Partial Differential Equations Commons^™

(Si14-11) Analysing Co-Current Imbibition Phenomenon In Heterogeneous Reservoir Using Multistep Hybrid Differential Transform Finite Difference Method, Aruna Sharma, Amit Parikh

Applications and Applied Mathematics: An International Journal (AAM)

The primary focus of this study is to analyse the co-current imbibition phenomenon in an inclined heterogeneous reservoir. This phenomenon occurs during the secondary oil recovery process. Capillary force is responsible for the displacement of a non-wetting phase by a wetting phase, and this phenomenon is called spontaneous imbibition. Imbibition is of two types and can be differentiated based on the direction in which the wetting phase (water) and non-wetting phase (oil) move. If the two phases flow in the same direction, it is called co-current imbibition, and if they flow in the opposite direction, it is called counter-current imbibition. …

(Si14-06) Influence Of Hall Effects On Rotating Mhd Casson Fluid Flow Over A Vertical Plate, Zankhana Mali, Akhil Mittal, Harshad Patel

Applications and Applied Mathematics: An International Journal (AAM)

Studies of Hall effects regarding motion of fluid due to some external forces in MHD transport of reacting Casson fluid with heat generation over an impulsively emerging vertical plate are considered in this work. This theory proposes that because of a sudden rise in temperature and an accompanying surface concentration profile, which shows an elevation with time, the boundary plate has endured rapid expansion. In a rotational environment, this characteristic occurs homogeneously inside a porous uniform material. It applies the Laplace transform method for determining the fundamental equations subject to imposed starting and side conditions. Under isothermal conditions, accurate formulae …

Investigation Of Node Refinement Methods In Local Adaptive Kernel Based Approximation, Shelby W. Woodrum

Theses and Dissertations

This thesis explores computational efficiency and accuracy of six node refinement methods for local adaptive kernel-based approximations of solutions to the two-dimensional Poisson equation. Using an adaptive kernel-based approximation algorithm, this research investigates performance of Delaunay triangulation-based methods (shifted barycenters and edge midpoints), refinement via approximate Fekete and discrete Leja points, and a meshless predefined shift refinement method across two domains with varying complexities. Computational experiments reveal that Delaunay triangulation-based methods achieve a practical balance between accuracy and efficiency, particularly in square domains. Refinement via approximate Fekete and discrete Leja points produce accurate results but incur greater computational costs, making …

Analyzing And Comparing Refinement Indicators For Rbf-Fd Adaptive Algorithms, Anders R. Johnson

Theses and Dissertations

Recent progress has been made in the development of collocation-based iterative algorithms that approximate solutions to PDEs. These algorithms rely on the ability to identify regions within a domain where a finer discretization is required. Such iterative algorithms are beneficial particularly when solution functions have highly localized behavior. This thesis proposes an indicator for node refinement that is constructed by approximating the forward error. This proposed indicator also helps to establish confidence in the accuracy of a given solution estimate. The proposed error estimator is theoretically examined and compared with contemporary refinement indicators. It is shown that an iterative algorithm, …

Mathematical Modelling Of Hybrid Photonic Structures For Holographic Sensors, Jack Lyons

Doctoral

This thesis outlines a mathematical framework for modelling the formation of holographic gratings in hybrid photopolymer based nanocomposites with the aim of optimising their holographic recording properties for optical sensing applications. Thus, the second aim of the work is to model the change in optical properties of the grating in response to exposure to a target analyte. This work has been a collaborative research project between the School of Mathematics & Statistics at Technological University Dublin and the Centre for Industrial and Engineering Optics that have done extensive experimental work with holographic gratings recorded in photopolymer materials.

In recent years, …

Long-Time Asymptotics For The Kadomtsev–Petviashvili I Equation With Small Initial Data, Samir Donmazov

Theses and Dissertations--Mathematics

We study the initial value problem for the Kadomtsev--Petviashvili I (KP I) equation (u_t + 6uu_x + u_xxx)_x = 3u_yy with small initial data belonging to a subspace of the energy space for the KP I equation. We establish the long-time asymptotics for solutions of the KP I equation using the inverse scattering transform formalism developed by Zhou. Within this framework, the inverse problem for the KP I equation is formulated as a nonlocal Riemann--Hilbert problem (RHP) in two spatial dimensions. As part of the asymptotic analysis, we determine the long-time behavior of the …

Ergodic Switching Control For Markov-Feller Processes Ii, Jose L. Menaldi, Maurice Robin

Mathematics Faculty Research Publications

This is the continuation of Part I [14], where we considered control problems with long term average (or ergodic) cost for Markov switching processes (zt , nt ), nt being a discrete component with values in a finite set N . The control acts only on this discrete component and consists of immediate switching actions. We solve the ergodic problem in several situations extending previous works, mainly when zt is a reflected diffusion with or without jumps and when the set of control values is strictly smaller than N . In this Part II, we conclude our theoretical analysis with …

Ergodic Switching Control For Markov-Feller Processes I, Jose L. Menaldi, Maurice Robin

Mathematics Faculty Research Publications

We consider control problems with long term average (or ergodic) cost for Markov switching processes (zt, nt ), nt being a discrete component with values in a finite set N . The control acts only on this discrete component and consists of immediate switching actions. We solve the ergodic problem in several situations extending previous works, mainly when zt is a reflected diffusion with or without jumps and when the set of control values is strictly smaller than N .

Horizontal Infiltration Of Water Through Porous Snow As A Gravity Current, Anthony Cheng

Dartmouth College Master’s Theses

On the surface of the Greenland ice sheet or around the margins of the Antarctic ice shelf, water infiltrates porous ice. It is important to understand this infiltration process since water populating the pore space of ice directly impacts the density, porosity, and wetness of ice. These properties influence the mechanics and tensile strength of ice, as greater amounts of infiltration result in faster or more widespread deformation events, which may lead to adverse climatic effects such as sea level rise and ocean current disruption. While studies have considered the thermodynamics and fluid mechanics of water vertically percolating through snow …

Bound Preserving Discontinuous Galerkin Methods For Euler Equations And Nonequilibrium Flows, Fangyao Zhu

Dissertations, Master's Theses and Master's Reports

This dissertation is composed of four chapters in which we will closely examine the high order bound preserving discontinuous Galerkin methods for solving partial differential equations, specifically non-equilibrium chemical reacting flows and Euler equations under gravitational fields. A shared requirement between the two is the necessity for positive values of both density and pressure. Due to this physical nature of the two systems, constructing a positivity preserving scheme become very essential in our research.

For non-equilibrium flows where multi-reactions and multi-species are involved, we are also required to keep the bounds of the mass fraction of each species in between …

0th Order Solutions Of The Wavefunctions For The Quantum Elliptical Box And Microstrip Antenna, Nishtha Tikalal

Honors Undergraduate Theses

For a quantum particle confined to a two-dimensional elliptical box or electromagnetic wave in a microstrip antenna, geometrical and boundary condition interplay result in a spectrum of spatial patterns. Due to the asymmetrical nature of the ellipse, we are faced with continuous symmetry reductions, leaving both degenerate and nondegenerate solutions. Here, we present a complete derivation of an analytical solution and visualizations of the fundamental wavefunctions for both Dirichlet and Neumann boundary conditions respectively corresponding to the quantum elliptical box and the elliptical microstrip antenna.

We demonstrate that the eigenmodes, governed by eccentricity, directly correspond to the modal field distributions …

The Inverse Scattering Transform For The Nonlinear Schrödinger Equation, Ivan Casas-Rocha

Honors Undergraduate Theses

The Nonlinear Schrödinger (NLS) Equation, iψ_t + 1/2 ψ_xx ± |ψ|²ψ = 0, is a nonlinear partial differential equation which is used to model several physical phenomena including nonlinear effects inside optical fibers and the formation of rogue waves in shallow water. It is particu- larly difficult to study solutions to this equation due to the nonlinearity, and the nonlinearity leads to incredibly interesting solutions not found in linear PDEs such as solitons. In this thesis, we highlight two methods of obtaining solutions to the (NLS) equation: the Inverse Scattering Transform and the Dressing Method. Furthermore, …

Modelling The Formation Of Unslanted Holographic Gratings In Hybrid Photopolymer Media, Jack Lyons, Dana Mackey, Izabela Naydenova

Articles

The theoretical modelling of holographic recording in photopolymers has been an important tool in their optimisation. More complex, hybrid organic/inorganic photopolymers have been developed in pursuit of materials with higher sensitivity, low shrinkage, high dynamic range and environmental stability. Recent attempts to augment the existing models for the redistribution of inorganic nanoparticles in holographic recording were successful but there is still a knowledge gap in regards to modelling optical losses, mutual cross-diffusion, the formation of slanted holographic gratings and polymerization induced shrinkage in hybrid photopolymer media. This paper will describe a novel approach to modelling the formation of unslanted holographic …

Applying The Lagrangian Variational Method To Atom Interferometry, Jeffrey W. Heward

College of Graduate Studies: Theses & Dissertations

Atom interferometry in Bose-Einstein condensate systems has emerged as a promising technique for precision metrology. The dynamics of these systems are well-described by the Gross-Pitaevskii equation (GPE), but direct numerical solution of this equation is infeasible for realistic systems. We use the Lagrangian Variational Method (LVM) to approximate solutions of the GPE in 1D and 3D, and we compare the LVM results to the exact solutions. We also present 3D LVM results for a case where numerical solution of the GPE is infeasible.

Weathering And Beyond: Leveraging Mathematical Modeling To Simulate Erosion In Digital Media, Fiona Irving-Beck

Scripps Senior Theses

How might we bring an idea to life from both a mathematical and an artistic perspective? Within Weathering, I use imagery of environmental erosion to explore the differences between physical and digital forms of representation. I created a physical painting of an abandoned copper mine, digitized the work, and then used a mathematical model to digitally “erode” it, which I re-translated into paintings. While the explicit texture present in physical work speaks best to my practice/intent, the mathematical framework that is the basis for my digital work affords a powerful mode of temporal flexibility. Used in conjunction, these two …

Wildfire Modeling Using Systems Of Odes And Pdes, Michael A. Quindlen

EWU Masters Thesis Collection

No abstract provided.

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 …

Modeling Energetic Electron Precipitation: Radiation Belt Loss, Its Drivers, And Atmospheric Impacts, Zhi Gu Li

Graduate Theses, Dissertations, and Problem Reports (ETD)

Energetic electrons in the terrestrial outer radiation belt present significant hazards to spacecraft systems and human operations in space. The intensity of these electrons can vary rapidly and dramatically during geomagnetic storms, governed by a complex competition between acceleration and loss processes. Among these, precipitation into the atmosphere via resonant wave-particle interaction acts as a key loss mechanism. This dissertation focuses on improving the quantification of energetic electron precipitation using physics-based modeling constrained by low-altitude satellite observations.

We begin by developing and validating the Drift-Diffusion model, which simulates low-altitude electron dynamics while accounting for azimuthal drift, pitch-angle diffusion, and atmospheric …

Individual And Collective Properties Of Tunable Photochemical Belousov-Zhabotinsky Micro-Reactors, Kudakwashe Benedict Shumba

Graduate Theses, Dissertations, and Problem Reports (ETD)

Cell-like model chemical systems are powerful tools that can be used to explore the role of intercellular coupling on population level behaviors in communities of biological cells. Firstly, we present a new method for fabricating such micro-reactors using the photosensitive Belousov–Zhabotinsky (BZ) reaction system employed in silica microparticles. These BZ micro-reactors have a tunable response to photochemical coupling, varying from a fully excitatory response to a fully inhibitory response. Their response can be tuned through variations in either the reactive mixture or, on an individual micro-reactor level, by changes in the synthesis temperature used during the fabrication of the silica …

An Introduction To The Time-Independent Schrödinger Equation And Methods To Solve It, Vu Giang, Alex Gnech

OUR Journal: ODU Undergraduate Research Journal

The Time-Independent Schrödinger Equation is a linear elliptic PDE that describes quantum-mechanical systems. Its significance in the science of submicroscopic phenomena, particularly quantum mechanics, is as central as Newton’s laws of motion are to classical mechanics. This study uses various methods, including novel neural networks and finite difference schemes, to solve the one-dimensional two-body equation.