Applied Mathematics Commons^™

Theoretical Analysis Of Nonlinear Differential Equations, Emily Jean Weymier

Electronic Theses and Dissertations

Nonlinear differential equations arise as mathematical models of various phenomena. Here, various methods of solving and approximating linear and nonlinear differential equations are examined. Since analytical solutions to nonlinear differential equations are rare and difficult to determine, approximation methods have been developed. Initial and boundary value problems will be discussed. Several linear and nonlinear techniques to approximate or solve the linear or nonlinear problems are demonstrated. Regular and singular perturbation theory and Magnus expansions are our particular focus. Each section offers several examples to show how each technique is implemented along with the use of visuals to demonstrate the accuracy, …

Modeling Public Opinion, Arden Baxter

Honors Program Theses

The population dynamics of public opinion have many similarities to those of epidemics. For example, models of epidemics and public opinion share characteristics like contact rates, incubation times, and recruitment rates. Generally, epidemic dynamics have been presented through epidemiological models. In this paper we adapt an epidemiological model to demonstrate the population dynamics of public opinion given two opposing viewpoints. We find equilibrium solutions for various cases of the system and examine the local stability. Overall, our system provides sociological insight on the spread and transition of a public opinion.

Parts Of The Whole: Why I Teach This Subject This Way, Dorothy Wallace

Numeracy

The importance of mathematics to biology is illustrated by search data from Google Scholar. I argue that a pedagogical approach based on student research projects is likely to improve retention and foster critical thinking about mathematical modeling, as well as reinforce quantitative reasoning and the appreciation of calculus as a tool. The usual features of a course (e.g., the instructor, assessment, text, etc.) are shown to have very different purposes in a research-based course.

Steady And Stable: Numerical Investigations Of Nonlinear Partial Differential Equations, R. Corban Harwood

Faculty Publications - Department of Mathematics

Excerpt: "Mathematics is a language which can describe patterns in everyday life as well as abstract concepts existing only in our minds. Patterns exist in data, functions, and sets constructed around a common theme, but the most tangible patterns are visual. Visual demonstrations can help undergraduate students connect to abstract concepts in advanced mathematical courses. The study of partial differential equations, in particular, benefits from numerical analysis and simulation."

Dynamics Of Gene Networks In Cancer Research, Paul Scott

College of Graduate Studies: Theses & Dissertations

Cancer prevention treatments are being researched to see if an optimized treatment schedule would decrease the likelihood of a person being diagnosed with cancer. To do this we are looking at genes involved in the cell cycle and how they interact with one another. Through each gene expression during the life of a normal cell we get an understanding of the gene interactions and test these against those of a cancerous cell. First we construct a simplified network model of the normal gene network. Once we have this model we translate it into a transition matrix and force changes on …

Complex Solutions Of The Time Fractional Gross-Pitaevskii (Gp) Equation With External Potential By Using A Reliable Method, Nasir Taghizadeh, Mona N. Foumani

Applications and Applied Mathematics: An International Journal (AAM)

In this article, modified (G'/G )-expansion method is presented to establish the exact complex solutions of the time fractional Gross-Pitaevskii (GP) equation in the sense of the conformable fractional derivative. This method is an effective method in finding exact traveling wave solutions of nonlinear evolution equations (NLEEs) in mathematical physics. The present approach has the potential to be applied to other nonlinear fractional differential equations. Based on two transformations, fractional GP equation can be converted into nonlinear ordinary differential equation of integer orders. In the end, we will discuss the solutions of the fractional GP equation with external potentials.

Spontaneous Calcium Release In Cardiac Myocytes: Store Overload And Electrical Dynamics, Amanda M. Alexander, Erin K. Denardo, Eric Frazier Iii, Michael Mccauley, Nicholas Rojina, Zana Coulibaly, Bradford E. Peercy, Leighton T. Izu

Spora: A Journal of Biomathematics

Heart disease is the leading cause of mortality in the United States. One cause of heart arrhythmia is calcium (Ca²⁺) mishandling in cardiac muscle cells. We adapt Izu's et al. mathematical reaction-diffusion model of calcium in cardiac muscle cells, or cardiomyocytes implemented by Gobbert, and analyzed in Coulibaly et al. to include calcium being released from the sarcoplasmic reticulum (SR), the effects of buffers in the SR, particularly calsequestrin, and the effects of Ca²⁺ influx due to voltage across the cell membrane. Based on simulations of the model implemented in parallel using MPI, our findings aligned with …

Newton's Law Of Cooling, Caleb J. Emmons

Journal of Humanistic Mathematics

A poem reflecting three different viewpoints on Newton's Law of Cooling.

4. Dragging Along, Ruth Dover

Differential Equations

More information on air drag.

2. Population, Ruth Dover

Differential Equations

Introduction to logistic population growth.

3: Drugs And De's, Ruth Dover

Differential Equations

Making a connection between discrete recursion and differential equations.

1. Coffee, Ruth Dover

Differential Equations

Newton’s Law of Cooling.

Use Of Cubic B-Spline In Approximating Solutions Of Boundary Value Problems, Maria Munguia, Dambaru Bhatta

Applications and Applied Mathematics: An International Journal (AAM)

Here we investigate the use of cubic B-spline functions in solving boundary value problems. First, we derive the linear, quadratic, and cubic B-spline functions. Then we use the cubic B-spline functions to solve second order linear boundary value problems. We consider constant coefficient and variable coefficient cases with non-homogeneous boundary conditions for ordinary differential equations. We also use this numerical method for the space variable to obtain solutions for second order linear partial differential equations. Numerical results for various cases are presented and compared with exact solutions.

A Comparison Of Obesity Interventions Using Energy Balance Models, Marcella Torres

Theses and Dissertations

An energy balance model of human metabolism developed by Hall et al. is extended to compare body composition outcomes among standard and proposed obesity interventions. Standard interventions include a drastic diet or a drastic diet with endurance training. Outcomes for these interventions are typically poor in clinical studies. Proposed interventions include a gradual diet and the addition of resistance training to preserve lean mass and metabolic rate. We see that resistance training, regardless of dietary strategy, achieves these goals. Finally, we observe that the optimal obesity intervention for continued maintenance of a healthy body composition following a diet includes a …

Mathematical Equations And System Identification Models For A Portable Pneumatic Bladder System Designed To Reduce Human Exposure To Whole Body Shock And Vibration, Ezzat Aziz Ayyad

UNLV Theses, Dissertations, Professional Papers, and Capstones

A mathematical representation is sought to model the behavior of a portable pneumatic foam bladder designed to mitigate the effects of human exposure to shock and whole body random vibration. Fluid Dynamics principles are used to derive the analytic differential equations used for the physical equations Model. Additionally, combination of Wiener and Hammerstein block oriented representation techniques have been selected to create system identification (SID) block oriented models. A number of algorithms have been iterated to obtain numerical solutions for the system of equations which was found to be coupled and non-linear, with no analytic closed form solution. The purpose …

On The Evolution Of Virulence, Thi Nguyen

Electronic Theses, Projects, and Dissertations

The goal of this thesis is to study the dynamics behind the evolution of virulence. We examine first the underlying mechanics of linear systems of ordinary differential equations by investigating the classification of fixed points in these systems, then applying these techniques to nonlinear systems. We then seek to establish the validity of a system that models the population dynamics of uninfected and infected hosts---first with one parasite strain, then n strains. We define the basic reproductive ratio of a parasite, and study its relationship to the evolution of virulence. Lastly, we investigate the mathematics behind superinfection.

Stability Aware Delaunay Refinement, Bishal Acharya

UNLV Theses, Dissertations, Professional Papers, and Capstones

Good quality meshes are extensively used for finding approximate solutions for partial differential equations for fluid flow in two dimensional surfaces. We present an overview of existing algorithms for refinement and generation of triangular meshes. We introduce the concept of node stability in the refinement of Delaunay triangulation. We present two algorithms for generating stable refinement of Delaunay triangulation. We also present an experimental investigation of a triangulation refinement algorithm based on the location of the center of gravity and the location of the center of circumcircle. The results show that the center of gravity based refinement is more effective …

Multi-Objective Preliminary Design Optimization Of Planing Craft Using Differential Evolution And Artificial Networks, Robert Eugene Weidman

Mechanical & Aerospace Engineering Theses & Dissertations

The differential evolution (DE) algorithm is an evolutionary algorithm that has been utilized as a metaheuristic optimizer in the last few years. A recently developed multi-objective version of DE is utilized in this research in order to optimize the preliminary design of planing craft. The craft performance is evaluated using the planing hull preliminary design software PHAST. The optimization problem objectives are the craft acquisition cost and the overall measure of effectiveness; the latter comprises various performance aspects: ride quality, transportability, payload, and mobility. Static stability and operating constraints are also considered. Each of the aforementioned four measures of effectiveness …

Population Models, Jessica Hauer

EWU Masters Thesis Collection

This thesis will examine mathematical interpretations of biological situations through the study of differential equations. It will first explore the interactions of the lynx and hare populations in Canada based on data retrieved by the Hudson Bay Company. The purpose of this study is to find a suitable mathematical model, namely that of a three-variable Lotka-Volterra system. Also, the paper will explore short-term infectious disease models as they relate to particular epidemics throughout history, including the Iowa Mumps outbreak of 1966 and the Bubonic Plaque. The thesis will then work to make sense of the rise and fall patterns in …

Periodic Solutions And Positive Solutions Of First And Second Order Logistic Type Odes With Harvesting, Cody Alan Palmer

UNLV Theses, Dissertations, Professional Papers, and Capstones

It was recently shown that the nonlinear logistic type ODE with periodic harvesting has a bifurcation on the periodic solutions with respect to the parameter ε:

u' = f (u) - ε h (t).

Namely, there exists an ε0 such that for 0 < ε < ε0 there are two periodic solutions, for ε = ε0 there is one periodic solution, and for ε >ε0 there are no periodic solutions, provided that....

In this paper we look at some numerical evidence regarding the behavior of this threshold for various types of harvesting terms, in particular we find evidence in the negative or a conjecture regarding the behavior of this threshold value.

Additionally, we look at analagous steady states for the reaction-diusion …

Delay-Periodic Solutions And Their Stability Using Averaging In Delay-Differential Equations, With Applications, Thomas W. Carr, Richard Haberman, Thomas Erneux

Mathematics Research

Using the method of averaging we analyze periodic solutions to delay-differential equations, where the period is near to the value of the delay time (or a fraction thereof). The difference between the period and the delay time defines the small parameter used in the perturbation method. This allows us to consider problems with arbitrarily size delay times or of the delay term itself. We present a general theory and then apply the method to a specific model that has application in disease dynamics and lasers.

Parallel And Distributed Simulation Of Parabolic And Telegraphic Equations., Ewedafe Simon Uzezi

Student Works (2010-2019)

In this thesis, a parallel implementation of explicit/implicit parallel algorithms such as the stationary iterative methods and the class of iterating alternating methods which includes: Alternating Direction Implicit (ADI), Iterative Alternating Direction Explicit (IADE), for D’Yakonov (IADE-DY), Double sweep Mitchell and Fairweather (MF-DS) and Alternating Group Explicit (AGE) method for solving 1-Dimensional (1-D), 2-Dimensional (2-D) Parabolic (special examples including 1-D, 2-D Bio-Heat Equation) and 1-D, 2-D and 3-D Telegraphic Equations on a distributed environment of Message Passing Interface (MPI) and Parallel Virtual Machine (PVM) platform is presented. To correlate the communication activity with computation, we counted events between significant MPI/PVM …

Quenching For Degenerate Semilinear Parabolic Problems With Insulated Boundary Conditions, Bernard Iyawe

Theses Digitization Project

This thesis studied the existence, uniqueness, and quenching behavior of the solution to a degenerate equation subject to the initial condition and the second boundary conditions.

Blow-Up Behavior Of Solutions For Some Ordinary And Partial Differential Equations, Sarah Y. Bahk

Theses Digitization Project

There are two parts in this project. Part 1 the Riccati initial-value problem is looked at. Part 2 considers blow-up property solutions for the degenerate semilinear parabolic initial-boundary value problem.

Multigrid Convergence For Second Order Elliptic Problems With Smooth Complex Coefficients, Jay Gopalakrishnan, Joseph E. Pasciak

Mathematics and Statistics Faculty Publications and Presentations

The finite element method when applied to a second order partial differential equation in divergence form can generate operators that are neither Hermitian nor definite when the coefficient function is complex valued. For such problems, under a uniqueness assumption, we prove the continuous dependence of the exact solution and its finite element approximations on data provided that the coefficients are smooth and uniformly bounded away from zero. Then we show that a multigrid algorithm converges once the coarse mesh size is smaller than some fixed number, providing an efficient solver for computing discrete approximations. Numerical experiments, while confirming the theory, …

Existence Of Large Solutions To Semilinear Elliptic Equations With Multiple Terms, David N. Smith

Theses and Dissertations

We consider the semilinear elliptic equation Δu = p(x)u^α + q(x)u^β on a domain Ω ⊆ Rⁿ, n ≥ 3, where p and q are nonnegative continuous functions with the property that each of their zeroes is contained in a bounded domain Ω_p or Ω_q, respectively in Ω such that p is positive on the boundary of Ω_p and q is positive on the boundary of Ω_q. For Ω bounded, we show that there exists a nonnegative solution u such that u(x) → ∞ as x → ∂Ω if …

Existence Of Explosive Solutions To Non-Monotone Semilinear Elliptic Equations, Zachary H. Proano

Theses and Dissertations

We consider the semilinear elliptic equation Δu = p(x)f(u) on a domain ­ Ω ⊆ Rⁿ, n ≥ 3, where f is a nonnegative function which vanishes at the origin and satisfies g₁ ≤ f ≤ g₂ where g₁; g₂ are nonnegative, nondecreasing functions which also vanish at the origin, and p is a nonnegative continuous function with the property that any zero of p is contained in a bounded domain in ­ such that p is positive on its boundary. For Ω­ bounded, we show that a nonnegative solution u satisfying u(x) …

The Motion Of A Thin Liquid Film Driven By Surfactant And Gravity, Michael Shearer, Rachel Levy

All HMC Faculty Publications and Research

We investigate wave solutions of a lubrication model for surfactant-driven flow of a thin liquid film down an inclined plane. We model the flow in one space dimension with a system of nonlinear PDEs of mixed hyperbolic-parabolic type in which the effects of capillarity and surface diffusion are neglected. Numerical solutions reveal distinct patterns of waves that are described analytically by combinations of traveling waves, some with jumps in height and surfactant concentration gradient. The various waves and combinations are strikingly different from what is observed in the case of flow on a horizontal plane. Jump conditions admit new shock …

Optimal Therapy Regimens For Treatment-Resistant Mutations Of Hiv, Weiqing Gu, Helen Moore

All HMC Faculty Publications and Research

In this paper, we use control theory to determine optimal treatment regimens for HIV patients, taking into account treatment-resistant mutations of the virus. We perform optimal control analysis on a model developed previously for the dynamics of HIV with strains of various resistance to treatment (Moore and Gu, 2005). This model incorporates three types of resistance to treatments: strains that are not responsive to protease inhibitors, strains not responsive to reverse transcriptase inhibitors, and strains not responsive to either of these treatments. We solve for the optimal treatment regimens analytically and numerically. We find parameter regimes for which optimal dosing …

On Doubly Periodic Solutions Of Quasilinear Hyperbolic Equations Of The Fourth Order, T. Kiguradze, T. Smith

Publications

The problem on doubly periodic solutions is considered for a class of quasilinear hyperbolic equations. Effective sufficient conditions of solvability and unique solvability of this problem are established.