Partial Differential Equations Commons^™

Similarity Analysis Of Three Dimensional Nanofluid Flow By Deductive Group Theoretic Method, Hemangini Shukla, Hema C. Surati, M. G. Timol

Applications and Applied Mathematics: An International Journal (AAM)

The objective of this paper is to obtain similarity solution of three-dimensional nanofluid flow over flat surface stretched continuously in two lateral directions. Two independent variables from governing equations are reduced by applying deductive two parameter group theoretical method. Partial differential equations with boundary conditions are converted into ordinary differential equations with appropriate boundary conditions. Obtained equations are solved for temperature and velocity. The effect of nanoparticles volume fraction on temperature and velocity profile is investigated.

Linear Stability Analysis With Solution Patterns Due To Varying Thermal Diffusivity For A Convective Flow In A Porous Medium, Dambaru Bhatta

Applications and Applied Mathematics: An International Journal (AAM)

Here we investigate the effect of the vertical rate of change in thermal diffusivity due to a hydrothermal convective flow in a horizontal porous medium. The continuity equation, the heat equation and the momentum-Darcy equation constitute the governing system for the flow in a porous medium. Assuming a vertically varying basic state, we derive the linear system and from this linear system, we compute the critical Rayleigh and wave numbers. Using fourth-order Runge-Kutta and shooting methods, we obtain the marginal stability curves and linear solutions to analyze the solution pattern for different diffusivity parameters.

Conformable Derivative Operator In Modelling Neuronal Dynamics, Mehmet Yavuz, Burcu Yaşkıran

Applications and Applied Mathematics: An International Journal (AAM)

This study presents two new numerical techniques for solving time-fractional one-dimensional cable differential equation (FCE) modeling neuronal dynamics. We have introduced new formulations for the approximate-analytical solution of the FCE by using modified homotopy perturbation method defined with conformable operator (MHPMC) and reduced differential transform method defined with conformable operator (RDTMC), which are derived the solutions for linear-nonlinear fractional PDEs. In order to show the efficiencies of these methods, we have compared the numerical and exact solutions of fractional neuronal dynamics problem. Moreover, we have declared that the proposed models are very accurate and illustrative techniques in determining to approximate-analytical …

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 …

Modeling The Transmission Of Wolbachia In Mosquitoes For Controlling Mosquito-Borne Diseases, Zhuolin Qu

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

Modeling The Impact Of The Latent Period On The Spatial Spread Of Dengue Fever, Juan Melendez-Alvarez

Annual Symposium on Biomathematics and Ecology Education and Research

No abstract provided.

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 …

Adaptive Meshfree Methods For Partial Differential Equations, Jaeyoun Oh

Dissertations

There are many types of adaptive methods that have been developed with different algorithm schemes and definitions for solving Partial Differential Equations (PDE). Adaptive methods have been developed in mesh-based methods, and in recent years, they have been extended by using meshfree methods, such as the Radial Basis Function (RBF) collocation method and the Method of Fundamental Solutions (MFS). The purpose of this dissertation is to introduce an adaptive algorithm with a residual type of error estimator which has not been found in the literature for the adaptive MFS. Some modifications have been made in developing the algorithm schemes depending …

Full Field Computing For Elastic Pulse Dispersion In Inhomogeneous Bars, A. Berezovski, R. Kolman, M. Berezovski, D. Gabriel, V. Adamek

Publications

In the paper, the finite element method and the finite volume method are used in parallel for the simulation of a pulse propagation in periodically layered composites beyond the validity of homogenization methods. The direct numerical integration of a pulse propagation demonstrates dispersion effects and dynamic stress redistribution in physical space on example of a one-dimensional layered bar. Results of numerical simulations are compared with analytical solution constructed specifically for the considered problem. Analytical solution as well as numerical computations show the strong influence of the composition of constituents on the dispersion of a pulse in a heterogeneous bar and …

Masked Instability: Within-Sector Financial Risk In The Presence Of Wealth Inequality, Youngna Choi

Department of Applied Mathematics and Statistics Faculty Scholarship and Creative Works

We investigate masked financial instability caused by wealth inequality. When an economic sector is decomposed into two subsectors that possess a severe wealth inequality, the sector in entirety can look financially stable while the two subsectors possess extreme financially instabilities of opposite nature, one from excessive equity, the other from lack thereof. The unstable subsector can result in further financial distress and even trigger a financial crisis. The market instability indicator, an early warning system derived from dynamical systems applied to agent-based models, is used to analyze the subsectoral financial instabilities. Detailed mathematical analysis is provided to explain what financial …

Spectra Of Quantum Trees And Orthogonal Polynomials, Zhaoxia Wang

LSU Doctoral Dissertations

We investigate the spectrum of regular quantum-graph trees, where the edges are endowed with a Schr\"odinger operator with self-adjoint Robin vertex conditions. It is known that, for large eigenvalues, the Robin spectrum approaches the Neumann spectrum. In this research, we compute the lower Robin spectrum. The spectrum can be obtained from the roots of a sequence of orthogonal polynomials involving two variables. As the length of the quantum tree increases, the spectrum approaches a band-gap structure. We find that the lowest band tends to minus infinity as the Robin parameter increases, whereas the rest of the bands remain positive. Unexpectedly, …

Transport Phenomena In Field Effect Transistors, Ryan M. Evans, Arvind Balijepalli, Anthony Kearsley

Biology and Medicine Through Mathematics Conference

No abstract provided.

Solution For System Of Fractional Partial Differential Equations, D. B. Dhaigude, Swati N. Kanade, C. D. Dhaigude

Applications and Applied Mathematics: An International Journal (AAM)

The purpose of this article is to discuss solutions of different initial value problems (IVPs) for system of fractional differential equations. These equations appear in physical processes such as transportation and anomalous diffusion. The iteration method is successfully developed and series solution of IVPs at hand are obtained which converges to a function known as solution function of the IVPs. Graphical representation of solution of some IVPs are given using Mathematical software “MATLAB”.

Generalized Problem Of Thermal Bending Analysis In The Cartesian Domain, V. S. Kulkarni, Vinayaki Parab

Applications and Applied Mathematics: An International Journal (AAM)

This is an attempt for mathematical formulation and general analytical solution of the most generalized thermal bending problem in the Cartesian domain. The problem has been formulated in the context of non-homogeneous transient heat equation subjected to Robin’s boundary conditions. The general solution of the generalized thermoelastic problem has been discussed for temperature change, displacements, thermal stresses, deflection, and deformation. The most important feature of this work is any special case of practical interest may be readily obtained by this most generalized mathematical formulation and its analytical solution. There are 729 such combinations of possible boundary conditions prescribed on parallelepiped …

Finite Element Solution Of The Two-Dimensional Incompressible Navier-Stokes Equations Using Matlab, Endalew G. Tsega, V. K. Katiyar

Applications and Applied Mathematics: An International Journal (AAM)

The Navier–Stokes equations are fundamental in fluid mechanics. The finite element method has become a popular method for the solution of the Navier-Stokes equations. In this paper, the Galerkin finite element method was used to solve the Navier-Stokes equations for two-dimensional steady flow of Newtonian and incompressible fluid with no body forces using MATLAB. The method was applied to the lid-driven cavity problem. The eight-noded rectangular element was used for the formulation of element equations. The velocity components were located at all of 8 nodes and the pressure variable is located at 4 corner of the element. From location of …

Spatial Spread Of Defective Interfering Particles And Its Role In Suppressing Viral Load, Qasim Ali Qa, Ruian Ke

Biology and Medicine Through Mathematics Conference

No abstract provided.

Automatic Construction Of Scalable Time-Stepping Methods For Stiff Pdes, Vivian Ashley Montiforte

Master's Theses

Krylov Subspace Spectral (KSS) Methods have been demonstrated to be highly scalable time-stepping methods for stiff nonlinear PDEs. However, ensuring this scalability requires analytic computation of frequency-dependent quadrature nodes from the coefficients of the spatial differential operator. This thesis describes how this process can be automated for various classes of differential operators to facilitate public-domain software implementation.

Properties And Convergence Of State-Based Laplacians, Kelsey Wells

Department of Mathematics: Dissertations, Theses, and Student Research

The classical Laplace operator is a vital tool in modeling many physical behaviors, such as elasticity, diffusion and fluid flow. Incorporated in the Laplace operator is the requirement of twice differentiability, which implies continuity that many physical processes lack. In this thesis we introduce a new nonlocal Laplace-type operator, that is capable of dealing with strong discontinuities. Motivated by the state-based peridynamic framework, this new nonlocal Laplacian exhibits double nonlocality through the use of iterated integral operators. The operator introduces additional degrees of flexibility that can allow better representation of physical phenomena at different scales and in materials with different …

Harmonic Functions And Harmonic Measure, David Mcdonald

Honors Scholar Theses

The purpose of this thesis is to give a brief introduction to the field of harmonic measure. In order to do this we first introduce a few important properties of harmonic functions and show how to find a Green’s function for a given domain. Following this we calculate the harmonic measure for some easy cases and end by examining the connection between harmonic measure and Brownian motion.

Physical Applications Of The Geometric Minimum Action Method, George L. Poppe Jr.

Dissertations, Theses, and Capstone Projects

This thesis extends the landscape of rare events problems solved on stochastic systems by means of the \textit{geometric minimum action method} (gMAM). These include partial differential equations (PDEs) such as the real Ginzburg-Landau equation (RGLE), the linear Schroedinger equation, along with various forms of the nonlinear Schroedinger equation (NLSE) including an application towards an ultra-short pulse mode-locked laser system (MLL).

Additionally we develop analytical tools that can be used alongside numerics to validate those solutions. This includes the use of instanton methods in deriving state transitions for the linear Schroedinger equation and the cubic diffusive NLSE.

These analytical solutions are …

Energy Calculations And Wave Equations, Ellen R. Hunter

Graduate Theses/Dissertations

The focus of this thesis is to show how methods of Fourier analysis, in particular Parseval’s equality, can be used to provide explicit energy calculations for solutions of wave equations in one dimension. These calculations are discussed for simple examples and then extended to fit the general wave equation with Robin boundary conditions. Ideas from Sobolev space theory are used to provide justification of the method.

The Advection-Diffusion Equation And The Enhanced Dissipation Effect For Flows Generated By Hamiltonians, Michael Kumaresan

Dissertations, Theses, and Capstone Projects

We study the Cauchy problem for the advection-diffusion equation when the diffusive parameter is vanishingly small. We consider two cases - when the underlying flow is a shear flow, and when the underlying flow is generated by a Hamiltonian. For the former, we examine the problem on a bounded domain in two spatial variables with Dirichlet boundary conditions. After quantizing the system via the Fourier transform in the first spatial variable, we establish the enhanced-dissipation effect for each mode. For the latter, we allow for non-degenerate critical points and represent the orbits by points on a Reeb graph, with vertices …

Godunov-Type Upwind Flux Schemes Of The Two-Dimensional Finite Volume Discrete Boltzmann Method, Leitao Chen, Laura Schaefer

Publications

A simple unified Godunov-type upwind approach that does not need Riemann solvers for the flux calculation is developed for the finite volume discrete Boltzmann method (FVDBM) on an unstructured cell-centered triangular mesh. With piecewise-constant (PC), piecewise-linear (PL) and piecewise-parabolic (PP) reconstructions, three Godunov-type upwind flux schemes with different orders of accuracy are subsequently derived. After developing both a semi-implicit time marching scheme tailored for the developed flux schemes, and a versatile boundary treatment that is compatible with all of the flux schemes presented in this paper, numerical tests are conducted on spatial accuracy for several single-phase flow problems. Four major …

The Pope's Rhinoceros And Quantum Mechanics, Michael Gulas

Honors Projects

In this project, I unravel various mathematical milestones and relate them to the human experience. I show and explain the solution to the Tautochrone, a popular variation on the Brachistochrone, which details a major battle between Leibniz and Newton for the title of inventor of Calculus. One way to solve the Tautochrone is using Laplace Transforms; in this project I expound on common functions that get transformed and how those can be used to solve the Tautochrone. I then connect the solution of the Tautochrone to clock making. From this understanding of clocks, I examine mankind’s understanding of time and …

Clark Measures And A Theorem Of Ritt, Isabelle Chalendar, Pamela Gorkin, Jonathan R. Partington, William T. Ross

Department of Math & Statistics Faculty Publications

We determine when a finite Blaschke product B can be written, in a non-trivial way, as a composition of two finite Blaschke products (Ritt's problem) in terms of the Clark measure for B. Our tools involve the numerical range of compressed shift operators and the geometry of certain polygons circumscribing the numerical range of the relevant operator. As a consequence of our results, we can determine, in terms of Clark measures, when two finite Blaschke products commute.

Swelling As A Stabilizing Mechanism During Ion Bombardment Of Thin Films: An Analytical And Numerical Study, Jennifer M. Swenson

Mathematics Theses and Dissertations

Irradiation of semiconductor surfaces often leads to the spontaneous formation of rippled structures at certain irradiation angles. However, at high enough energies, these structures are observed to vanish for all angles, despite the absence of any identified, universally-stabilizing physical mechanisms in operation. Here, we examine the effect on pattern formation of radiation-induced swelling, which has been excluded from prior treatments of stress in irradiated films. After developing a suitable continuum model, we perform a linear stability analysis to determine its effect on stability. Under appropriate simplifying assumptions, we find swelling indeed to be stabilizing at wavenumbers typical of experimental observations. …