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 …

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, …

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 …

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 …

Numerical Analysis Of A Combustion Model For Layered Media Via Mathematical Homogenization, 2023 United States Naval Academy

#### Numerical Analysis Of A Combustion Model For Layered Media Via Mathematical Homogenization, Jessica M. Riggs, Ana Maria Soane

*Mathematica Militaris*

We propose to investigate a mathematical model

for combustion in a rod made of periodically alternating thin

layers of two combustible materials such as those occurring in

gun propellants. We apply the homogenization theory to resolve

the fast oscillations of the model’s coefficients across adjacent

layers, and set up numerical simulations to better understand

the reactions occurring in such media.

(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 …

(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 …

(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, …

Pde Model For Protocell Evolution And The Origin Of Chromosomes Via Multilevel Selection, 2023 University of Pennsylvania

#### Pde Model For Protocell Evolution And The Origin Of Chromosomes Via Multilevel Selection, Daniel B. Cooney, Fernando W. Rossine, Dylan H. Morris, Simon A. Levin

*Biology and Medicine Through Mathematics Conference*

No abstract provided.

Reaction-Diffusion System On Irregular Boundaries Reproduces Multiple Generations Of Petal Spot Patterns In Monkeyflower Hybrids, 2023 William & Mary

#### Reaction-Diffusion System On Irregular Boundaries Reproduces Multiple Generations Of Petal Spot Patterns In Monkeyflower Hybrids, Emily Simmons

*Biology and Medicine Through Mathematics Conference*

No abstract provided.

Helices In Fluids And Their Applications, 2023 James Madison University

#### Helices In Fluids And Their Applications, Eva M. Strawbridge

*Biology and Medicine Through Mathematics Conference*

No abstract provided.

Monolithic Multiphysics Simulation Of Hypersonic Aerothermoelasticity Using A Hybridized Discontinuous Galerkin Method, 2023 Mississippi State University

#### Monolithic Multiphysics Simulation Of Hypersonic Aerothermoelasticity Using A Hybridized Discontinuous Galerkin Method, William Paul England

*Theses and Dissertations*

This work presents implementation of a hybridized discontinuous Galerkin (DG) method for robust simulation of the hypersonic aerothermoelastic multiphysics system. Simulation of hypersonic vehicles requires accurate resolution of complex multiphysics interactions including the effects of high-speed turbulent flow, extreme heating, and vehicle deformation due to considerable pressure loads and thermal stresses. However, the state-of-the-art procedures for hypersonic aerothermoelasticity are comprised of low-fidelity approaches and partitioned coupling schemes. These approaches preclude robust design and analysis of hypersonic vehicles for a number of reasons. First, low-fidelity approaches limit their application to simple geometries and lack the ability to capture small scale flow …

An Integrated Experimental And Modeling Approach To Design Rotating Algae Biofilm Reactors (Rabrs) Via Optimizing Algae Biofilm Productivity, Nutrient Recovery, And Energy Efficiency, 2023 Utah State University

#### An Integrated Experimental And Modeling Approach To Design Rotating Algae Biofilm Reactors (Rabrs) Via Optimizing Algae Biofilm Productivity, Nutrient Recovery, And Energy Efficiency, Gerald Benjamin Jones

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

Microalgae biofilms have been demonstrated to recover nutrients from wastewater and serve as biomass feedstock for bioproducts. However, there is a need to develop a platform to quantitatively describe microalgae biofilm production, which can provide guidance and insights for improving biomass areal productivity and nutrient uptake efficiency. This paper proposes a unified experimental and theoretical framework to investigate algae biofilm growth on a rotating algae biofilm reactor (RABR). The experimental laboratory setups are used to conduct controlled experiments on testing environmental and operational factors for RABRs. We propose a differential-integral equation-based mathematical model for microalgae biofilm cultivation guided by laboratory …

Advancements In Fluid Simulation Through Enhanced Conservation Schemes, 2023 Clemson University

#### Advancements In Fluid Simulation Through Enhanced Conservation Schemes, Sean Ingimarson

*All Dissertations*

To better understand and solve problems involving the natural phenomenon of fluid and air flows, one must understand the Navier-Stokes equations. Branching several different fields including engineering, chemistry, physics, etc., these are among the most important equations in mathematics. However, these equations do not have analytic solutions save for trivial solutions. Hence researchers have striven to make advancements in varieties of numerical models and simulations. With many variations of numerical models of the Navier-Stokes equations, many lose important physical meaningfulness. In particular, many finite element schemes do not conserve energy, momentum, or angular momentum. In this thesis, we will study …

The Magnetic Field Of Protostar-Disk-Outflow Systems, 2023 Western University

#### The Magnetic Field Of Protostar-Disk-Outflow Systems, Mahmoud Sharkawi

*Electronic Thesis and Dissertation Repository*

Recent observations of protostellar cores reveal complex magnetic field configurations that are distorted in the innermost disk region. Unlike the prestellar phase, where the magnetic field geometry is simpler with an hourglass configuration, magnetic fields in the protostellar phase are sculpted by the formation of outflows and rapid rotation. This gives rise to a significant azimuthal (or toroidal) component that has not yet been analytically modelled in the literature. Moreover, the onset of outflows, which act as angular momentum transport mechanisms, have received considerable attention in the past few decades. Two mechanisms: 1) the driving by the gradient of a …

Data-Driven Computational Methods For Quasi-Stationary Distribution And Sensitivity Analysis, 2023 University of Massachusetts Amherst

#### Data-Driven Computational Methods For Quasi-Stationary Distribution And Sensitivity Analysis, Yaping Yuan

*Doctoral Dissertations*

The goal of the dissertation is to develop the computational methods for quasi-stationary- distributions(QSDs) and the sensitivity analysis of a QSD against the modification of the boundary conditions and against the diffusion approximation.

Many models in various applications are described by Markov chains with absorbing states. For example, any models with mass-action kinetics, such as ecological models, epidemic models, and chemical reaction models, are subject to the population-level randomness called the demographic stochasticity, which may lead to extinction in finite time. There are also many dynamical systems that have interesting short term dynamics but trivial long term dynamics, such as …

Fourth Order Dispersion In Nonlinear Media, 2023 University of Massachusetts Amherst

#### Fourth Order Dispersion In Nonlinear Media, Georgios Tsolias

*Doctoral Dissertations*

In recent years, there has been an explosion of interest in media bearing quartic

dispersion. After the experimental realization of so-called pure-quartic solitons, a

significant number of studies followed both for bright and for dark solitonic struc-

tures exploring the properties of not only quartic, but also setic, octic, decic etc.

dispersion, but also examining the competition between, e.g., quadratic and quartic

dispersion among others.

In the first chapter of this Thesis, we consider the interaction of solitary waves in

a model involving the well-known φ^{4} Klein-Gordon theory, bearing both Laplacian and biharmonic terms with different prefactors. As a …

Using Modflow To Assess Groundwater Storage Enhancement Via A Floodplain Infiltration Basin, 2023 Central Washington University

#### Using Modflow To Assess Groundwater Storage Enhancement Via A Floodplain Infiltration Basin, Lindsay Henning

*All Master's Theses*

Delaying groundwater discharge into rivers until it is critically needed during baseflow conditions provides promise for lowering elevated stream temperatures and improving habitat for aquatic species. Increasing groundwater storage may accomplish this in locations where excess spring runoff can be captured and allowed to infiltrate into the subsurface for later beneficial use, a process known as Managed Aquifer Recharge (MAR). Here, MAR via an infiltration basin is considered at a site along the Teanaway River in central Washington State. The effects of simulated ephemeral ponds of sizes varying from 554 m^{3} to 2430 m^{3} (0.449 acre-feet to 1.97 …