Numerical Analysis and Computation Commons^™

Dynamics Of Discontinuities In Elastic Solids, Arkadi Berezovski, Mihhail Berezovski

Publications

The paper is devoted to evolving discontinuities in elastic solids. A discontinuity is represented as a singular set of material points. Evolution of a discontinuity is driven by the configurational force acting at such a set. The main attention is paid to the determination of the velocity of a propagating discontinuity. Martensitic phase transition fronts and brittle cracks are considered as representative examples.

A New Class Of Discontinuous Galerkin Methods For Wave Equations In Second-Order Form, Lu Zhang

Mathematics Theses and Dissertations

Discontinuous Galerkin methods are widely used in many practical fields. In this thesis, we focus on a new class of discontinuous Galerkin methods for second-order wave equations. This thesis is constructed by three main parts. In the first part, we study the convergence properties of the energy-based discontinuous Galerkin proposed in [3] for wave equations. We improve the existing suboptimal error estimates to an optimal convergence rate in the energy norm. In the second part, we generalize the energy-based discontinuous Galerkin method proposed in [3] to the advective wave equation and semilinear wave equation in second-order form. Energy-conserving or energy-dissipating …

A Time Integration Method Of Approximate Particular Solutions For Nonlinear Ordinary Differential Equations, Cyril Ocloo

Master's Theses

We consider a time-dependent method which is coupled with the method of approximate particular solutions (MAPS) of Delta-shaped basis functions and the method of fundamental solutions (MFS) to solve nonlinear ordinary differential equations. Firstly, we convert a nonlinear problem into a sequence of time-dependent non-homogeneous boundary value problems through a fictitious time integration method. The superposition principle is applied to split the numerical solution at each time step into an approximate particular solution and a homogeneous solution. Delta-shaped basis functions are used to provide an approximation of the source function at each time step. The purpose of this is to …

The Dual Of The Compressed Shift, M. C. Câmara, William T. Ross

Department of Math & Statistics Faculty Publications

For an inner function u, we discuss the dual operator for the compressed shift P_uS∣_Ku, where K_u is the model space for u. We describe the unitary equivalence/similarity classes for these duals as well as their invariant subspaces.

Teaching And Learning Of Fluid Mechanics, Ashwin Vaidya

Department of Mathematics Faculty Scholarship and Creative Works

Fluid mechanics occupies a privileged position in the sciences; it is taught in various science departments including physics, mathematics, environmental sciences and mechanical, chemical and civil engineering, with each highlighting a different aspect or interpretation of the foundation and applications of fluids. Doll’s fluid analogy [5] for this idea is especially relevant to this issue: “Emergence of creativity from complex flow of knowledge—example of Benard convection pattern as an analogy—dissipation or dispersal of knowledge (complex knowledge) results in emergent structures, i.e., creativity which in the context of education should be thought of as a unique way to arrange information so …

Numerical Analysis Of Three-Dimensional Mhd Flow Of Casson Nanofluid Past An Exponentially Stretching Sheet, Madhusudan Senapati, Kharabela Swain, Sampad Kumar Parida

Karbala International Journal of Modern Science

The convective three dimensional electrically conducting Casson nanofluid flow over an exponentially stretching sheet embedded in a saturated porous medium and subjected to thermal as well as solutal slip in the presence of externally applied transverse magnetic field (force-at-a-distance) is studied. The heat transfer phenomenon includes the viscous dissipation, Joulian dissipation, thermal radiation, contribution of nanofluidity and temperature dependent volumetric heat source. The study of mass diffusion in the presence of chemically reactive species enriches the analysis. The numerical solutions of coupled nonlinear governing equations bring some earlier reported results as particular cases providing a testimony of validation of the …

Investigating The Solution Properties Of Population Model Of Cross-Diffusion Model With Double Nonlinearity And With Variable Density, Dildora Kabilovna Muhamediyeva

Chemical Technology, Control and Management

The models of two competing populations with double nonlinear diffusion and three types of functional dependencies are considered. The first dependence corresponds to the Malthusian type, the second to the Verhühlst type (logistic population), and the third to Olli-type populations. A common element of this kind of description is the presence of a linear source. Nonlinear sinks are also present in descriptions of populations of the Verhulst and Ollie type. Suitable initial approximations for a rapidly converging iterative process are proposed. Based on a self-similar analysis and comparison of the solutions of the Cauchy problem in the domain for an …

Higher Accuracy Methods For Fluid Flows In Various Applications: Theory And Implementation, Dilek Erkmen

Dissertations, Master's Theses and Master's Reports

This dissertation contains research on several topics related to Defect-deferred correction (DDC) method applying to CFD problems. First, we want to improve the error due to temporal discretization for the problem of two convection dominated convection-diffusion problems, coupled across a joint interface. This serves as a step towards investigating an atmosphere-ocean coupling problem with the interface condition that allows for the exchange of energies between the domains.

The main diffuculty is to decouple the problem in an unconditionally stable way for using legacy code for subdomains. To overcome the issue, we apply the Deferred Correction (DC) method. The DC method …

Determinism Of Stochastic Processes Through The Relationship Between The Heat Equation And Random Walks, Gurmehar Singh Makker

Publications and Research

We study the deterministic characteristics of stochastic processes through investigation of random walks and the heat equation. The relationship is confirmed by discretizing the heat equation in time and space and determining the probability distribution function for random walks in dimension d = 1, 2. The existence of the relationship is presented both through theoretical analysis and numerical computation.

Numerical Solution Of Fractional Partial Differential Equations With Normalized Bernstein Wavelet Method, Mahsa Entezari, Saeid Abbasbandy, Esmail Babolian

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, normalized Bernstein wavelets are presented. Next, the fractional order integration and Bernstein wavelets operational matrices of integration are derived and finally are used for solving fractional partial differential equations. The operational matrices merged with the collocation method are used in order to convert fractional problems to a number of algebraic equations. In the suggested method the boundary conditions are automatically taken into consideration. An assessment of the error of function approximation based on the normalized Bernstein wavelet is also presented. Some numerical instances are given to manifest the versatility and applicability of the suggested method. Founded numerical …

Multi-Point Flux Approximations Via The O-Method, Christen Leggett

Master's Theses

When an oil refining company is drilling for oil, much of the oil gets left behind after the first drilling. Enhanced oil recovery techniques can be used to recover more of that oil, but these methods are quite expensive. When a company is deciding if it is worth their time and money to use enhanced oil recovery methods, simulations can be used to model oil flow, showing the behavior and location of the oil. While methods do exist to model this flow, these methods are often very slow and inaccurate due to a large domain and wide variance in coefficients. …

Local Non-Similar Solution Of Powell-Eyring Fluid Flow Over A Vertical Flat Plate, Hemangini Shukla, Hema C. Surati, M. G. Timol

Applications and Applied Mathematics: An International Journal (AAM)

Our objective is to obtain the non-similarity solution of non-Newtonian fluid for Powell-Eyring model by a local non-similarity method. Here, free stream velocity is considered in power-law form (𝑈=𝑥^m). The governing equations are transformed using non-similar transformations and derived equations are treated as ordinary differential equations. Non-similar solutions are obtained for different values of power-law index 𝑚 and stream-wise location 𝜉. Influence of various parameters on velocity and temperature field are presented graphically using MATLAB bvp4c solver.

Sperm Motility In Groups, Julie Simons

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Multidisciplinary Education And Research In Biomathematics For Solving Global Challenges, Padmanabhan Seshaiyer

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Parallel Multipole Expansion Algorithms And Their Biology Applications, Jiahui Chen

Mathematics Theses and Dissertations

N-body pairwise interactions are ubiquitous in scientific areas such as astrophysics, fluids mechanics, electrical engineering, molecular biology, etc. Computing these interactions using direct sum of an O(N) cost is expensive, whereas multipole expansion methods, such as the fast multipole method (FMM) or treecode, can reduce the cost to O(N) or O(N log N). This thesis focuses on developing numerical algorithms of Cartesian FMM and treecode, as well as using these algorithms to directly or implicitly solve biological problems involving pairwise interactions. This thesis consists of the following topics. 1) A cyclic parallel scheme is developed to handle the load balancing …

Krylov Subspace Spectral Methods With Non-Homogenous Boundary Conditions, Abbie Hendley

Master's Theses

For this thesis, Krylov Subspace Spectral (KSS) methods, developed by Dr. James Lambers, will be used to solve a one-dimensional, heat equation with non-homogenous boundary conditions. While current methods such as Finite Difference are able to carry out these computations efficiently, their accuracy and scalability can be improved. We will solve the heat equation in one-dimension with two cases to observe the behaviors of the errors using KSS methods. The first case will implement KSS methods with trigonometric initial conditions, then another case where the initial conditions are polynomial functions. We will also look at both the time-independent and time-dependent …

Radial Basis Function Finite Difference Approximations Of The Laplace-Beltrami Operator, Sage Byron Shaw

Boise State University Theses and Dissertations

Partial differential equations (PDEs) are used throughout science and engineering for modeling various phenomena. Solutions to PDEs cannot generally be represented analytically, and therefore must be approximated using numerical techniques; this is especially true for geometrically complex domains. Radial basis function generated finite differences (RBF-FD) is a recently developed mesh-free method for numerically solving PDEs that is robust, accurate, computationally efficient, and geometrically flexible. In the past seven years, RBF-FD methods have been developed for solving PDEs on surfaces, which have applications in biology, chemistry, geophysics, and computer graphics. These methods are advantageous, as they are mesh-free, operate on arbitrary …

Solitary And Periodic Exact Solutions Of The Viscosity-Capillarity Van Der Waals Gas Equations, Emad A. Az-Zo’Bi

Applications and Applied Mathematics: An International Journal (AAM)

Periodic and soliton solutions are derived for the (1+1)-dimensional van der Waals gas system in the viscosity-capillarity regularization form. The system is handled via the e-^φ(ξ) -expansion method. The obtained solutions have been articulated by the hyperbolic, trigonometric, exponential and rational functions with arbitrary constants. Mathematical analysis and numerical graphs are provided for some solitons, periodic and kink solitary wave solutions to visualize the dynamics of equations. Obtained results reveal that the method is very influential and effective tool for solving nonlinear partial differential equations in applied mathematics.

Enhancement Of Krylov Subspace Spectral Methods Through The Use Of The Residual, Haley Dozier

Dissertations

Depending on the type of equation, finding the solution of a time-dependent partial differential equation can be quite challenging. Although modern time-stepping methods for solving these equations have become more accurate for a small number of grid points, in a lot of cases the scalability of those methods leaves much to be desired. That is, unless the timestep is chosen to be sufficiently small, the computed solutions might exhibit unreasonable behavior with large input sizes. Therefore, to improve accuracy as the number of grid points increases, the time-steps must be chosen to be even smaller to reach a reasonable solution. …

Mathematical Models: The Lanchester Equations And The Zombie Apocalypse, Hailey Bauer

Undergraduate Theses and Capstone Projects

This research study used mathematical models to analyze and depicted specific battle situations and the outcomes of the zombie apocalypse. The original models that predicted warfare were the Lanchester models, while the zombie apocalypse models were fictional expansions upon mathematical models used to examine infectious diseases. In this paper, I analyzed and compared different mathematical models by examining each model’s set of assumptions and the impact of the change in variables on the population classes. The purpose of this study was to understand the basics of the discrete dynamical systems and to determine the similarities between imaginary and realistic models. …

Partially Isometric Matrices: A Brief And Selective Survey, Stephan Ramon Garcia, Matthew Okubo Patterson, William T. Ross

Department of Math & Statistics Faculty Publications

We survey a variety of results about partially isometric matrices. We focus primarily on results that are distinctly finite-dimensional. For example, we cover a recent solution to the similarity problem for partial isometries. We also discuss the unitary similarity problem and several other results.

Algorithms To Approximate Solutions Of Poisson's Equation In Three Dimensions, Ray Dambrose

Rose-Hulman Undergraduate Mathematics Journal

The focus of this research was to develop numerical algorithms to approximate solutions of Poisson's equation in three dimensional rectangular prism domains. Numerical analysis of partial differential equations is vital to understanding and modeling these complex problems. Poisson's equation can be approximated with a finite difference approximation. A system of equations can be formed that gives solutions at internal points of the domain. A computer program was developed to solve this system with inputs such as boundary conditions and a nonhomogenous source function. Approximate solutions are compared with exact solutions to prove their accuracy. The program is tested with an …

Surface Waves Over Currents And Uneven Bottom, Alan Compelli, Rossen Ivanov, Calin I. Martin, Michail D. Todorov

Articles

The propagation of surface water waves interacting with a current and an uneven bottom is studied. Such a situation is typical for ocean waves where the winds generate currents in the top layer of the ocean. The role of the bottom topography is taken into account since it also influences the local wave and current patterns. Specific scaling of the variables is selected which leads to approximations of Boussinesq and KdV types. The arising KdV equation with variable coefficients, dependent on the bottom topography, is studied numerically when the initial condition is in the form of the one soliton solution …

Discontinuous Galerkin Methods For Convection-Diffusion Equations And Applications In Petroleum Engineering, Nattaporn Chuenjarern

Dissertations, Master's Theses and Master's Reports

This dissertation contains research in discontinuous Galerkin (DG) methods applying to convection-diffusion equations. It contains both theoretical analysis and applications. Initially, we develop a conservative local discontinuous Galerkin (LDG) method for the coupled system of compressible miscible displacement problem in two space dimensions. The main difficulty is how to deal with the discontinuity of approximations of velocity, u, in the convection term across the cell interfaces. To overcome the problems, we apply the idea of LDG with IMEX time marching using the diffusion term to control the convection term. Optimal error estimates in L^infinity(0, T; L^{2 …}

Numerical Study Of Singular And Delta Shock Solutions Using A Large Time Step Method, Ilija Jegdic

Applications and Applied Mathematics: An International Journal (AAM)

We illustrate recently proposed large time step method for hyperbolic conservation laws. In the scalar case, it was proved earlier that if the approximate solutions converge boundedly, then they converge to the entropy solution. The main goal of this paper is to consider the large time step method for several systems of hyperbolic conservation laws. We compute approximate solutions to Riemann problems for three genuinely nonlinear one-dimensional systems (the Keyfitz-Kranzer system, the isentropic generalized Chaplygin gas dynamics equations, and the isentropic gas dynamics equations for polytropic gases with vanishing pressure). Besides approximating solutions that contain shocks and rarefaction waves, the …

Creating A Computational Tool To Simulate Vibration Control For Piezoelectric Devices, Ahmet Ozkan Ozer, Emma J. Moore

Posters-at-the-Capitol

Piezoelectric materials have the unique ability to convert electrical energy to mechanical vibrations and vice versa. This project takes a stab to develop a reliable computational tool to simulate the vibration control of a novel “partial differential equation” model for a piezoelectric device, which is designed by integrating electric conducting piezoelectric layers constraining a viscoelastic layer to provide an active and lightweight intelligent structure. Controlling unwanted vibrations on piezoelectric devices (or harvesting energy from ambient vibrations) through piezoelectric layers has been the major focus in cutting-edge engineering applications such as ultrasonic welders and inchworms. The corresponding mathematical models for piezoelectric …

Two Applications Of High Order Methods: Wave Propagation And Accelerator Physics, Oleksii Beznosov

Shared Knowledge Conference

Numerical simulations of partial differential equations (PDE) are used to predict the behavior of complex physics phenomena when the real life experiments are expensive. Discretization of a PDE is the representation of the continuous problem as a discrete problem that can be solved on a computer. The discretization always introduces a certain inaccuracy caused by the numerical approximation. By increasing the computational cost of the numerical algorithm the solution can be computed more accurately. In the theory of numerical analysis this fact is called the convergence of the numerical algorithm. The idea behind high order methods is to improve the …

Boundary Homogenization And Capture Time Distributions Of Semipermeable Membranes With Periodic Patterns Of Reactive Sites, Andrew J. Bernoff, Daniel Schmidt, Alan E. Lindsay

All HMC Faculty Publications and Research

We consider the capture dynamics of a particle undergoing a random walk in a half- space bounded by a plane with a periodic pattern of absorbing pores. In particular, we numerically measure and asymptotically characterize the distribution of capture times. Numerically we develop a kinetic Monte Carlo (KMC) method that exploits exact solutions to create an efficient particle- based simulation of the capture time that deals with the infinite half-space exactly and has a run time that is independent of how far from the pores one begins. Past researchers have proposed homogenizing the surface boundary conditions, replacing the reflecting (Neumann) …

Inner Vectors For Toeplitz Operators, Raymond Cheng, Javad Mashreghi, William T. Ross

Department of Math & Statistics Faculty Publications

In this paper we survey and bring together several approaches to obtaining inner functions for Toeplitz operators. These approaches include the classical definition, the Wold decomposition, the operator-valued Poisson Integral, and Clark measures. We then extend these notions somewhat to inner functions on model spaces. Along the way we present some novel examples.

On The Well-Posedness And Global Boundary Controllability Of A Nonlinear Beam Model, Jessie Jamieson

Department of Mathematics: Dissertations, Theses, and Student Research

The theory of beams and plates has been long established due to works spanning many fields, and has been explored through many investigations of beam and plate mechanics, controls, stability, and the well-posedness of systems of equations governing the motions of plates and beams. Additionally, recent investigations of flutter phenomena by Dowell, Webster et al. have reignited interest into the mechanics and stability of nonlinear beams. In this thesis, we wish to revisit the seminal well-posedness results of Lagnese and Leugering for the one dimensional, nonlinear beam from their 1991 paper, "Uniform stabilization of a nonlinear beam by nonlinear boundary …