Total Variation Flow In R^N Dimensions With Examples Relating To Perimeters Of Level Sets, 2024 University of Missouri-St. Louis

#### Total Variation Flow In R^N Dimensions With Examples Relating To Perimeters Of Level Sets, Luis Schneegans, Victoria Shumakovich

*Undergraduate Research Symposium*

In this project, we explore radial solutions to the Total Variation Flow (TVF) equation with the help of the Sign Fast Diffusion Equation (SFDE) and prior results in the 1-dimensional case. Specifically for radial solutions, we derive equations and explicit solutions relating to the n-dimensional case. Lastly, we look at how level sets and (time) profiles change.

Year-2 Progress Report On Numerical Methods For Bgk-Type Kinetic Equations, 2024 University of Tennessee, Knoxville

#### Year-2 Progress Report On Numerical Methods For Bgk-Type Kinetic Equations, Steven M. Wise, Evan Habbershaw

*Faculty Publications and Other Works -- Mathematics*

In this second progress report we expand upon our previous report and preliminary work. Specifically, we review some work on the numerical solution of single- and multi-species BGK-type kinetic equations of particle transport. Such equations model the motion of fluid particles via a density field when the kinetic theory of rarefied gases must be used in place of the continuum limit Navier-Stokes and Euler equations. The BGK-type equations describe the fluid in terms of phase space variables, and, in three space dimensions, require 6 independent phase-space variables (3 for space and 3 for velocity) for each species for accurate simulation. …

Using A Sand Tank Groundwater Model To Investigate A Groundwater Flow Model, 2024 Ball State University

#### Using A Sand Tank Groundwater Model To Investigate A Groundwater Flow Model, Christopher Evrard, Callie Johnson, Michael A. Karls, Nicole Regnier

*CODEE Journal*

A Sand Tank Groundwater Model is a tabletop physical model constructed of plexiglass and filled with sand that is typically used to illustrate how groundwater water flows through an aquifer, how water wells work, and the effects of contaminants introduced into an aquifer. Mathematically groundwater flow through an aquifer can be modeled with the heat equation. We will show how a Sand Tank Groundwater Model can be used to simulate groundwater flow through an aquifer with a no flow boundary condition.

Multiscale Modelling Of Brain Networks And The Analysis Of Dynamic Processes In Neurodegenerative Disorders, 2024 Wilfrid Laurier University

#### Multiscale Modelling Of Brain Networks And The Analysis Of Dynamic Processes In Neurodegenerative Disorders, Hina Shaheen

*Theses and Dissertations (Comprehensive)*

The complex nature of the human brain, with its intricate organic structure and multiscale spatio-temporal characteristics ranging from synapses to the entire brain, presents a major obstacle in brain modelling. Capturing this complexity poses a significant challenge for researchers. The complex interplay of coupled multiphysics and biochemical activities within this intricate system shapes the brain's capacity, functioning within a structure-function relationship that necessitates a specific mathematical framework. Advanced mathematical modelling approaches that incorporate the coupling of brain networks and the analysis of dynamic processes are essential for advancing therapeutic strategies aimed at treating neurodegenerative diseases (NDDs), which afflict millions of …

An Exposition Of The Curvature Of Warped Product Manifolds, 2023 California State University - San Bernardino

#### An Exposition Of The Curvature Of Warped Product Manifolds, Angelina Bisson

*Electronic Theses, Projects, and Dissertations*

The field of differential geometry is brimming with compelling objects, among which are warped products. These objects hold a prominent place in differential geometry and have been widely studied, as is evident in the literature. Warped products are topologically the same as the Cartesian product of two manifolds, but with distances in one of the factors in skewed. Our goal is to introduce warped product manifolds and to compute their curvature at any point. We follow recent literature and present a previously known result that classifies all flat warped products to find that there are flat examples of warped products …

Effects Of A Protection Zone In A Reaction-Diffusion Model With Strong Allee Effect., 2023 University of Louisville

#### Effects Of A Protection Zone In A Reaction-Diffusion Model With Strong Allee Effect., Isaac Johnson

*Electronic Theses and Dissertations*

A protection zone model represents a patchy environment with positive growth over the protection zone and strong Allee effect growth outside the protection zone. Generally, these models are considered through the corresponding eigenvalue problem, but that has certain limitations. In this thesis, a general protection zone model is considered. This model makes no assumption on the direction of the traveling wave solution over the Strong Allee effect patch. We use phase portrait analysis of this protection zone model to draw conclusions about the existence of equilibrium solutions. We establish the existence of three types of equilibrium solutions and the necessary …

Computational Modeling Using A Novel Continuum Approach Coupled With Pathway-Informed Neural Networks To Optimize Dynein-Mediated Centrosome Positioning In Polarized Cells, 2023 George Mason University

#### Computational Modeling Using A Novel Continuum Approach Coupled With Pathway-Informed Neural Networks To Optimize Dynein-Mediated Centrosome Positioning In Polarized Cells, Arkaprovo Ghosal, Padmanabhan Seshaiyar Dr., Adriana Dawes Dr., General Genomics Inc.

*Annual Symposium on Biomathematics and Ecology Education and Research*

No abstract provided.

Lnksc Method On Pde-Constrained Optimization For Mcf-7 Breast Cancer Cell Growth Predictions And Treatment Response With Gold Nanoparticles, 2023 Jarvis Christian College

#### Lnksc Method On Pde-Constrained Optimization For Mcf-7 Breast Cancer Cell Growth Predictions And Treatment Response With Gold Nanoparticles, Widodo Samyono, Shakhawat Bhuiyan

*Annual Symposium on Biomathematics and Ecology Education and Research*

No abstract provided.

Helices In Fluids And Applications To Modeling In Biology, 2023 James Madison University

#### Helices In Fluids And Applications To Modeling In Biology, Eva M. Strawbridge

*Annual Symposium on Biomathematics and Ecology Education and Research*

No abstract provided.

Effects Of Slip On Highly Viscous Thin-Film Flows Inside A Vertical Tube Of Constant Radius, 2023 Illinois State University

#### Effects Of Slip On Highly Viscous Thin-Film Flows Inside A Vertical Tube Of Constant Radius, Mark S. Schwitzerlett

*Annual Symposium on Biomathematics and Ecology Education and Research*

No abstract provided.

Computational Study Of Twin Circular Particles Settling In Fluid Using A Fictitious Boundary Approach, 2023 Department of Mathematics, Air University, PAF Complex, Islamabad 44000, Pakistan

#### Computational Study Of Twin Circular Particles Settling In Fluid Using A Fictitious Boundary Approach, Imran Abbas, Kamran Usman

*International Journal of Emerging Multidisciplinaries: Mathematics*

The objective of this study is to examine the performance of two adjacent solid particles as they settle in close nearness, with a focus on comprehending the intricate interactions between the particles and the surrounding fluid during the process of sediment transport. Simulations are conducted with different initial horizontal spacing between particles and Reynolds numbers (Re). The findings of the simulations highlight the impact of the initial spacing between particles and Reynolds numbers (Re) as key factors influencing the ultimate settling velocity and separation distance. In general, when the initial spacing between particles is small and the Reynolds number (Re) …

Data-Driven Exploration Of Coarse-Grained Equations: Harnessing Machine Learning, 2023 The University of Western Ontario

#### Data-Driven Exploration Of Coarse-Grained Equations: Harnessing Machine Learning, Elham Kianiharchegani

*Electronic Thesis and Dissertation Repository*

In scientific research, understanding and modeling physical systems often involves working with complex equations called Partial Differential Equations (PDEs). These equations are essential for describing the relationships between variables and their derivatives, allowing us to analyze a wide range of phenomena, from fluid dynamics to quantum mechanics. Traditionally, the discovery of PDEs relied on mathematical derivations and expert knowledge. However, the advent of data-driven approaches and machine learning (ML) techniques has transformed this process. By harnessing ML techniques and data analysis methods, data-driven approaches have revolutionized the task of uncovering complex equations that describe physical systems. The primary goal in …

Null Space Removal In Finite Element Discretizations, 2023 Clemson University

#### Null Space Removal In Finite Element Discretizations, Pengfei Jia

*All Theses*

Partial differential equations are frequently utilized in the mathematical formulation of physical problems. Boundary conditions need to be applied in order to obtain the unique solution to such problems. However, some types of boundary conditions do not lead to unique solutions because the continuous problem has a null space. In this thesis, we will discuss how to solve such problems effectively. We first review the foundation of all three problems and prove that Laplace problem, linear elasticity problem and Stokes problem can be well posed if we restrict the test and trial space in the continuous and discrete finite element …

Recovering Coefficients Of Second-Order Hyperbolic And Plate Equations Via Finite Measurements On The Boundary, 2023 Clemson University

#### Recovering Coefficients Of Second-Order Hyperbolic And Plate Equations Via Finite Measurements On The Boundary, Scott Randall Scruggs

*All Dissertations*

Abstract In this dissertation, we consider the inverse problem for a second-order hyperbolic equation of recovering n + 3 unknown coefficients defined on an open bounded domain with a smooth enough boundary. We also consider the inverse problem of recovering an unknown coefficient on the Euler- Bernoulli plate equation on a lower-order term again defined on an open bounded domain with a smooth enough boundary. For the second-order hyperbolic equation, we show that we can uniquely and (Lipschitz) stably recover all these coefficients from only using half of the corresponding boundary measurements of their solutions, and for the plate equation, …

Neural Network Learning For Pdes With Oscillatory Solutions And Causal Operators, 2023 Southern Methodist University

#### Neural Network Learning For Pdes With Oscillatory Solutions And Causal Operators, Lizuo Liu

*Mathematics Theses and Dissertations*

In this thesis, we focus on developing neural networks algorithms for scientific computing. First, we proposed a phase shift deep neural network (PhaseDNN), which provides a uniform wideband convergence in approximating high frequency functions and solutions of wave equations. Several linearized learning schemes have been proposed for neural networks solving nonlinear Navier-Stokes equations. We also proposed a causality deep neural network (Causality-DeepONet) to learn the causal response of a physical system. An extension of the Causality-DeepONet to time-dependent PDE systems is also proposed. The PhaseDNN makes use of the fact that common DNNs often achieve convergence in the low frequency …

On Solutions Of First Order Pde With Two-Dimensional Dirac Delta Forcing Terms, 2023 Murray State University

#### On Solutions Of First Order Pde With Two-Dimensional Dirac Delta Forcing Terms, Ian Robinson

*Rose-Hulman Undergraduate Mathematics Journal*

We provide solutions of a first order, linear partial differential equation of two variables where the nonhomogeneous term is a two-dimensional Dirac delta function. Our results are achieved by applying the unilateral Laplace Transform, solving the subsequently transformed PDE, and reverting back to the original space-time domain. A discussion of existence and uniqueness of solutions, a derivation of solutions of the PDE coupled with a boundary and initial condition, as well as a few worked examples are provided.

Solving The Cable Equation, A Second-Order Time Dependent Pde For Non-Ideal Cables With Action Potentials In The Mammalian Brain Using Kss Methods, 2023 The University of Southern Mississippi

#### Solving The Cable Equation, A Second-Order Time Dependent Pde For Non-Ideal Cables With Action Potentials In The Mammalian Brain Using Kss Methods, Nirmohi Charbe

*Master's Theses*

In this thesis we shall perform the comparisons of a Krylov Subspace Spectral method with Forward Euler, Backward Euler and Crank-Nicolson to solve the Cable Equation. The Cable Equation measures action potentials in axons in a mammalian brain treated as an ideal cable in the first part of the study. We shall subject this problem to the further assumption of a non-ideal cable. Assume a non-uniform cross section area along the longitudinal axis. At the present time, the effects of torsion, curvature and material capacitance are ignored. There is particular interest to generalize the application of the PDEs including and …

(R1966) Semi Analytical Approach To Study Mathematical Model Of Atmospheric Internal Waves Phenomenon, 2023 P D Patel Institute of Applied Sciences

#### (R1966) Semi Analytical Approach To Study Mathematical Model Of Atmospheric Internal Waves Phenomenon, Patel Yogeshwari, Jayesh M. Dhodiya

*Applications and Applied Mathematics: An International Journal (AAM)*

This research aims to study atmospheric internal waves which occur within the fluid rather than on the surface. The mathematical model of the shallow fluid hypothesis leads to a coupled nonlinear system of partial differential equations. In the shallow flow model, the primary assumption is that vertical size is smaller than horizontal size. This model can precisely replicate atmospheric internal waves because waves are dispersed over a vast horizontal area. A semi-analytical approach, namely modified differential transform, is applied successfully in this research. The proposed method obtains an approximate analytical solution in the form of convergent series without any linearization, …

(R2028) A Brief Note On Space Time Fractional Order Thermoelastic Response In A Layer, 2023 Shri Lemdeo Patil Mahavidyalaya,Mandhal

#### (R2028) A Brief Note On Space Time Fractional Order Thermoelastic Response In A Layer, Navneet Lamba, Jyoti Verma, Kishor Deshmukh

*Applications and Applied Mathematics: An International Journal (AAM)*

In this study, a one-dimensional layer of a solid is used to investigate the exact analytical solution of the heat conduction equation with space-time fractional order derivatives and to analyze its associated thermoelastic response using a quasi-static approach. The assumed thermoelastic problem was subjected to certain initial and boundary conditions at the initial and final ends of the layer. The memory effects and long-range interaction were discussed with the help of the Caputo-type fractional-order derivative and finite Riesz fractional derivative. Laplace transform and Fourier transform techniques for spatial coordinates were used to investigate the solution of the temperature distribution and …

(R2052) Flow Patterns For Newtonian And Non-Newtonian Fluids In A Cylindrical Pipe, 2023 The University of Texas Rio Grande Valley

#### (R2052) Flow Patterns For Newtonian And Non-Newtonian Fluids In A Cylindrical Pipe, Erick Sanchez, Dambaru Bhatta

*Applications and Applied Mathematics: An International Journal (AAM)*

A fully developed laminar steady flow of an incompressible, viscous fluid in a horizontal cylindrical pipe is considered here. Flow patterns for an incompressible, viscous fluid for both Newtonian and non-Newtonian fluids such as shear-thinning, shear-thickening and Bingham plastic fluids are analyzed in this study. Assuming that the flow is only due to the wall shear stress and the pressure drop, the velocity component in the axial direction for these cases is derived. Computational results of the velocity profiles for various cases are obtained using MATLAB and presented in graphical forms. It is observed that the velocity profile is parabolic …