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Partial Differential Equations Commons™

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2016

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Articles 31 - 50 of 50

Full-Text Articles in Partial Differential Equations

Models Of Internal Waves In The Presence Of Currents, Alan Compelli, Rossen Ivanov Jan 2016

Models Of Internal Waves In The Presence Of Currents, Alan Compelli, Rossen Ivanov

Conference papers

A fluid system consisting of two domains is examined. The system is considered as being bounded at the bottom and top by a flatbed and wave-free surface respectively. An internal wave propagating in one direction, driven by gravity, acts as a free common interface between the fluids. Various current profiles are considered. The Hamiltonian of the system is determined and expressed in terms of canonical wave-related variables. Limiting behaviour is examined and compared to that of other known models. The linearised equations as well as long-wave approximations are formulated. The presented models provide potential applications to modelling of internal geophysical …

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The Global Stability Of The Solution To The Morse Potential In A Catastrophic Regime, Weerapat Pittayakanchit Jan 2016

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 …

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4. Dragging Along, Ruth Dover Jan 2016

4. Dragging Along, Ruth Dover

Differential Equations

More information on air drag.

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Steady State Solutions For A System Of Partial Differential Equations Arising From Crime Modeling, Bo Li Jan 2016

Steady State Solutions For A System Of Partial Differential Equations Arising From Crime Modeling, Bo Li

HMC Senior Theses

I consider a model for the control of criminality in cities. The model was developed during my REU at UCLA. The model is a system of partial differential equations that simulates the behavior of criminals and where they may accumulate, hot spots. I have proved a prior bounds for the partial differential equations in both one-dimensional and higher dimensional case, which proves the attractiveness and density of criminals in the given area will not be unlimitedly high. In addition, I have found some local bifurcation points in the model.

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Project Oasis: Optimizing Aquaponic Systems To Improve Sustainability, Siddharth Nigam, Paige Balcom Jan 2016

Project Oasis: Optimizing Aquaponic Systems To Improve Sustainability, Siddharth Nigam, Paige Balcom

Honors Theses and Capstones

Started in Fall 2015, Project OASIS (Optimizing Aquaponic Systems to Improve Sustainability) is an interdisciplinary capstone project with the goal of designing a sustainable and affordable small-scale aquaponic system for use in developing nations to tackle the problems of malnutrition and food insecurity. Aquaponics is a symbiotic relationship between fish and vegetables growing together in a recirculating system. The project’s goals were to minimize energy consumption and construction costs while using universally available materials. The computational fluid dynamics (CFD) software OpenFOAM was used to create transient and steady-state models of fish tanks to visualize velocity profiles, streamlines, and particle movement. …

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Homogenization Of Stokes Systems With Periodic Coefficients, Shu Gu Jan 2016

Homogenization Of Stokes Systems With Periodic Coefficients, Shu Gu

Theses and Dissertations--Mathematics

In this dissertation we study the quantitative theory in homogenization of Stokes systems. We study uniform regularity estimates for a family of Stokes systems with rapidly oscillating periodic coefficients. We establish interior Lipschitz estimates for the velocity and L∞ estimates for the pressure as well as Liouville property for solutions in ℝd. We are able to obtain the boundary W{1,p} estimates in a bounded C1 domain for any 1 < p < ∞. We also study the convergence rates in L2 and H1 of Dirichlet and Neumann problems for Stokes systems with rapidly oscillating periodic coefficients, without any regularity assumptions on the coefficients.

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Black-Scholes Equation And Heat Equation, Charles D. Joyner Jan 2016

Black-Scholes Equation And Heat Equation, Charles D. Joyner

Honors College Theses

First, we present and define the Black-Scholes equation which is used to model assets on the stock market. After that, we derive the heat equation that describes how the temperature increases through a homogeneous material. Finally, we detail how the two equations are related. We introduce and relate the Black-Scholes equation and Heat Equation.

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Stereographic Visualization Of Bose-Einstein Condensate Clouds To Measure The Gravitational Constant, Ed Wesley Wells Jan 2016

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 …

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Hamiltonian Formulation For Wave-Current Interactions In Stratified Rotational Flows, Adrian Constantin, Rossen Ivanov, Calin-Iulian Martin Jan 2016

Hamiltonian Formulation For Wave-Current Interactions In Stratified Rotational Flows, Adrian Constantin, Rossen Ivanov, Calin-Iulian Martin

Articles

We show that the Hamiltonian framework permits an elegant formulation of the nonlinear governing equations for the coupling between internal and surface waves in stratified water flows with piecewise constant vorticity.

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On The N-Wave Equations With Pt-Symmetry, Vladimir Gerdjikov, Georgi Grahovski, Rossen Ivanov Jan 2016

On The N-Wave Equations With Pt-Symmetry, Vladimir Gerdjikov, Georgi Grahovski, Rossen Ivanov

Articles

We study extensions of N-wave systems with PT-symmetry. The types of (nonlocal) reductions leading to integrable equations invariant with respect to P- (spatial reflection) and T- (time reversal) symmetries is described. The corresponding constraints on the fundamental analytic solutions and the scattering data are derived. Based on examples of 3-wave (related to the algebra sl(3,C)) and 4-wave (related to the algebra so(5,C)) systems, the properties of different types of 1- and 2-soliton solutions are discussed. It is shown that the PT symmetric 3-wave equations may have regular multi-soliton solutions for some specific choices of their parameters.

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The Dynamics Of Flat Surface Internal Geophysical Waves With Currents, Alan Compelli, Rossen Ivanov Jan 2016

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 …

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Controllability Of Neutral Stochastic Integro-Differential Evolution Equations Driven By A Fractional Brownian Motion, El Hassan Lakhel, Mark A. Mckibben Jan 2016

Controllability Of Neutral Stochastic Integro-Differential Evolution Equations Driven By A Fractional Brownian Motion, El Hassan Lakhel, Mark A. Mckibben

Mathematics Faculty Publications

We establish sufficient conditions for the controllability of a certain class of neutral stochastic functional integro-differential evolution equations in Hilbert spaces. The results are obtained using semigroup theory, resolvent operators and a fixed-point technique. An application to neutral partial integro-differential stochastic equations perturbed by fractional Brownian motion is given.

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A Discontinuous Galerkin Method For Unsteady Two-Dimensional Convective Flows, Andreas C. Aristotelous, N. C. Papanicolaou Jan 2016

A Discontinuous Galerkin Method For Unsteady Two-Dimensional Convective Flows, Andreas C. Aristotelous, N. C. Papanicolaou

Mathematics Faculty Publications

We develop a High-Order Symmetric Interior Penalty (SIP) Discontinuous Galerkin (DG) Finite Element Method (FEM) to investigate two-dimensional in space natural convective flows in a vertical cavity. The physical problem is modeled by a coupled nonlinear system of partial differential equations and admits various solutions including stable and unstable modes in the form of traveling and/or standing waves, depending on the governing parameters. These flows are characterized by steep boundary and internal layers which evolve with time and can be well-resolved by high-order methods that also are adept to adaptive meshing. The standard no-slip boundary conditions which apply on the …

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2. Population, Ruth Dover Jan 2016

2. Population, Ruth Dover

Differential Equations

Introduction to logistic population growth.

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3: Drugs And De's, Ruth Dover Jan 2016

3: Drugs And De's, Ruth Dover

Differential Equations

Making a connection between discrete recursion and differential equations.

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1. Coffee, Ruth Dover Jan 2016

1. Coffee, Ruth Dover

Differential Equations

Newton’s Law of Cooling.

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A High Accuracy Minimally Invasive Regularization Technique For Navier-Stokes Equations At High Reynolds Number, Mustafa Aggul Jan 2016

A High Accuracy Minimally Invasive Regularization Technique For Navier-Stokes Equations At High Reynolds Number, Mustafa Aggul

Dissertations, Master's Theses and Master's Reports

A method is presented, that combines the defect and deferred correction approaches to approximate solutions of Navier-Stokes equations at high Reynolds number. The method is of high accuracy in both space and time, and it allows for the usage of legacy codes (a frequent requirement in the simulation of turbulent flows in complex geometries). The two-step method is considered here; in order to obtain a regularization that is second order accurate in space and time, the method computes a low-order accurate, stable and computationally inexpensive approximation (Backward Euler with artificial viscosity) twice. The results are readily extendable to the higher …

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Galvanically Induced/Accelerated Crevice Corrosion, Zachary R. Roland Jan 2016

Galvanically Induced/Accelerated Crevice Corrosion, Zachary R. Roland

Williams Honors College, Honors Research Projects

In this thesis, a one dimensional model is developed to investigate the initial stages of corrosion in a fastener assembly consisting of a stainless steel fastener and aluminum 7075 as the plate. Differential equations are formulated and solved to determine the profiles for the potential, the oxygen concentration, and the aluminum ion concentration in the crevice, and also the potential in the bulk electrolyte. This fastener system exhibits galvanic corrosion, pitting corrosion, and crevice corrosion. It is found that the potential decreases monotonically down the length of the crevice, the oxygen concentration decreases exponentially down the length of the crevice, …

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Effects Of Invasion Timing In A One-Dimensional Model Of Competing Species With An Infectious Disease, Eliza Jacops Jan 2016

Effects Of Invasion Timing In A One-Dimensional Model Of Competing Species With An Infectious Disease, Eliza Jacops

Williams Honors College, Honors Research Projects

In combining two classes of models, we are able to analyze the dynamics of two species that compete for the same resources while fighting a disease. The native species is the disease host and the invasive species enters their habitat and encounters the disease for the first time. Their natural response is to evolve resistance to the disease, and this can assist in their invasion of the natives' habitat. We find conditions for coexistence of the two species, conditions under which an invasion would succeed and wipe out all native individuals, and conditions under which the invasion fails. We explore …

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Defect-Deferred Correction Method For The Two-Domain Convection-Dominated Convection-Diffusion Problem, Dilek Erkmen Jan 2016

Defect-Deferred Correction Method For The Two-Domain Convection-Dominated Convection-Diffusion Problem, Dilek Erkmen

Dissertations, Master's Theses and Master's Reports

We present a method for solving a fluid-fluid interaction problem (two convection-dominated convection-diusion problems adjoined by an interface), which is a simplifed version of the atmosphere ocean coupling problem. The method resolves some of the issues that can be crucial to the fluid-fluid interaction problems: it is a partitioned time stepping method, yet it is of high order accuracy in both space and time (the two-step algorithm considered in this report provides second order accuracy); it allows for the usage of the legacy codes (which is a common requirement when resolving flows in complex geometries), yet it can be applied …

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