Applied Mathematics Commons^™

Newton's Law Of Cooling, Caleb J. Emmons

Journal of Humanistic Mathematics

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

The Global Stability Of The Solution To The Morse Potential In A Catastrophic Regime, Weerapat Pittayakanchit

HMC Senior Theses

Swarms of animals exhibit aggregations whose behavior is a challenge for mathematicians to understand. We analyze this behavior numerically and analytically by using the pairwise interaction model known as the Morse potential. Our goal is to prove the global stability of the candidate local minimizer in 1D found in A Primer of Swarm Equilibria. Using the calculus of variations and eigenvalues analysis, we conclude that the candidate local minimizer is a global minimum with respect to all solution smaller than its support. In addition, we manage to extend the global stability condition to any solutions whose support has a single …

Topological Data Analysis For Systems Of Coupled Oscillators, Alec Dunton

HMC Senior Theses

Coupled oscillators, such as groups of fireflies or clusters of neurons, are found throughout nature and are frequently modeled in the applied mathematics literature. Earlier work by Kuramoto, Strogatz, and others has led to a deep understanding of the emergent behavior of systems of such oscillators using traditional dynamical systems methods. In this project we outline the application of techniques from topological data analysis to understanding the dynamics of systems of coupled oscillators. This includes the examination of partitions, partial synchronization, and attractors. By looking for clustering in a data space consisting of the phase change of oscillators over a …

1-65-S-Algal Blooms: Algal Blooms Threatening Lake Chapala, R. Corban Harwood

Faculty Publications - Department of Mathematics

This modeling scenario investigates the massive algal blooms that struck Lake Chapala, Mexico, starting in 1994. After reading a summary of articles written on the incidents, students are guided through the process of creating a first order differential equation from a verbal model of the factors and analyze the nonautonomous ODE using direction field, parameter evaluation, and exact solution computation to fully describe the population behavior. Students are expected to be familiar with the separable method and direction fields. Students will learn building and improving a model from qualitative descriptions, nondimensionalization, evaluating parameters, and how to use DFIELD software to …

Development And Separation Of Forced Convective Flow, Adefemi Sunmonu

Publications and Research

Singularities are considered in the solution of the laminar bound- ary - layer equation at a position of separation. The works of Howarth (1938). Goldstein (1948), Stewartson (1958), Terrill (1960) and Akin- relere [(1981), (1982)] are reviewed to fully establish the existence of singularity in the incompressible boundary layer at separation for both the velocity and thermal fields. A ow at a large Reynold's number along an immersed solid surface around which bounda y layer is formed through which the velocity rises rapidly from zero at the surface to its value in the main stream is considered. It is found …

The Dynamics Of Vector-Borne Relapsing Diseases, Cody Palmer

Graduate Student Theses, Dissertations, & Professional Papers

We begin this dissertation with a review of the relevant history and theory behind disease modeling, investigating important motivating examples. The concept of the fundamental reproductive ratio of a disease, $R_0$, is introduced through these examples. The compartmental theory of disease spread and its results are introduced, particularly the next-generation method of computing $R_0$. We review center manifold theory, as it is critical to the reduction of the dimension of our problems. We review diseases that have a relapsing character and focus in on relapsing diseases that are spread by vectors in a host population. The primary example of such …

An Interactive Tool For The Computational Exploration Of Integrodifference Population Models, Kennedy Agwamba

HMC Senior Theses

Mathematical modeling of population dynamics can provide novel insight to the growth and dispersal patterns for a variety of species populations, and has become vital to the preservation of biodiversity on a global-scale. These growth and dispersal stages can be modeled using integrodifference equations that are discrete in time and continuous in space. Previous studies have identified metrics that can determine whether a given species will persist or go extinct under certain model parameters. However, a need for computational tools to compute these metrics has limited the scope and analysis within many of these studies. We aim to create computational …

A Physiologically-Based Pharmacokinetic Model For Vancomycin, Rebekah White

Undergraduate Honors Theses

Vancomycin is an antibiotic used for the treatment of systemic infections. It is given

intravenously usually every twelve or twenty-four hours. This particular drug has a

medium level of boundedness, with approximately fty percent of the drug being free

and thus physiologically eective. A physiologically-based pharmacokinetic (PBPK)

model was used to better understand the absorption, distribution, and elimination of

the drug. Using optimal parameters, the model could be used in the future to test

how various factors, such as BMI or excretion levels, might aect the concentration

of the antibiotic.

Dynamics Of An Sir Model With Nonlinear Incidence And Treatment Rate, Balram Dubey, Preeti Dubey, Uma S. Dubey

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, global dynamics of an SIR model are investigated in which the incidence rate is being considered as Beddington-DeAngelis type and the treatment rate as Holling type II (saturated). Analytical study of the model shows that the model has two equilibrium points (diseasefree equilibrium (DFE) and endemic equilibrium (EE)). The disease-free equilibrium (DFE) is locally asymptotically stable when reproduction number is less than one. Some conditions on the model parameters are obtained to show the existence as well as nonexistence of limit cycle. Some sufficient conditions for global stability of the endemic equilibrium using Lyapunov function are obtained. …

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.

The Shifted Jacobi Polynomial Integral Operational Matrix For Solving Riccati Differential Equation Of Fractional Order, A. Neamaty, B. Agheli, R. Darzi

Applications and Applied Mathematics: An International Journal (AAM)

In this article, we have applied Jacobi polynomial to solve Riccati differential equation of fractional order. To do so, we have presented a general formula for the Jacobi operational matrix of fractional integral operator. Using the Tau method, the solution of this problem reduces to the solution of a system of algebraic equations. The numerical results for the examples presented in this paper demonstrate the efficiency of the present method.

Local Fractional Variational Iteration Method For Solving Nonlinear Partial Differential Equations Within Local Fractional Operators, Hossein Jafari, Hassan K. Jassim

Applications and Applied Mathematics: An International Journal (AAM)

In this article, the local fractional variational iteration method is proposed to solve nonlinear partial differential equations within local fractional derivative operators. To illustrate the ability and reliability of the method, some examples are illustrated. A comparison between local fractional variational iteration method with the other numerical methods is given, revealing that the proposed method is capable of solving effectively a large number of nonlinear differential equations with high accuracy. In addition, we show that local fractional variational iteration method is able to solve a large class of nonlinear problems involving local fractional operators effectively, more easily and accurately, and …

A Boundedness And Stability Results For A Kind Of Third Order Delay Differential Equations, Moussadek Remili, Djamila Beldjerd

Applications and Applied Mathematics: An International Journal (AAM)

The objective of this study was to get some sufficient conditions which guarantee the asymptotic stability and uniform boundedness of the null solution of some differential equations of third order with the variable delay. The most efficient tool for the study of the stability and boundedness of solutions of a given nonlinear differential equation is provided by Lyapunov theory. However the construction of such functions which are positive definite with corresponding negative definite derivatives is in general a difficult task, especially for higher-order differential equations with delay. Such functions and their time derivatives along the system under consideration must satisfy …

Numerical Solutions Of Generalized Burgers' Equations For Some Incompressible Non-Newtonian Fluids, Yupeng Shu

LSU New Orleans Theses and Dissertations

The author presents some generalized Burgers' equations for incompressible and isothermal flow of viscous non-Newtonian fluids based on the Cross model, the Carreau model, and the Power-Law model and some simple assumptions on the flows. The author numerically solves the traveling wave equations for the Cross model, the Carreau model, the Power-Law model by using industrial data. The author proves existence and uniqueness of solutions to the traveling wave equations of each of the three models. The author also provides numerical estimates of the shock thickness as well as maximum strain $\varepsilon_{11}$ for each of the fluids.

Spontaneous Synchrony On Graphs And The Emergence Of Order From Disorder, Dylan Linville, Daniel Trugillo Martins Fontes

Mathematical Sciences Technical Reports (MSTR)

From pulsars to pedestrians and bacteria to brain cells, objects that exhibit cyclical behavior, called oscillators, are found in a variety of different settings. When oscillators adjust their behavior in response to nearby oscillators, they often achieve a state of synchrony, in which they all have the same phase and frequency. Here, we explore the Kuramoto model, a simple and general model which describes oscillators as dynamical systems on a graph and has been used to study synchronization in systems ranging from firefly swarms to the power grid. We discuss analytical and numerical methods used to investigate the governing system …

Spontaneous Synchrony On Graphs And The Emergence Of Order From Disorder, Dylan Linville, Daniel Trugillo Martins Fontes

Rose-Hulman Undergraduate Research Publications

From pulsars to pedestrians and bacteria to brain cells, objects that exhibit cyclical behavior, called oscillators, are found in a variety of different settings. When oscillators adjust their behavior in response to nearby oscillators, they often achieve a state of synchrony, in which they all have the same phase and frequency. Here, we explore the Kuramoto model, a simple and general model which describes oscillators as dynamical systems on a graph and has been used to study synchronization in systems ranging from firefly swarms to the power grid. We discuss analytical and numerical methods used to investigate the governing system …

On A Convex Set With Nondifferentiable Metric Projection, Shyan S. Akmal, Nguyen Mau Nam, J. J. P. Veerman

Mathematics and Statistics Faculty Publications and Presentations

A remarkable example of a nonempty closed convex set in the Euclidean plane for which the directional derivative of the metric projection mapping fails to exist was constructed by A. Shapiro. In this paper, we revisit and modify that construction to obtain a convex set with smooth boundary which possesses the same property.

Comparison Of Two Parameter Estimation Techniques For Stochastic Models, Thomas C. Robacker

Electronic Theses and Dissertations

Parameter estimation techniques have been successfully and extensively applied to deterministic models based on ordinary differential equations but are in early development for stochastic models. In this thesis, we first investigate using parameter estimation techniques for a deterministic model to approximate parameters in a corresponding stochastic model. The basis behind this approach lies in the Kurtz limit theorem which implies that for large populations, the realizations of the stochastic model converge to the deterministic model. We show for two example models that this approach often fails to estimate parameters well when the population size is small. We then develop a …

New Exact Solutions Of The Perturbed Nonlinear Fractional Schr¨Odinger Equation Using Two Reliable Methods, Nasir Taghizadeh, Mona N. Foumani, Vahid S. Mohammadi

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, the fractional derivatives in the sense of the modified Riemann-Liouville derivative and the first integral method and the Bernoulli sub-ODE method are employed for constructing the exact complex solutions of the perturbed nonlinear fractional Schr ¨odinger equation and comparing the solutions.

Hyers-Ulam And Hyers-Ulam-Aoki-Rassias Stability For Linear Ordinary Differential Equations, A. N. Mohapatra

Applications and Applied Mathematics: An International Journal (AAM)

Here we prove the Hyers-Ulam stability and Hyers-Ulam-Aoki-Rassias stability of the n-th order ordinary linear differential equation with smooth coefficients on compact and semi-bounded intervals using successive integration by parts. Keywords:

Solution Of Fractional Drinfeld-Sokolov-Wilson Equation Using Homotopy Perturbation Transform Method, P. K. Singh, K. Vishal, T. Som

Applications and Applied Mathematics: An International Journal (AAM)

In this article, the approximate solutions of the non-linear Drinfeld-Sokolov-Wilson equation with fractional time derivative have been obtained. The fractional derivative is described in the Caputo sense. He’s polynomial is used to tackle the nonlinearity which arise in our considered problems. A time fractional nonlinear partial differential equation has been computed numerically. The numerical procedures illustrate the effectiveness and reliability of the method. Effects of fractional order time derivatives on the solutions for different particular cases are presented through graphs.

Application Of Reduced Differential Transform Method For Solving Nonlinear Reaction-Diffusion-Convection Problems, A. Taghavi, A. Babaei, A.` Mohammadpour

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, Reduced differential transform method is presented for solving nonlinear reactiondiffusion- convection initial value problems. The methodology with some known techniques shows that the present approach is simple and effective.To show the efficiency of the present method, four interesting examples is given.

A Hybrid Variational Iteration Method For Blasius Equation, M. Sajid, N. Ali, T. Javed

Applications and Applied Mathematics: An International Journal (AAM)

The objective of this paper is to present the hybrid variational iteration method. The proposed algorithm is based on the combination of variational iteration and shooting methods. In the proposed algorithm the entire domain is divided into subintervals to establish the accuracy and convergence of the approximate solution. It is found that in each subinterval a three term approximate solution using variational iteration method is sufficient. The proposed hybrid variational iteration method offers not only numerical values, but also closed form analytic solutions in each subinterval. The method is implemented using an example of the Blasius equation. The results show …

Thermal Stresses In Functionally Graded Hollow Sphere Due To Non-Uniform Internal Heat Generation, S. P. Pawar, K. C. Deshmukh, G. D. Kedar

Applications and Applied Mathematics: An International Journal (AAM)

In this article, the thermal stresses in a hollow thick sphere of functionally graded material subjected to non-uniform internal heat generation are obtained as a function of radius to an exact solution by using the theory of elasticity. Material properties and heat generation are assumed as a function of radius of sphere and Poisson’s ratio as constant. The distribution of thermal stresses for different values of the powers of the module of elasticity and varying power law index of heat generation is studied. The results are illustrated numerically and graphically.

A Stage-Structured Two Species Competition Mathematical Model Under The Effect Of Disease, Manju Agarwal, Vinay Verma

Applications and Applied Mathematics: An International Journal (AAM)

In this paper we study the dynamics of two competing species model; one of this competing species is stage structured and the disease spreads only in the other competing specie. In order to keep the model simple, we present it under the strong assumption that the disease can not cross the species barrier. Dynamical behaviors such as positivity, boundedness, stability, bifurcation and persistence of the model are studied analytically using the theory of differential equations. Computer simulations are carried out to substantiate the analytical findings. It is noted that c the loss rate of the population, T the maturation time …

Unsteady Boundary Layer Flow Of Thermophoretic Mhd Nanofluid Past A Stretching Sheet With Space And Time Dependent Internal Heat Source/Sink, N. Sandeep, C. Sulochana, C. S. K. Raju, M. J. Babu, V. Sugunamma

Applications and Applied Mathematics: An International Journal (AAM)

In this study we analyze the boundary layer flow of a thermophoretic magnetohydrodynamic dissipative nanofluid over an unsteady stretching sheet in a porous medium with space and time dependent internal heat source/sink. The governing equations are transformed to ordinary differential equations by using similarity transformation. Numerical solutions of these equations are obtained by using the Shooting Technique. The effects of non-dimensional governing parameters on the velocity, temperature, concentration profiles, friction factor, Nusselt and Sherwood numbers are discussed and presented through graphs and tables. Accuracy of the results compared with the existing ones. Excellent agreement is found with earlier studies.

Controllability Of An Eco-Epidemiological System With Disease Transmission Delay: A Theoretical Study, Samadyuti Haldar, Kunal Chakraborty, T. K. Kar

Applications and Applied Mathematics: An International Journal (AAM)

This paper deals with the qualitative analysis of a disease transmission delay induced prey preda-tor system in which disease spreads among the predator species only. The growth of the preda-tors’ susceptible and infected subpopulations is assumed as modified Leslie–Gower type. Suffi-cient conditions for the persistence, permanence, existence and stability of equilibrium points are obtained. Global asymptotic stability of the system is investigated around the coexisting equilib-rium using a geometric approach. The existence of Hopf bifurcation phenomenon is also exam-ined with respect to some important parameters of the system. The criterion for disease a trans-mission delay the induced Hopf bifurcation phenomenon …

Mathematical Modeling And Optimal Control Of Alternative Pest Management For Alfalfa Agroecosystems, Cara Sulyok

Mathematics Honors Papers

This project develops mathematical models and computer simulations for cost-effective and environmentally-safe strategies to minimize plant damage from pests with optimal biodiversity levels. The desired goals are to identify tradeoffs between costs, impacts, and outcomes using the enemies hypothesis and polyculture in farming. A mathematical model including twelve size- and time-dependent parameters was created using a system of non-linear differential equations. It was shown to accurately fit results from open-field experiments and thus predict outcomes for scenarios not covered by these experiments.

The focus is on the application to alfalfa agroecosystems where field experiments and data were conducted and provided …

Solving Ordinary Differential Equations Using Differential Forms And Lie Groups, Richard M. Shumate

Senior Honors Theses

Differential equations have bearing on practically every scientific field. Though they are prevalent in nature, they can be challenging to solve. Most of the work done in differential equations is dependent on the use of many methods to solve particular types of equations. Sophus Lie proposed a modern method of solving ordinary differential equations in the 19th century along with a coordinate free variation of finding the infinitesimal generator by combining the influential work of Élie Cartan among others in the field of differential geometry. The driving idea behind using symmetries to solve differential equations is that there exists a …

Solutions Of A Logistic Equation On Varying Time Scales: A Quantitative And Qualitative Analysis, Alexandria Amity Amorim

Theses, Dissertations and Capstones

Time Scale Calculus, introduced by Dr. Stefan Hilger in 1988, combines the study of differential and difference equations into a single topic. We begin with an introduction of sets used in this field, time scales, and build up to the definition of the exponential function on a time scale. The main focus of this work is a study of the solutions of a particular logistic dynamic equation on varying time scales. We study both the analytical and graphical solutions of this equation. Analytical solutions are worked out using theorems from Time Scale Calculus, including properties of the exponential function. Graphical …