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Stability Of Cauchy's Equation On Δ+., Holden Wells

Electronic Theses and Dissertations

The most famous functional equation f(x+y)=f(x)+f(y) known as Cauchy's equation due to its appearance in the seminal analysis text Cours d'Analyse (Cauchy 1821), was used to understand fundamental aspects of the real numbers and the importance of regularity assumptions in mathematical analysis. Since then, the equation has been abstracted and examined in many contexts. One such examination, introduced by Stanislaw Ulam and furthered by Donald Hyers, was that of stability. Hyers demonstrated that Cauchy's equation exhibited stability over Banach Spaces in the following sense: functions that approximately satisfy Cauchy's equation are approximated with the same level of error by functions …

Analysis Of Covid-19 And Vaccine Administration In Mississippi, Megan Sickinger

Honors Theses

In this work, we develop a simple mathematical model to observe the spread of COVID-19 and vaccine administration in Mississippi. Based on the well-known Kermack-McKendrick Susceptible-Infected-Removed epidemiological model, the ASIRD−V model has eight ordinary differential equations that split infected populations and recovered populations into vaccinated and unvaccinated populations. After determining that the system is reliable for real-world applications, we investigate and determine the stability and equilibrium points of this system. The system is found to be disease-free when R0 < 1 and endemic when R0 > 1. We use MATLAB to numerically solve the system and optimize the model’s parameters over four short periods, two with the …

A Numerical Method For Solving Fuzzy Initial Value Problems, Safa Emad Al-Refai

Theses

In this thesis, the optimized one-step methods based on the hybrid block method (HBM) are derived for solving first and second-order fuzzy initial value problems. The off-step points are chosen to minimize the local truncation error of the proposed methods. Several theoretical properties of the proposed methods, such as stability, convergence, and consistency are investigated. Moreover, the regions of absolute stability of the proposed methods are plotted. Numerical results indicate that the proposed methods have order three and they are stable and convergent. In addition, several numerical examples are presented to show the efficiency and accuracy of the proposed methods. …

Stability Theory Of Nonlinear Differential Equations, Jiaxiao Wei

MSU Graduate Theses

Nonlinear differential equations are often effective tools in modeling some important phenomena in nature. However, most of the nonlinear ordinary differential equation cannot be solved by analytical methods. A more effective way is to explore the prop- erties of critical point and the trajectory around it. In this study, I will focus on sys- tems of autonomous differential equations, linear as well as nonlinear. I will not only focus on the stability of equilibria but also the orbital stability of nonlinear differential equations. I will introduce various approaches to the study of equilibrium points of the system in terms of …

Stability Analysis Of Krylov Subspace Spectral Methods For The 1-D Wave Equation In Inhomogeneous Media, Bailey Rester

Master's Theses

Krylov subspace spectral (KSS) methods are high-order accurate, explicit time-stepping methods for partial differential equations (PDEs) that also possess the stability characteristic of implicit methods. Unlike other time-stepping approaches, KSS methods compute each Fourier coefficient of the solution from an individualized approximation of the solution operator of the PDE. As a result, KSS methods scale effectively to higher spatial resolution. This thesis will present a stability analysis of a first-order KSS method applied to the wave equation in inhomogeneous media.

Stochastic Delay Differential Equations With Applications In Ecology And Epidemics, Hebatallah Jamil Alsakaji

Dissertations

Mathematical modeling with delay differential equations (DDEs) is widely used for analysis and predictions in various areas of life sciences, such as population dynamics, epidemiology, immunology, physiology, and neural networks. The memory or time-delays, in these models, are related to the duration of certain hidden processes like the stages of the life cycle, the time between infection of a cell and the production of new viruses, the duration of the infectious period, the immune period, and so on. In ordinary differential equations (ODEs), the unknown state and its derivatives are evaluated at the same time instant. In DDEs, however, the …

Bifurcation Analysis Of Two Biological Systems: A Tritrophic Food Chain Model And An Oscillating Networks Model, Xiangyu Wang

Electronic Thesis and Dissertation Repository

In this thesis, we apply bifurcation theory to study two biological systems. Main attention is focused on complex dynamical behaviors such as stability and bifurcation of limit cycles. Hopf bifurcation is particularly considered to show bistable or even tristable phenomenon which may occur in biological systems. Recurrence is also investigated to show that such complex behavior is common in biological systems.

First we consider a tritrophic food chain model with Holling functional response types III and IV for the predator and superpredator, respectively. Main attention is focused on the sta- bility and bifurcation of equilibria when the prey has a …

Switching Diffusion Systems With Past-Dependent Switching Having A Countable State Space, Hai Dang Nguyen

Wayne State University Dissertations

Emerging and existing applications in wireless communications, queueing networks, biological models, financial engineering, and social networks demand the

mathematical modeling and analysis of hybrid models in which continuous dynamics and discrete events coexist.

Assuming that the systems are in continuous times,

stemming from stochastic-differential-equation-based models and random discrete events,

switching diffusions come into being. In such systems, continuous states and discrete events

(discrete states)

coexist and interact.

A switching diffusion is a two-component process $(X(t),\alpha(t))$, a continuous component and a discrete component taking values in a discrete set (a set consisting of isolated points).

When the discrete component takes a …

Orbital Stability Results For Soliton Solutions To Nonlinear Schrödinger Equations With External Potentials, Joseph B. Lindgren

Theses and Dissertations--Mathematics

For certain nonlinear Schroedinger equations there exist solutions which are called solitary waves. Addition of a potential $V$ changes the dynamics, but for small enough $||V||_{L^\infty}$ we can still obtain stability (and approximately Newtonian motion of the solitary wave's center of mass) for soliton-like solutions up to a finite time that depends on the size and scale of the potential $V$. Our method is an adaptation of the well-known Lyapunov method.

For the sake of completeness, we also prove long-time stability of traveling solitons in the case $V=0$.

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 …

Computational Analysis Of The Sir Mathematical Model For The Dengue Fever, Joseph Phillip Diaz

Theses and Dissertations

Dengue fever is a disease affecting people in more than 100 countries. Here we consider a host and vector model for the transmission of dengue fever. This SIR model consists of three compartments of susceptible, infective and removed for host (human) and two compartments of susceptible and infective for vector (dengue mosquitos). These five compartments yield five coupled nonlinear ordinary differential equations (ODEs). After non-dimensionalization, we have a system of three nonlinear ODEs. Reproductive number and two equilibrium points are calculated for various cases. Simulation is carried out for susceptible, infective and removed and the results are presented in graphical …

Eigenvalue Methods For Interpolation Bases, Piers W. Lawrence

Electronic Thesis and Dissertation Repository

This thesis investigates eigenvalue techniques for the location of roots of polynomials expressed in the Lagrange basis. Polynomial approximations to functions arise in almost all areas of computational mathematics, since polynomial expressions can be manipulated in ways that the original function cannot. Polynomials are most often expressed in the monomial basis; however, in many applications polynomials are constructed by interpolating data at a series of
points. The roots of such polynomial interpolants can be found by computing the eigenvalues of a generalized companion matrix pair constructed directly from the values of the interpolant. This affords the opportunity to work with …

Mathematical Modeling And Analysis Of Asthma Stability And Severity, Arezoo Hanifi

Electronic Theses and Dissertations

Asthma is one of the most common chronic conditions in the United States. Asthma affects about one in fifteen people. It affects children more than adults and blacks more than whites. People with asthma experience attacks of wheezing, breathlessness, chest tightness, and coughing. Asthma can be fatal and the costs for the disease (direct and indirect) are approximated to be tens of billions of dollars each year.

There is no cure for asthma. However; for most people if asthma is controlled well they can lead normal, active lives. Therefore asthma controllability is a main factor in clinical practice. In order …

Bifurcations And Stability In Models Of Infectious Diseases, Bernard S. Chan

Electronic Thesis and Dissertation Repository

This work is concerned with bifurcation and stability in models related to various aspects of infections diseases.

First, we study the dynamics of a mathematical model on primary and secondary cytotoxic T-lymphocyte responses to viral infections by Wodarz et al. This model has three equilibria and the stability criteria of them are discussed. We analytically show that periodic solutions may arise from the third equilibrium via Hopf bifurcation. Numerical simulations of the model agree with the theoretical results. These dynamical behaviours occur within biologically realistic parameter range.

After studying the single-strain model, we analyze the bifurcation dynamics of an …

Asymptotic Expansions And Stability Of Hybrid Systems With Two-Time Scales, Dung Tien Nguyen

Wayne State University Dissertations

In this dissertation, we consider solutions of hybrid systems in which both continuous dynamics and discrete events coexists. One

of the main ingredients of our models is the two-time-scale formulation. Under broad conditions, asymptotic expansions are developed for the solutions of the systems of backward equations for switching diffusion in two classes of models, namely, fast switching systems and fast diffusion systems. To prove the validity of the asymptotic expansions, uniform error bounds are obtained.

In the second part of the dissertation, a singular linear system is considered. Again a two-time-scale formulation is used. Under suitable conditions, the system has …

Mathematical Models And Stability Analysis Of Cholera Dynamics, Shu Liao

Mathematics & Statistics Theses & Dissertations

In this dissertation, we present a careful mathematical study of several epidemic cholera models, including the model of Codeco [11] in 2001, that of Hartley, Morris and Smith [22] in 2006, and that of Mukandavire, Liao, Wang and Gaff et al. [60] in 2010. We formally derive the basic reproduction number R0 for each model by computing the spectral radius of the next generation matrix. We focus our attention on the stability analysis at the disease-free equilibrium which determines the short-term epidemic behavior, and the endemic equilibrium which determines the long-term disease dynamics. Particularly, we incorporate the Volterra-Lyapunov matrix …

Stability In Dynamical Polysystems, George Cazacu

LSU Doctoral Dissertations

A dynamical polysystem consists of a family of continuous dynamical systems, all acting on a given metric space. The first chapter of the present thesis shows a generalization of control systems via dynamical polysystems and establishes the equivalence of the two notions under certain lipschitz condition on the function defining the dynamics. The remaining chapters are focused on a basic theory of dynamical polysystems. Some topological properties of limit sets are described in Chapter 2. Chapters 3 and 4 provide characterizations for various notions of strong stability. Chapter 5 makes use of the theory of closed relations to study Lyapunov …