Applied Mathematics Commons^™

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 …

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 .

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 …

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 …

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

EWU Masters Thesis Collection

No abstract provided.

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 …

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 …

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, …

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 …

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.

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 …

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 …

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.

Asymptotic Expansion Of A Maier-Saupe Type Potential Near The Nematic-Isotropic Transition Point In Liquid Crystals, Ryan P. Oneill, Giangvuthanh Nguyen, Xiang Xu

OUR Journal: ODU Undergraduate Research Journal

In this paper we study a Maier-Saupe type bulk potential (Maier & Saupe, 1959) in the Landau-de Gennes free energy in the Q-tensor theory modeling nematic liquid crystal configurations. This potential was originally introduced in Katriel et al. (1986), which is considered as a natural enforcement of a physical constraint on the eigenvalues of symmetric, traceless Q-tensors. More specifically, we present a rigorous derivation of the asymptotic expansion of this singular potential near the nematic-isotropic transition point up to the 4-th order.

An Efficient Fourier Caching Algorithm For Walk On Spheres, Zihong Zhou

Dartmouth College Master’s Theses

Walk on Spheres (WoS) is a grid-free Monte Carlo method for solving elliptic partial differential equations (PDEs).
Rather than discretizing the domain, WoS leverages the mean-value principle to obtain Monte Carlo estimates by recursively averaging the solution over the largest contained sphere, terminating upon reaching the boundary.
Unfortunately, WoS requires many independent estimates to achieve noise-free results.

We propose an acceleration technique for WoS, inspired by irradiance caching methods, that computes the solution at a sparse set of locations, and extrapolates these cached values to local neighborhoods. A key insight is that WoS can be extended to compute not only …

Solving Fractional Differential Equations On A Quantum Computer: A Variational Approach, Fong Yew Leong, Dax Enshan Koh, Jian Feng Kong, Siong Thye Goh, Jun Yong Khoo, Wei Bin Ewe, Hongying Li, Jayne Thompson, Dario Poletti

Research Collection School Of Computing and Information Systems

We introduce an efficient variational hybrid quantum-classical algorithm designed for solving Caputo time-fractional partial differential equations. Our method employs an iterable cost function incorporating a linear combination of overlap history states. The proposed algorithm is not only efficient in terms of time complexity but also has lower memory costs compared to classical methods. Our results indicate that solution fidelity is insensitive to the fractional index and that gradient evaluation costs scale economically with the number of time steps. As a proof of concept, we apply our algorithm to solve a range of fractional partial differential equations commonly encountered in engineering …

Delta-Shaped Approximation Based Homotopy Analysis Method For Nonlinear Poisson-Type Partial Differential Equations, Cyril Ocloo

Dissertations

This research aims to solve nonlinear Poisson-type partial differential equations (PDEs) by the approach of the homotopy analysis method (HAM) incorporated with approximate particular solutions (APS) using Delta-shaped basis (DSB) approximations.

With the inclusion of the h auxiliary parameters, we tackle nonlinear problems by studying the mathematical characteristics of the h curve. This is to ensure the numerical convergence of the HAM.

In the solution process, we use the homotopy analysis method to convert a nonlinear PDE into linear inhomogeneous PDEs, which are solved using the method of approximate particular solutions with DSB.

A proper value of the h is …

On Weak Solutions And The Navier-Stokes Equations, Aryan Prabhudesai

Mathematical Sciences Undergraduate Honors Theses

In this paper, I will discuss a partial differential equation that has solutions that are discontinuous. This example motivates the need for distribution theory, which will provide an interpretation of what it means for a discontinuous function to be a “solution” to a PDE. Then I will give a detailed foundation of distributions, including the definition of the derivative of a distribution. Then I will introduce and give background on the Navier-Stokes equations. Following that, I will explain the Millennium Problem concerning global regularity for the Navier-Stokes equations and share mathematical results regarding weak solutions. Finally, I will go over …

Scalable Solution Of Time-Dependent Pdes Through Component-Wise Exponential Integrator, Chelsea Drum

Dissertations

Exponential integrators, such as exponential Runge-Kutta or Rosenbrock methods, are designed specifically for the time integration of stiff systems of ordinary differential equations (ODEs) and allow the use of larger time steps than other general-purpose ODE solvers. However, these methods rely on computing matrix function-vector products that are traditionally computed using a Krylov projection, such as Lanczos or Arnoldi iteration, that involves substantial computational expense at high spatial resolution. Krylov Subspace Spectral (KSS) methods' frequency-dependent approach, designed to circumvent stiffness in linear problems, computes these products with greater scalability. We propose the combination of such KSS methods with exponential integrators …

Resonant Solutions Of The Non-Linear Schrödinger Equation With Periodic Potential, Arein Duaibes, Yulia Karpeshina

Mathematics Faculty Publications

The goal is construction of stationary solutions close to non-trivial combinations of two plane waves at high energies for a periodic non-linear Schrödinger Equation in dimension two. The corresponding isoenergetic surface is described for any sufficiently large energy k2. It is shown that the isoenergetic surface corresponding to k2 is essentially different from that for the zero potential even for small potentials. We use a combination of the perturbative results obtained earlier for the linear case and a method of successive approximation.

Asymptotic Formula For Scattering Problems Related To Thin Metasurfaces, Zachary Jermain

LSU Doctoral Dissertations

The goal of this work is to develop an asymptotic formula for the behavior of a scattered electromagnetic field in the presence of a thin metamaterial known as a metasurface. By using a carefully chosen Green’s function and the single and double layer potentials we analyze the perturbed scattering problem in the presence of the metamaterial and a background scattering problem. By using Lippman-Schwinger type representation formulas for the two fields we develop the asymptotic formula for the perturbed field. From here we prove the asymptotic formula holds up to a specific error term based on the size of the …

Existence Of Smooth Solutions For The Landau Equation With Hard Potentials, Shelly Ann Taylor

Theses and Dissertations

This dissertation is concerned with the Landau equation, an integro-differential equation that models the particle density of a plasma as it evolves in phase space. The main topic is the (large-data) local existence of classical solutions to the Landau equation in the case of hard potentials (γ ∈ (0, 1]). Solutions have previously been constructed by Chaturvedi [SIAM. J. Math. Anal., 55(5), 5345–5385, 2023] for initial data in an exponentially-weighted Sobolev space of order 10, but it is not a priori clear whether these solutions have more regularity than the initial data. We improve Chaturvedi’s existence result in two ways: …

(R2076) New Exact Solution Of Gilson–Pickering Equation In Plasma, Bingnuo Yang, Weinan Wu, Hongfeng Yu, Peng Guo

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we use Paul-Painlev´e approach method, extended rational sine-cosine method and extended rational sinh-cosh method to construct the exact solution of the nonlinear Gilson-Pickering (GP) equation in plasma. The exact solution of GP equation obtained by the above three methods is new, and we use mathematical software to draw the two-dimensional and three-dimensional graphs of the new exact solutions. Through the study of nonlinear equations in plasma, this study will enrich the research and connotation of nonlinear development equations in plasma.

(R2067) Solutions Of Hyperbolic System Of Time Fractional Partial Differential Equations For Heat Propagation, Sagar Sankeshwari, Vinayak Kulkarni

Applications and Applied Mathematics: An International Journal (AAM)

Hyperbolic linear theory of heat propagation has been established in the framework of a Caputo time fractional order derivative. The solution of a system of integer and fractional order initial value problems is achieved by employing the Adomian decomposition approach. The obtained solution is in convergent infinite series form, demonstrating the method’s strengths in solving fractional differential equations. Moreover, the double Laplace transform method is employed to acquire the solution of a system of integer and fractional order boundary conditions in the Laplace domain. An inversion of double Laplace transforms has been achieved numerically by employing the Xiao algorithm in …

(R2074) A Comparative Study Of Two Novel Analytical Methods For Solving Time-Fractional Coupled Boussinesq-Burger Equation, Jyoti U. Yadav, Twinkle R. Singh

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, a comparative study between two different methods for solving nonlinear timefractional coupled Boussinesq-Burger equation is conducted. The techniques are denoted as the Natural Transform Decomposition Method (NTDM) and the Variational Iteration Transform Method (VITM). To showcase the efficacy and precision of the proposed approaches, a pair of different numerical examples are presented. The outcomes garnered indicate that both methods exhibit robustness and efficiency, yielding approximations of heightened accuracy and the solutions in a closed form. Nevertheless, the VITM boasts a distinct advantage over the NTDM by addressing nonlinear predicaments without recourse to the application of Adomian polynomials. …

(R2085) Heat And Mass Transport Characteristics In Williamson Fluid Flow Over A Permeable Stretching Cylinder, Amala Olkha, Mukesh Kumar

Applications and Applied Mathematics: An International Journal (AAM)

The intention of this research endeavor is to examine heat and mass transport in Williamson fluid flow induced by a permeable stretching cylinder in a porous medium. Various physical factors (like viscous dissipation, chemical reaction, etc.) affecting the relevant fields (flow, temperature and concentration) are incorporated in the investigation. The governing PDEs are turned into nondimensional ODEs using adequate similarity transformation relations, and then tackled numerically using MATLAB based Bvp4c technique along with shooting method. The impacts of various parameters arising in the problem are exhibited on fluid flow, temperature and concentration distribution by drawing portraits and discussed. Moreover, impressions …