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Articles 1 - 16 of 16
Full-Text Articles in Applied Mathematics
Numerical Methods For Optimal Transport And Optimal Information Transport On The Sphere, Axel G. R. Turnquist
Numerical Methods For Optimal Transport And Optimal Information Transport On The Sphere, Axel G. R. Turnquist
Dissertations
The primary contribution of this dissertation is in developing and analyzing efficient, provably convergent numerical schemes for solving fully nonlinear elliptic partial differential equation arising from Optimal Transport on the sphere, and then applying and adapting the methods to two specific engineering applications: the reflector antenna problem and the moving mesh methods problem. For these types of nonlinear partial differential equations, many numerical studies have been done in recent years, the vast majority in subsets of Euclidean space. In this dissertation, the first major goal is to develop convergent schemes for the sphere. However, another goal of this dissertation is …
Nystrom Methods For High-Order Cq Solutions Of The Wave Equation In Two Dimensions, Erli Wind-Andersen
Nystrom Methods For High-Order Cq Solutions Of The Wave Equation In Two Dimensions, Erli Wind-Andersen
Dissertations
An investigation of high order Convolution Quadratures (CQ) methods for the solution of the wave equation in unbounded domains in two dimensions is presented. These rely on Nystrom discretizations for the solution of the ensemble of associated Laplace domain modified Helmholtz problems. Two classes of CQ discretizations are considered: one based on linear multistep methods and the other based on Runge-Kutta methods. Both are used in conjunction with Nystrom discretizations based on Alpert and QBX quadratures of Boundary Integral Equation (BIE) formulations of the Laplace domain Helmholtz problems with complex wavenumbers. CQ in conjunction with BIE is an excellent candidate …
Positive Solutions To Semilinear Elliptic Equations With Logistic-Type Nonlinearities And Harvesting In Exterior Domains, Eric Jameson
Positive Solutions To Semilinear Elliptic Equations With Logistic-Type Nonlinearities And Harvesting In Exterior Domains, Eric Jameson
UNLV Theses, Dissertations, Professional Papers, and Capstones
Existing results provide the existence of positive solutions to a class of semilinear elliptic PDEs with logistic-type nonlinearities and harvesting terms both in RN and in bounded domains U ⊂ RN with N ≥ 3, when the carrying capacity of the environment is not constant. We consider these same equations in the exterior domain Ω, defined as the complement of the closed unit ball in RN , N ≥ 3, now with a Dirichlet boundary condition. We first show that the existing techniques forsolving these equations in the whole space RN can be applied to the exterior domain with some …
A Direct Method For Modeling And Simulations Of Elliptic And Parabolic Interface Problems, Kumudu Janani Gamage
A Direct Method For Modeling And Simulations Of Elliptic And Parabolic Interface Problems, Kumudu Janani Gamage
Mathematics & Statistics Theses & Dissertations
Interface problems have many applications in physics. In this dissertation, we develop a direct method for solving three-dimensional elliptic interface problems and study their application in solving parabolic interface problems. As many of the physical applications of interface problems can be approximated with partial differential equations (PDE) with piecewise constant coefficients, our derivation of the model is focused on interface problems with piecewise constant coefficients but have a finite jump across the interface. The critical characteristic of the method is that our computational framework is based on a finite difference scheme on a uniform Cartesian grid system and does not …
Mathematical And Statistical Modeling With Deep Neural Networks, Albert Romero
Mathematical And Statistical Modeling With Deep Neural Networks, Albert Romero
Theses and Dissertations
General adversarial networks (GANs) are a form of machine learning that includes two neural networks competing in a zero-sum game. One network produces artificial, while the other tries to distinguish artificial data from real. The Wasserstein general adversarial network with gradient penalty (WGAN-GP) variant of this technique is used to produce solutions for ordinary and partial differential equations.
Modeling Dewetting, Demixing, And Thermal Effects In Nanoscale Metal Films, Ryan Howard Allaire
Modeling Dewetting, Demixing, And Thermal Effects In Nanoscale Metal Films, Ryan Howard Allaire
Dissertations
Thin film dynamics, particularly on the nanoscale, is a topic of extensive interest. The process by which thin liquids evolve is far from trivial and can lead to dewetting and drop formation. Understanding this process involves not only resolving the fluid mechanical aspects of the problem, but also requires the coupling of other physical processes, including liquid-solid interactions, thermal transport, and dependence of material parameters on temperature and material composition. The focus of this dissertation is on the mathematical modeling and simulation of nanoscale liquid metal films, which are deposited on thermally conductive substrates, liquefied by laser heating, and subsequently …
Finite Difference Schemes For Integral Equations With Minimal Regularity Requirements, Wesley Cameron Davis
Finite Difference Schemes For Integral Equations With Minimal Regularity Requirements, Wesley Cameron Davis
Mathematics & Statistics Theses & Dissertations
Volterra integral equations arise in a variety of applications in modern physics and engineering, namely in interactions that contain a memory term. Classical formulations of these problems are largely inflexible when considering non-homogeneous media, which can be problematic when considering long term interactions of real-world applications. The use of fractional derivative and integral terms naturally relax these restrictions in a natural way to consider these problems in a more general setting. One major drawback to the use of fractional derivatives and integrals in modeling is the regularity requirement for functions, where we can no longer assume that functions are as …
The Effect Of Initial Conditions On The Weather Research And Forecasting Model, Aaron D. Baker
The Effect Of Initial Conditions On The Weather Research And Forecasting Model, Aaron D. Baker
Electronic Theses and Dissertations
Modeling our atmosphere and determining forecasts using numerical methods has been a challenge since the early 20th Century. Most models use a complex dynamical system of equations that prove difficult to solve by hand as they are chaotic by nature. When computer systems became more widely adopted and available, approximating the solution of these equations, numerically, became easier as computational power increased. This advancement in computing has caused numerous weather models to be created and implemented across the world. However a challenge of approximating these solutions accurately still exists as each model have varying set of equations and variables to …
Convolution Inequalities And Applications To Partial Differential Equations., Matthew Reynolds
Convolution Inequalities And Applications To Partial Differential Equations., Matthew Reynolds
Electronic Theses and Dissertations
In this dissertation we develop methods for obtaining the existence of mild solutions to certain partial differential equations with initial data in weighted L p spaces and apply them to some examples as well as improve the solutions to some known PDEs studied extensively in the literature. We begin by obtaining a version of a Stein-Weiss integral inequality which we will use to obtain general convolution inequalities in weighted L p spaces using the techniques of interpolation. We will then use these convolution inequalities to make estimates on PDEs that will help us obtain mild solutions as fixed points of …
Homogenization In Perforated Domains And With Soft Inclusions, Brandon C. Russell
Homogenization In Perforated Domains And With Soft Inclusions, Brandon C. Russell
Theses and Dissertations--Mathematics
In this dissertation, we first provide a short introduction to qualitative homogenization of elliptic equations and systems. We collect relevant and known results regarding elliptic equations and systems with rapidly oscillating, periodic coefficients, which is the classical setting in homogenization of elliptic equations and systems. We extend several classical results to the so called case of perforated domains and consider materials reinforced with soft inclusions. We establish quantitative H1-convergence rates in both settings, and as a result deduce large-scale Lipschitz estimates and Liouville-type estimates for solutions to elliptic systems with rapidly oscillating periodic bounded and measurable coefficients. Finally, …
A Numerical Study Of Construction Of Honey Bee Comb, Pamela Guerrero, Pamela C. Guerrero
A Numerical Study Of Construction Of Honey Bee Comb, Pamela Guerrero, Pamela C. Guerrero
Murray State Theses and Dissertations
We use finite difference methods in the treatment of an existing system of partial differential equations that captures the dynamics of parallel honeycomb construction in a bee hive. We conduct an uncertainty analysis by calculating the partial rank correlation coefficient for the parameters to find which are most important to the outcomes of the model. We then use an eFAST method to determine both the individual and total sensitivity index for the parameters. Afterwards we examine our numerical model under varying initial conditions and parameter values, and compare ratios found from local data with the golden mean by fitting images …
Nonlinear Partial Differential Equations, Their Solutions, And Properties, Prasanna Bandara
Nonlinear Partial Differential Equations, Their Solutions, And Properties, Prasanna Bandara
Boise State University Theses and Dissertations
Although valuable understanding of real-world phenomena can be gained experimentally, it is often the case that experimental investigations can be found to be limited by financial, ethical or other constraints making such an approach impractical or, in some cases, even impossible. To nevertheless understand and make predictions of the natural world around us, countless processes encountered in the physical and biological sciences, engineering, economics and medicine can be efficiently described by means of mathematical models written in terms of ordinary or/and partial differential equations or their systems. Fundamental questions that arise in the modeling process need care that relies on …
A Local Radial Basis Function Method For The Numerical Solution Of Partial Differential Equations, Maggie Elizabeth Chenoweth
A Local Radial Basis Function Method For The Numerical Solution Of Partial Differential Equations, Maggie Elizabeth Chenoweth
Theses, Dissertations and Capstones
Most traditional numerical methods for approximating the solutions of problems in science, engineering, and mathematics require the data to be arranged in a structured pattern and to be contained in a simply shaped region, such as a rectangle or circle. In many important applications, this severe restriction on structure cannot be met, and traditional numerical methods cannot be applied. In the 1970s, radial basis function (RBF) methods were developed to overcome the structure requirements of existing numerical methods. RBF methods are applicable with scattered data locations. As a result, the shape of the domain may be determined by the application …
Boundary Element Method (Bem) And Method Of Fundamental Solutions (Mfs) For The Boundary Value Problems Of The 2-D Laplace's Equation, Ermes Anthony Salgado-Ibarra
Boundary Element Method (Bem) And Method Of Fundamental Solutions (Mfs) For The Boundary Value Problems Of The 2-D Laplace's Equation, Ermes Anthony Salgado-Ibarra
UNLV Theses, Dissertations, Professional Papers, and Capstones
In this thesis we study the solution of the two dimensional Laplace equation by the boundary Element method (BEM) and the method of fundamental solutions (MFS). Both the BEM and MFS used to solve boundary value problems involving the Laplace equation 2-D settings. Both methods rely on the use of fundamental solution of the Laplace's equation (the solution of Laplace's equation in the distributional sense). We will contrast and compare the results we get using the BEM with results we get using the MFS.
Source Optimization In Abstract Function Spaces For Maximizing Distinguishability: Applications To The Optical Tomography Inverse Problem, Bonnie Jacob
All Dissertations
The focus of this thesis is to formulate an optimal source problem for the medical imaging technique of optical tomography by maximizing certain distinguishability criteria. We extend the concept of distinguishability in electrical impedance tomography to the frequency-domain diffusion approximation model used in optical tomography.
We consider the dependence of the optimal source on the choice of appropriate function spaces, which can be chosen from certain Sobolev or Lp spaces. All of the spaces we consider are Hilbert spaces; we therefore exploit the inner product in several ways. First, we define and use throughout an inner product on the Sobolev …
High Accuracy Multiscale Multigrid Computation For Partial Differential Equations, Yin Wang
High Accuracy Multiscale Multigrid Computation For Partial Differential Equations, Yin Wang
University of Kentucky Doctoral Dissertations
Scientific computing and computer simulation play an increasingly important role in scientific investigation and engineering designs, supplementing traditional experiments, such as in automotive crash studies, global climate change, ocean modeling, medical imaging, and nuclear weapons. The numerical simulation is much cheaper than experimentation for these application areas and it can be used as the third way of science discovery beyond the experimental and theoretical analysis. However, the increasing demand of high resolution solutions of the Partial Differential Equations (PDEs) with less computational time has increased the importance for researchers and engineers to come up with efficient and scalable computational techniques …