Physical Sciences and Mathematics Commons^™

Applying The Sir Model: Can Students Advise The Mayor Of A Small Community?, Carrin Goosen, Mark I. Nelson, Mahime Watanabe

CODEE Journal

This is an account of a modelling scenario that uses the sir epidemic model. It was used in a third year applied mathematics subject. All students were enrolled in a mathematics degree of some type. Students are presented with the results of a test carried out on 100 individuals in a community containing 3000 people. From this they determined the number of infectious and recovered individuals in the population. Given the per capita recovery rate and making a suitable assumption about the number of infectious individuals at the start of the epidemic, they then estimate the infectious contact rate and …

Solar Panels, Euler’S Method And Community-Based Projects: Connecting Differential Equations With Climate Change, Victor J. Donnay

CODEE Journal

How does mathematics connect with the search for solutions to the climate emergency? One simple connection, which can be explored in an introductory differential equations course, can be found by analyzing the energy generated by solar panels or wind turbines. The power generated by these devices is typically recorded at standard time intervals producing a data set which gives a discrete approximation to the power function $P(t)$. Using numerical techniques such as Euler’s method, one can determine the energy generated. Here we describe how we introduce the topic of solar power, apply Euler’s method to determine the energy generated, and …

A Scattering Result For The Fifth-Order Kp-Ii Equation, Camille Schuetz

Theses and Dissertations--Mathematics

We will prove scattering for the fifth-order Kadomtsev-Petviashvilli II (fifth-order KP-II) equation. The fifth-order KP-II equation is an example of a nonlinear dispersive equation which takes the form $u_t=Lu + NL(u)$ where $L$ is a linear differential operator and $NL$ is a nonlinear operator. One looks for solutions $u(t)$ in a space $C(\R,X)$ where $X$ is a Banach space. For a nonlinear dispersive differential equation, the associated linear problem is $v_t=Lv$. A solution $u(t)$ of the nonlinear equation is said to scatter if as $t \to \infty$, the solution $u(t)$ approaches a solution $v(t)$ to the linear problem in the …

On The Spatial Modelling Of Biological Invasions, Tedi Ramaj

Electronic Thesis and Dissertation Repository

We investigate problems of biological spatial invasion through the use of spatial modelling. We begin by examining the spread of an invasive weed plant species through a forest by developing a system of partial differential equations (PDEs) involving an invasive weed and a competing native plant species. We find that extinction of the native plant species may be achieved by increasing the carrying capacity of the forest as well as the competition coefficient between the species. We also find that the boundary conditions exert long-term control on the biomass of the invasive weed and hence should be considered when implementing …

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 …

Resources For Supporting Mathematics And Data Science Instructors During Covid-19, Eduardo C. Balreira, C. Hawthorne, G. Stadnyk, Z. Teymuroglu, M. Torres, J. R. Wares

Mathematics Faculty Research

In late May of 2020, a few months after the raging COVID-19 pandemic forced university faculty to quickly switch to online teaching, the Associated Colleges of the South (ACS) released a call for grant applications to support working groups "to help faculty within our consortium who will be teaching during the pandemic (e.g., from hybrid courses with some remote/online components to fully remote/online courses; socially distanced face-to-face courses)." We replied to this call and the ACS awarded the six of us (from four ACS schools) a Summer Rapid Response Grant in early June. The grant funded our efforts to create …

Some Results On A Two Variables Pell Polynomials, Mohammed Abdulhadi Sarhan, Suha Shihab, Mohammed Rasheed

Al-Qadisiyah Journal of Pure Science

In this study, Pell polynomials in two variables, and their properties are investigated. Some formulas for two variables Pell polynomials are derived by matrices. By defining special formula for Pell polynomials in one variable, new important properties of Pell polynomials in two variables can be enabled to derive. A new exact formula expressing the partial derivatives of Pell polynomials explicitly of any degree in terms of Pell polynomials themselves is proved. A novel explicit formula, which constructs the two explicit formulas, which construct the two-dimension Pell polynomials expansion coefficients of a first partial derivative of a differentiable function in terms …

Analysis Of An Ode Model For Sea Turtle Populations With Temperature-Dependent Sex Determination, Lindsey A. Ukishima

Student Publications

The sex of green sea turtles is determined by the temperature at which the eggs are incubated. Recent studies have shown that the sex ratios of sea turtle populations have changed over recent years, likely due to climate change, which has produced a more female-biased population. This paper finds the nonzero equilibrium point of the novel system developed by Herrera et a. (2019) and attempts to determine the stability of the population at that point.

Climate Change Models, Lauren Fie

Capstone Showcase

As a result of the changing climate, global temperatures and global mean sea levels (GMSL) have been increasing rapidly. The complex physical systems surrounding this growth make it difficult to form an accurate model. This paper looks at a simplified model proposed and supported by Aral, Guan, and Chang. This model consists of a system of ordinary differential equations that are simplified and solved theoretically, then applied using python to calculate precise values and form predictions.

Adjoint Appell-Euler And First Kind Appell-Bernoulli Polynomials, Pierpaolo Natalini, Paolo E. Ricci

Applications and Applied Mathematics: An International Journal (AAM)

The adjunction property, recently introduced for Sheffer polynomial sets, is considered in the case of Appell polynomials. The particular case of adjoint Appell-Euler and Appell-Bernoulli polynomials of the first kind is analyzed.

Stability Analysis For The Equilibria Of A Monkeypox Model, Rachel Elizabeth Tewinkel

Theses and Dissertations

Monkeypox virus was first identified in 1958 and has since been an ongoing problem in Central and Western Africa. Although the smallpox vaccine provides partial immunity against monkeypox, the number of cases has greatly increased since the eradication of smallpox made its vaccination unnecessary. Although studied by epidemiologists, monkeypox has not been thoroughly studied by mathematicians to the extent of other serious diseases. Currently, to our knowledge, only three mathematical models of monkeypox have been proposed and studied. We present the first of these models, which is related to the second, and discuss the global and local asymptotic stability of …

On The Complexity Of Computing Galois Groups Of Differential Equations, Mengxiao Sun

Dissertations, Theses, and Capstone Projects

The differential Galois group is an analogue for a linear differential equation of the classical Galois group for a polynomial equation. An important application of the differential Galois group is that a linear differential equation can be solved by integrals, exponentials and algebraic functions if and only if the connected component of its differential Galois group is solvable. Computing the differential Galois groups would help us determine the existence of the solutions expressed in terms of elementary functions (integrals, exponentials and algebraic functions) and understand the algebraic relations among the solutions.

Hrushovski first proposed an algorithm for computing the differential …

Modeling The Spread And Prevention Of Malaria In Central America, Michael Huber

CODEE Journal

In 2016, the World Health Organization (WHO) estimated that there were 216 million cases of Malaria reported in 91 countries around the world. The Central American country of Honduras has a high risk of malaria exposure, especially to United States soldiers deployed in the region. This article will discuss various aspects of the disease, its spread and its treatment and the development of models of some of these aspects with differential equations. Exercises are developed which involve, respectively, exponential growth, logistics growth, systems of first-order equations and Laplace transforms. Notes for instructors are included.

Linking Differential Equations To Social Justice And Environmental Concerns

CODEE Journal

Special issue of the CODEE Journal in honor of its founder, Professor Robert Borrelli.

A Model Of The Transmission Of Cholera In A Population With Contaminated Water, Therese Shelton, Emma Kathryn Groves, Sherry Adrian

CODEE Journal

Cholera is an infectious disease that is a major concern in countries with inadequate access to clean water and proper sanitation. According to the World Health Organization (WHO), "cholera is a disease of inequity--an ancient illness that today sickens and kills only the poorest and most vulnerable people\dots The map of cholera is essentially the same as a map of poverty." We implement a published model (Fung, "Cholera Transmission Dynamic Models for Public Health Practitioners," Emerging Themes in Epidemiology, 2014) of a SIR model that includes a bacterial reservoir. Bacterial concentration in the water is modeled by the Monod …

Sir Models: Differential Equations That Support The Common Good, Lorelei Koss

CODEE Journal

This article surveys how SIR models have been extended beyond investigations of biologically infectious diseases to other topics that contribute to social inequality and environmental concerns. We present models that have been used to study sustainable agriculture, drug and alcohol use, the spread of violent ideologies on the internet, criminal activity, and health issues such as bulimia and obesity.

A Note On Equity Within Differential Equations Education By Visualization, Younes Karimifardinpour

CODEE Journal

The growing importance of education equity is partly based on the premise that an individual's level of education directly correlates to future quality of life. Educational equity for differential equations (DEs) is related to achievement, fairness, and opportunity. Therefore, a pedagogy that practices DE educational equity gives a strong foundation of social justice. However, linguistic barriers pose a challenge to equity education in DEs. For example, I found myself teaching DEs either in classrooms with a low proficiency in the language of instruction or in multilingual classrooms. I grappled with a way to create an equity educational environment that supported …

Experiences Using Inquiry-Oriented Instruction In Differential Equations, Keith Nabb

CODEE Journal

Student-centered instruction can be a challenging endeavor for teachers and students. This article reports on the use of the Inquiry-Oriented Differential Equations (IO-DE) curriculum (Rasmussen, 2002) in an undergraduate differential equations course. Examples of student work are shared with specific reference to research in mathematics education.

Teaching Differential Equations Without Computer Graphics Solutions Is A Crime, Beverly H. West

CODEE Journal

In the early 1980s computer graphics revolutionized the teaching of ordinary differential equations (ODEs). Yet the movement to teach and learn the qualitative methods that interactive graphics affords seems to have lost momentum. There still exist college courses, even at big universities, being taught without the immense power that computer graphics has brought to differential equations. The vast majority of ODEs that arise in mathematical models are nonlinear, and linearization only approximates solutions sufficiently near an equilibrium. Introductory courses need to include nonlinear DEs. Graphs of phase plane trajectories and time series solutions allow one to see and analyze the …

An Accurate Method For Solving Higher Order Differential Equations Based On The Implicit Hybrid Method, Ahlam Mohammed S Almokhmari

Mathematical Sciences Theses

In this thesis, we present numerical techniques to solve higher order differential equations based on the implicit Hybrid method. In these methods, we use the collocation and interpolating methods. Then, we derive the main schemes and their block methods. We investigate some theoretical results such as the order of the method, consistency, convergence, and region of absolute stability. Some numerical results and simulations are provided to show the efficiency of the proposed methods using Mathematica.

Galois Groups Of Differential Equations And Representing Algebraic Sets, Eli Amzallag

Dissertations, Theses, and Capstone Projects

The algebraic framework for capturing properties of solution sets of differential equations was formally introduced by Ritt and Kolchin. As a parallel to the classical Galois groups of polynomial equations, they devised the notion of a differential Galois group for a linear differential equation. Just as solvability of a polynomial equation by radicals is linked to the equation’s Galois group, so too is the ability to express the solution to a linear differential equation in "closed form" linked to the equation’s differential Galois group. It is thus useful even outside of mathematics to be able to compute and represent these …

Differential Equations (Fvsu), Samuel Cartwright, Bhavana Burell, Patcharin Marion, Jianmin Zhu

Mathematics Grants Collections

This Grants Collection for Differential Equations was created under a Round Six ALG Textbook Transformation Grant.

Affordable Learning Georgia Grants Collections are intended to provide faculty with the frameworks to quickly implement or revise the same materials as a Textbook Transformation Grants team, along with the aims and lessons learned from project teams during the implementation process.

Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials:

• Linked Syllabus
• Initial Proposal
• Final Report

Differential Equations (Ung), Hashim Saber, Beata Hebda, Piotr Hebda, Benkam Bobga

Mathematics Grants Collections

This Grants Collection for Differential Equations was created under a Round Seven ALG Textbook Transformation Grant.

Affordable Learning Georgia Grants Collections are intended to provide faculty with the frameworks to quickly implement or revise the same materials as a Textbook Transformation Grants team, along with the aims and lessons learned from project teams during the implementation process.

Documents are in .pdf format, with a separate .docx (Word) version available for download. Each collection contains the following materials:

• Linked Syllabus
• Initial Proposal
• Final Report

Radial Basis Function Generated Finite Differences For The Nonlinear Schrodinger Equation, Justin Ng

Theses and Dissertations

Solutions to the one-dimensional and two-dimensional nonlinear Schrodinger (NLS) equation are obtained numerically using methods based on radial basis functions (RBFs). Periodic boundary conditions are enforced with a non-periodic initial condition over varying domain sizes. The spatial structure of the solutions is represented using RBFs while several explicit and implicit iterative methods for solving ordinary differential equations (ODEs) are used in temporal discretization for the approximate solutions to the NLS equation. Splitting schemes, integration factors and hyperviscosity are used to stabilize the time-stepping schemes and are compared with one another in terms of computational efficiency and accuracy. This thesis shows …

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, …

The Mathematics Of Cancer: Fitting The Gompertz Equation To Tumor Growth, Dyjuan Tatro

Senior Projects Spring 2018

Mathematical models are finding increased use in biology, and partuculary in the field of cancer research. In relation to cancer, systems of differential equations have been proven to model tumor growth for many types of cancer while taking into account one or many features of tumor growth. One feature of tumor growth that models must take into account is that tumors do not grow exponentially. One model that embodies this feature is the Gomperts Model of Cell Growth. By fitting this model to long-term breast cancer study data, this project ascertains gompertzian parameters that can be used to predicts tumor …

A Unified Inter-Host And In-Host Model Of Antibiotic Resistance And Infection Spread In A Hospital Ward, Lester Caudill, Barry Lawson

Department of Math & Statistics Faculty Publications

As the battle continues against hospital-acquired infections and the concurrent rise in antibiotic resistance among many of the major causative pathogens, there is a dire need to conduct controlled experiments, in order to compare proposed control strategies. However, cost, time, and ethical considerations make this evaluation strategy either impractical or impossible to implement with living patients. This paper presents a multi-scale model that offers promise as the basis for a tool to simulate these (and other) controlled experiments. This is a “unified” model in two important ways: (i) It combines inter-host and in-host dynamics into a single model, and (ii) …

Models Of Nation-Building Via Systems Of Differential Equations, Carissa F. Slone, Darryl K. Ahner, Mark E. Oxley, William P. Baker

The Research and Scholarship Symposium (2013-2019)

Nation-building modeling is an important field of research given the increasing number of candidate nations and the limited resources available. A modeling methodology and a system of differential equations model are presented to investigate the dynamics of nation-building. The methodology is based upon parameter identification techniques applied to a system of differential equations, to evaluate nation-building operations. Data from Operation Iraqi Freedom (OIF) and Afghanistan are used to demonstrate the validity of different models as well as the comparison of models.

Weighing Fog: Hands On Modeling For Day 1 Of Differential Equations, Tom Clark

Faculty Work Comprehensive List

The first day of many mathematics classes contains formalities and very little mathematics. Here an alternative is presented where modeling is placed as the centerpiece to orient students to the real work of differential equations. Namely, to capture as beautifully and compactly as possible through the process of conjecture and investigation, the deep and interesting aspects of the physical world. A demonstration of the sublimation of dry ice sits at the center of the lesson. Students collaborate in groups to design an experiment that could measure the change in mass of a piece of dry ice that is dropped into …

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."