we examine the effect on pattern formation of radiation-induced swelling, which has been excluded from prior treatments of stress in irradiated films. After developing a suitable continuum model, we perform a linear stability analysis to determine its effect on stability. Under appropriate simplifying assumptions, we find swelling indeed to be stabilizing at wavenumbers typical of experimental observations. …

Evaporation Of A Sessile Droplet On A Slope, Mitch Timm

Dissertations, Master's Theses and Master's Reports

We theoretically examine the drying of a stationary liquid droplet on an inclined surface. Both analytical and numerical approaches are considered, while assuming that the evaporation results from a purely diffusive transport of the liquid vapor and that the contact line is a pinned circle. For the purposes of our analytical calculations, we suppose that the effect of gravity relative to the surface tension is weak, i.e. the Bond number (Bo) is small. Then, we express the shape of the drop and the vapor concentration field as perturbation expansions in terms of Bo. When the Bond number is zero, the …

High Accuracy Methods And Regularization Techniques For Fluid Flows And Fluid-Fluid Interaction, Mustafa Aggul

Dissertations, Master's Theses and Master's Reports

This dissertation contains several approaches to resolve irregularity issues of CFD problems, including a decoupling of non-linearly coupled fluid-fluid interaction, due to high Reynolds number. New models present not only regularize the linear systems but also produce high accurate solutions both in space and time. To achieve this goal, methods solve a computationally attractive artificial viscosity approximation of the target problem, and then utilize a correction approach to make it high order accurate. This way, they all allow the usage of legacy code | a frequent requirement in the simulation of fluid flows in complex geometries. In addition, they all …

Nonlinear Coupled Effects In Nanomaterials, Sia Bhowmick

Theses and Dissertations (Comprehensive)

Materials at the nanoscale have different chemical, structural, and optoelectrical properties compared to their bulk counterparts. As a result, such materials, called nanomaterials, exhibit observable differences in certain physical phenomena. One such resulting phenomenon called the piezoelectric effect has played a crucial role in miniature self-powering electronic devices called nanogenerators which are fabricated by using nanostructures, such as nanowires, nanorods, and nanofilms. These devices are capable of harvesting electrical energy by inducing mechanical strain on the individual nanostructures. Electrical energy created in this manner does not have environmental limitations. In this thesis, important coupled effects, such as the nonlinear piezoelectric …

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

LSU 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 …

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 …

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.

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.

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 …

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 …

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. …

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 …

Asymptotic Behavior Of Waves In A Nonuniform Medium, Nezam Iraniparast, Lan Nguyen, Mikhail Khenner

Applications and Applied Mathematics: An International Journal (AAM)

An incoming wave on an infinite string, that has uniform density except for one or two jump discontinuities, splits into transmitted and reflected waves. These waves can explicitly be described in terms of the incoming wave with changes in the amplitude and speed. But when a string or membrane has continuous inhomogeneity in a finite region the waves can only be approximated or described asymptotically. Here, we study the cases of monochromatic waves along a nonuniform density string and plane waves along a membrane with nonuniform density. In both cases the speed of the physical system is assumed to tend …

Effect Of Nonlinear Thermal Radiation On Mhd Chemically Reacting Maxwell Fluid Flow Past A Linearly Stretching Sheet, A. M. Ramireddy, J. V. Ramana Reddy, N. Sandeep, V. Sugunamma

Applications and Applied Mathematics: An International Journal (AAM)

This communication addresses the influence of nonlinear thermal radiation on magneto hydrodynamic Maxwell fluid flow past a linearly stretching surface with heat and mass transfer. The effects of heat generation/absorption and chemical reaction are taken into account. At first, we converted the governing partial differential equations into nonlinear ordinary differential equations with the help of suitable similarity transformations and solved by using Runge-Kutta based shooting technique. Further, the effects of various physical parameters on velocity, temperature and concentration fields were discussed thoroughly with the help of graphs obtained by using bvp5c MATLAB package. In view of many engineering applications we …

New Structure For Exact Solutions Of Nonlinear Time Fractional Sharma-Tasso-Olver Equation Via Conformable Fractional Derivative, Hadi Rezazadeh, Farid S. Khodadad, Jalil Manafian

Applications and Applied Mathematics: An International Journal (AAM)

In this paper new fractional derivative and direct algebraic method are used to construct exact solutions of the nonlinear time fractional Sharma-Tasso-Olver equation. As a result, three families of exact analytical solutions are obtained. The results reveal that the proposed method is very effective and simple for obtaining approximate solutions of nonlinear fractional partial differential equations.

An Rbf Interpolation Blending Scheme For Effective Shock-Capturing, M. Harris, Eduardo Divo, Alain J. Kassab

Publications

In recent years significant focus has been given to the study of Radial basis functions (RBF), especially in their use on solving partial differential equations (PDE). RBF have an impressive capability of inter- polating scattered data, even when this data presents localized discontinuities. However, for infinitely smooth RBF such as the Multiquadrics, inverse Multiquadrics, and Gaussian, the shape parameter must be chosen properly to obtain accurate approximations while avoiding ill-conditioning of the interpolating matrices. The optimum shape parameter can vary significantly depending on the field, particularly in locations of steep gradients, shocks, or discontinuities. Typically, the shape parameter is chosen …

A Coupled Localized Rbf Meshless/Drbem Formulation For Accurate Modeling Of Incompressible Fluid Flows, Leonardo Bueno, Eduardo Divo, Alain J. Kassab

Publications

Velocity-pressure coupling schemes for the solution of incompressible fluid flow problems in Computational Fluid Dynamics (CFD) rely on the formulation of Poisson-like equations through projection methods. The solution of these Poisson-like equations represent the pressure correction and the velocity correction to ensure proper satisfaction of the conservation of mass equation at each step of a time-marching scheme or at each level of an iteration process. Inaccurate solutions of these Poisson-like equations result in meaningless instantaneous or intermediate approximations that do not represent the proper time-accurate behavior of the flow. The fact that these equations must be solved to convergence at …

Orbital Stability Results For Soliton Solutions To Nonlinear Schrödinger Equations With External Potentials, Joseph B. Lindgren

Theses and Dissertations--Mathematics

For certain nonlinear Schroedinger equations there exist solutions which are called solitary waves. Addition of a potential $V$ changes the dynamics, but for small enough $||V||_{L^\infty}$ we can still obtain stability (and approximately Newtonian motion of the solitary wave's center of mass) for soliton-like solutions up to a finite time that depends on the size and scale of the potential $V$. Our method is an adaptation of the well-known Lyapunov method.

For the sake of completeness, we also prove long-time stability of traveling solitons in the case $V=0$.

A Physics-Based Approach To Modeling Wildland Fire Spread Through Porous Fuel Beds, Tingting Tang

Theses and Dissertations--Mechanical and Aerospace Engineering

Wildfires are becoming increasingly erratic nowadays at least in part because of climate change. CFD (computational fluid dynamics)-based models with the potential of simulating extreme behaviors are gaining increasing attention as a means to predict such behavior in order to aid firefighting efforts. This dissertation describes a wildfire model based on the current understanding of wildfire physics. The model includes physics of turbulence, inhomogeneous porous fuel beds, heat release, ignition, and firebrands. A discrete dynamical system for flow in porous media is derived and incorporated into the subgrid-scale model for synthetic-velocity large-eddy simulation (LES), and a general porosity-permeability model is …

Complex Solutions Of The Time Fractional Gross-Pitaevskii (Gp) Equation With External Potential By Using A Reliable Method, Nasir Taghizadeh, Mona N. Foumani

Applications and Applied Mathematics: An International Journal (AAM)

In this article, modified (G'/G )-expansion method is presented to establish the exact complex solutions of the time fractional Gross-Pitaevskii (GP) equation in the sense of the conformable fractional derivative. This method is an effective method in finding exact traveling wave solutions of nonlinear evolution equations (NLEEs) in mathematical physics. The present approach has the potential to be applied to other nonlinear fractional differential equations. Based on two transformations, fractional GP equation can be converted into nonlinear ordinary differential equation of integer orders. In the end, we will discuss the solutions of the fractional GP equation with external potentials.

Impact Of Permeable Lining Of The Wall On The Peristaltic Flow Of Herschel Bulkley Fluid, G. C. Sankad, Asha Patil

Applications and Applied Mathematics: An International Journal (AAM)

The peristaltic motion is modeled for the Herschel Bulkley fluid, considered to flow in a non-uniform inclined channel. The channel wall is supposed to be lined with a non-erodible porous material. The flow is considered to be moving in a wave frame of reference moving with same velocity as of the sinusoidal wave. Low Reynolds number and long wave length assumptions are made to solve the model. Analytical solution is obtained for the pressure difference and also for the frictional force. Graphs are plotted, using Mathematica software, for both the results of pressure difference and frictional force against time average …

On The Propagation Of Atmospheric Gravity Waves In A Non-Uniform Wind Field: Introducing A Modified Acoustic-Gravity Wave Equation, Ahmad Talaei

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

Atmospheric gravity waves play fundamental roles in a broad-range of dynamical processes extending throughout the Earth’s neutral atmosphere and ionosphere. In this paper, we present a modified form for the acoustic-gravity wave equation and its dispersion relationships for a compressible and non-stationary atmosphere in hydrostatic balance. Importantly, the solutions have been achieved without the use of the well-known Boussinesq approximation which have been used extensively in previous studies.

We utilize the complete set of governing equations for a compressible atmosphere with non-uniform airflows to determine an equation for vertical velocity of possible atmospheric waves. This intricate wave equation is simplified …

Model For Computing Kinetics Of The Graphene Edge Epitaxial Growth On Copper, Mikhail Khenner

Mathematics Faculty Publications

A basic kinetic model that incorporates a coupled dynamics of the carbon atoms and dimers ona copper surface is used to compute growth of a single-layer graphene island. The speed of theisland's edge advancement on Cu[111] and Cu[100] surfaces is computed as a function of the growthtemperature and pressure. Spatially resolved concentration pro les of the atoms and dimers aredetermined, and the contributions provided by these species to the growth speed are discussed.Island growth in the conditions of a thermal cycling is studied.

Approximate Analytical Solution Of Boussinesq Equation In Homogeneous Medium With Leaky Base, Rajeev K. Bansal

Applications and Applied Mathematics: An International Journal (AAM)

Approximate analytical solutions of Boussinesq equation are widely used for approximation of subsurface seepage flow in confined and unconfined aquifers under varying hydrological conditions. In this paper, we use a 2-dimensional linearized Boussinesq equation to simulate the water table fluctuations in an isotropic aquifer overlying a semi pervious bed under multiple localized recharge and withdrawal. The unconfined aquifer is considered to be in contact with two water bodies of constant water head along opposite cost lines, while the remaining two faces have no flow condition. The mathematical model is solved analytically using finite Fourier sine transform and the application of …

Theoretical Modeling Of The Effect Of Polymer Chain Immobilization Rates On Holographic Recording In Photopolymers, Dana Mackey, Paul O'Reilly, Izabela Naydenova

Articles

This paper introduces an improved mathematical model for holographic grating formation in an acrylamide-based photopolymer, which consists of partial differential equations derived from physical laws. The model is based on the two-way diffusion theory of \cite{izabela}, which assumes short polymer chains are free to diffuse, and generalizes a similar model presented in \cite{josab} by introducing an immobilization rate governed by chain growth and cross-linking. Numerical simulations were carried out in order to investigate the behaviour of the photopolymer system for short and long exposures, with particular emphasis on the effect of recording parameters (such as illumination frequency and intensity), as …

The Global Stability Of The Solution To The Morse Potential In A Catastrophic Regime, Weerapat Pittayakanchit

HMC Senior Theses

Swarms of animals exhibit aggregations whose behavior is a challenge for mathematicians to understand. We analyze this behavior numerically and analytically by using the pairwise interaction model known as the Morse potential. Our goal is to prove the global stability of the candidate local minimizer in 1D found in A Primer of Swarm Equilibria. Using the calculus of variations and eigenvalues analysis, we conclude that the candidate local minimizer is a global minimum with respect to all solution smaller than its support. In addition, we manage to extend the global stability condition to any solutions whose support has a single …

Stereographic Visualization Of Bose-Einstein Condensate Clouds To Measure The Gravitational Constant, Ed Wesley Wells

College of Graduate Studies: Theses & Dissertations

This thesis describes a set of tools that can be used for the rapid design of atom interferometer schemes suitable for measuring Newton's Universal Gravitation constant also known as "Big G". This tool set is especially applicable to Bose--Einstein--condensed systems present in NASA's Cold Atom Laboratory experiment to be deployed to the International Space Station in 2017. These tools include a method of approximating the solutions of the nonlinear Schrödinger or Gross--Pitaevskii equation (GPE) using the Lagrangian Variational Method. They also include a set of software tools for translating the approximate solutions of the GPE into images of the optical …

The Dynamics Of Flat Surface Internal Geophysical Waves With Currents, Alan Compelli, Rossen Ivanov

Articles

A two-dimensional water wave system is examined consisting of two discrete incompressible fluid domains separated by a free common interface. In a geophysical context this is a model of an internal wave, formed at a pycnocline or thermocline in the ocean. The system is considered as being bounded at the bottom and top by a flatbed and wave-free surface respectively. A current profile with depth-dependent currents in each domain is considered. The Hamiltonian of the system is determined and expressed in terms of canonical wave-related variables. Limiting behavior is examined and compared to that of other known models. The linearised …