Applied Mathematics Commons^™

A Study Of Nonlinear Dynamics In Mathematical Biology, Joseph Ferrara

UNF Graduate Theses and Dissertations

We first discuss some fundamental results such as equilibria, linearization, and stability of nonlinear dynamical systems arising in mathematical modeling. Next we study the dynamics in planar systems such as limit cycles, the Poincaré-Bendixson theorem, and some of its useful consequences. We then study the interaction between two and three different cell populations, and perform stability and bifurcation analysis on the systems. We also analyze the impact of immunotherapy on the tumor cell population numerically.

Nonlinear Techniques For Stochastic Systems Of Differential Equations, Tadesse G. Zerihun

USF Tampa Graduate Theses and Dissertations

Two of the most well-known nonlinear methods for investigating nonlinear dynamic processes in sciences and engineering are nonlinear variation of constants parameters and comparison method. Knowing the existence of solution process, these methods provide a very powerful tools for investigating variety of problems, for example, qualitative and quantitative properties of solutions, finding error estimates between solution processes of stochastic system and the corresponding nominal system, and inputs for the designing engineering and industrial problems. The aim of this work is to systematically develop mathematical tools to undertake the mathematical frame-work to investigate a complex nonlinear nonstationary stochastic systems of differential …

Project Haiti 2012: Providing An Experiential Learning Experience Through The Design And Delivery Of A Water Purifier In Haiti, Yung Wong, Johnathon Camp, Shavin Pinto, Kyle Fennesy, Marc Compere, Yan Tang

Publications

In this paper, we share our experiences and lessons learned from Project Haiti 2012, a project to design and install a water purification system serving 20,000 people per day in the largest tent city in Haiti. Project Haiti 2012 was the third and largest system we have built for Haitians and represents a huge success for all participants and stakeholders. This paper discusses the unique experiential learning opportunity involved in the design and delivery of the water purifier in a foreign developing country. Multiple positive educational, social, and economic outcomes were achieved including students applying knowledge gained from coursework towards …

Mathematical Modelling Of Internal Heat Recovery In Flash Tank Heat Exchanger Cascades, Andrei Korobeinikov, John E. Mccarthy, Emma Mooney, Krum Semkov, James Varghese

Mathematics Faculty Research

Flash tank evaporation combined with a condensing heat exchanger can be used when heat exchange is required between two streams and where at least one of these streams is difficult to handle (tends severely to scale, foul, causing blockages). To increase the efficiency of heat exchange, a cascade of these units in series can be used. Heat transfer relationships in such a cascade are very complex due to their interconnectivity, thus the impact of any changes proposed is difficult to predict. Moreover, the distribution of loads and driving forces in different stages and the number of designed stages faces tradeoffs …

Finite Element Methods For Fourth Order Variational Inequalities, Yi Zhang

LSU Doctoral Dissertations

In this work we study finite element methods for fourth order variational inequalities. We begin with two model problems that lead to fourth order obstacle problems and a brief survey of finite element methods for these problems. Then we review the fundamental results including Sobolev spaces, existence and uniqueness results of variational inequalities, regularity results for biharmonic problems and fourth order obstacle problems, and finite element methods for the biharmonic problem. In Chapter 2 we also include three types of enriching operators which are useful in the convergence analysis. In Chapter 3 we study finite element methods for the displacement …

Application Of Helmholtz/Hodge Decomposition To Finite Element Methods For Two-Dimensional Maxwell's Equations, Zhe Nan

LSU Doctoral Dissertations

In this work we apply the two-dimensional Helmholtz/Hodge decomposition to develop new finite element schemes for two-dimensional Maxwell's equations. We begin with the introduction of Maxwell's equations and a brief survey of finite element methods for Maxwell's equations. Then we review the related fundamentals in Chapter 2. In Chapter 3, we discuss the related vector function spaces and the Helmholtz/Hodge decomposition which are used in Chapter 4 and 5. The new results in this dissertation are presented in Chapter 4 and Chapter 5. In Chapter 4, we propose a new numerical approach for two-dimensional Maxwell's equations that is based on …

Lattice Boltzmann Simulations Of Thermal Convective Flows In Two Dimensions, Jia Wang, Donghai Wang, Pierre Lallemand, Li-Shi Luo

Mathematics & Statistics Faculty Publications

In this paper we study the lattice Boltzmann equation (LBE) with multiple-relaxation-time (MRT) collision model for incompressible thermo-hydrodynamics with the Boussinesq approximation. We use the MRT thermal LBE (TLBE) to simulate the following two flows in two dimensions: the square cavity with differentially heated vertical walls and the Rayleigh-Benard convection in a rectangle heated from below. For the square cavity, the flow parameters in this study are the Rayleigh number Ra = 10³-10⁶, and the Prandtl number Pr = 0.71; and for the Rayleigh-Benard convection in a rectangle, Ra = 2 ^. 10³, 10 …

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 …

Multiple Subject Barycentric Discriminant Analysis (Musubada): How To Assign Scans To Categories Without Using Spatial Normalization, Hervé Abdi, Lynne J. Williams, Andrew C. Connolly, M. Ida Gobbini

Dartmouth Scholarship

We present a new discriminant analysis (DA) method called Multiple Subject Barycentric Discriminant Analysis (MUSUBADA) suited for analyzing fMRI data because it handles datasets with multiple participants that each provides different number of variables (i.e., voxels) that are themselves grouped into regions of interest (ROIs). Like DA, MUSUBADA (1) assigns observations to predefined categories, (2) gives factorial maps displaying observations and categories, and (3) optimally assigns observations to categories. MUSUBADA handles cases with more variables than observations and can project portions of the data table (e.g., subtables, which can represent participants or ROIs) on the factorial maps. Therefore MUSUBADA can …

On The Peakon And Soliton Solutions Of An Integrable Pde With Cubic Nonlinearities, Rossen Ivanov, Tony Lyons

Conference papers

The interest in the singular solutions (peakons) has been inspired by the Camassa-Holm (CH) equation and its peakons. An integrable peakon equation with cubic nonlinearities was first discovered by Qiao. Another integrable equation with cubic nonlinearities was introduced by V. Novikov . We investigate the peakon and soliton solutions of the Qiao equation.

Title Ix Compliance: A Comparison Of Division I Equality, Jacqueline Leake

Honors Theses

The passage of Title IX of the Education Amendments of 1972 has had a significant impact on college athletics. However, there is still a large disparity between opportunities offered for men and women. This study determined the true gender equality within Division I athletics. Inequalities were assessed in the areas of athletic participation, athletically related student aid, recruiting expenses, and total expenses. Data from these areas were gathered from the Equity in Athletics Disclosure Analysis Cutting Tool. Ratios and the difference between the ideal and current values were calculated for each category. Institutions were ranked in each category, as well …

Blow-Up Of Solutions To The Generalized Inviscid Proudman-Johnson Equation, Alejandro Sarria

LSU New Orleans Theses and Dissertations

The generalized inviscid Proudman-Johnson equation serves as a model for n-dimensional incompressible Euler flow, gas dynamics, high-frequency waves in shallow waters, and orientation of waves in a massive director field of a nematic liquid crystal. Furthermore, the equation also serves as a tool for studying the role that the natural fluid processes of convection and stretching play in the formation of spontaneous singularities, or of their absence.

In this work, we study blow-up, and blow-up properties, in solutions to the generalized, inviscid Proudman-Johnson equation endowed with periodic or Dirichlet boundary conditions. More particularly,regularity of solutions in an Lp setting will …

Decision Making Under Interval Uncertainty (And Beyond), Vladik Kreinovich

Departmental Technical Reports (CS)

To make a decision, we must find out the user's preference, and help the user select an alternative which is the best -- according to these preferences. Traditional utility-based decision theory is based on a simplifying assumption that for each two alternatives, a user can always meaningfully decide which of them is preferable. In reality, often, when the alternatives are close, the user is often unable to select one of these alternatives. In this chapter, we show how we can extend the utility-based decision theory to such realistic (interval) cases.

Nabla Fractional Calculus And Its Application In Analyzing Tumor Growth Of Cancer, Fang Wu

Masters Theses & Specialist Projects

This thesis consists of six chapters. In the first chapter, we review some basic definitions and concepts of fractional calculus. Then we introduce fractional difference equations involving the Riemann-Liouville operator of real number order between zero and one. In the second chapter, we apply the Brouwer fixed point and Contraction Mapping Theorems to prove that there exists a solution for up to the first order nabla fractional difference equation with an initial condition. In chapter three, we define a lower and an upper solution for up to the first order nabla fractional difference equation with an initial condition. Under certain …

Thermalization And Initial State-Recurrence In Discrete Kdv-Like Lattices, Garrett Taylor Nieddu

Theses, Dissertations and Culminating Projects

Three discretizations of the Korteweg de-Vries equation are studied; convergence rate, initial state-recurrence, and the energy distribution of the three schemes are all considered. For each discrete scheme over 300 lattices with varying grid sizes were investigated, and the solutions were compared with other lattices from the same scheme, as well as solutions from the other two. It is found that the two schemes that are least accurate display the best recurrence at intermediate grid sizes, away from convergence. This is a notable result because the best recurrence is expected to be found in the most accurate, and converged lattices. …

Convex Hull Characterization Of Special Polytopes In N-Ary Variables, Ruobing Shen

All Theses

This paper characterizes the convex hull of the set of n-ary vectors that are lexicographically less than or equal to a given such vector. A polynomial number of facets is shown to be sufficient to describe the convex hull. These facets generalize the family of cover inequalities for the binary case. They allow for advances relative to both the modeling of integer variables using base-n expansions, and the solving of n-ary knapsack problems with weakly super-decreasing coefficients.

Finding Unpredictable Behaviors Of Periodic Bouncing For Forced Nonlinear Spring Systems When Oscillating Time Is Large, Yanyue Ning

Honors Scholar Theses

The model of nonlinear spring systems can be applied to deal with different aspect of mechanical problems, such as oscillations in periodic flexing in bridges and ships. The concentration of this research is the bouncing behaviors of nonlinear spring system when the processing time is large, therefore nonlinear ordinary differential equations (ODE) are suitable since researchers can add different variables into the models and solve them by computational methods. Benefit from this, it is easy to check the oscillations or bouncing behaviors that each variable contributes to the model and find the relationship between some important factors: vibrating frequency, external …

Stability And Convergence For Nonlinear Partial Differential Equations, Oday Mohammed Waheeb

Boise State University Theses and Dissertations

If used cautiously, numerical methods can be powerful tools to produce solutions to partial differential equations with or without known analytic solutions. The resulting numerical solutions may, with luck, produce stable and accurate solutions to the problem in question, or may produce solutions with no resemblance to the problem in question at all. More such numerical computations give no hope of solving this troublesome feature and one needs to resort to investing time in a theoretical approach. This thesis is devoted not solely to computations, but also to a theoretical analysis of the numerical methods used to generate computationally the …

On Numerical Solution For Optimal Allocation Of Investment Funds In Portfolio Selection Problem, Yahaya Abubakar

CBN Journal of Applied Statistics (JAS)

In this article, we present a procedure for obtaining an optimal solution to the Markowitz’s mean-variance portfolio selection problem based on the analytical solution developed in a previous research that lead to the emergence of an important model known as the Black Model. The procedure is well presented, illustrated and validated by a numerical example from real stocks dataset obtainable from a popular European stock market.

On The Geometrıc Interpretatıons Of The Kleın-Gordon Equatıon And Solution Of The Equation By Homotopy Perturbation Method, Hasan Bulut, H. M. Başkonuş

Applications and Applied Mathematics: An International Journal (AAM)

This paper is organized in the following ways: In the first part, we obtained the Klein Gordon Equation (KGE) in the Galilean space. In the second part, we applied Homotopy Perturbation Method (HPM) to this differential equation. In the third part, we gave two examples for the Klein Gordon equation. Finally, We compared the numerical results of this differential equation with their exact results. We also showed that approach used is easy and highly accurate.

Numerical Studies For Solving Fractional Riccati Differential Equation, N. H. Sweilam, M. M. Khader, A. M. S. Mahdy

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, finite difference method (FDM) and Pade'-variational iteration method (Pade'- VIM) are successfully implemented for solving the nonlinear fractional Riccati differential equation. The fractional derivative is described in the Caputo sense. The existence and the uniqueness of the proposed problem are given. The resulting nonlinear system of algebraic equations from FDM is solved by using Newton iteration method; moreover the condition of convergence is verified. The convergence's domain of the solution is improved and enlarged by Pade'-VIM technique. The results obtained by using FDM is compared with Pade'-VIM. It should be noted that the Pade'-VIM is preferable because …

On The Numerical Solution Of Linear Fredholm-Volterra İntegro Differential Difference Equations With Piecewise İntervals, Mustafa Gülsu, Yalçın Öztürk

Applications and Applied Mathematics: An International Journal (AAM)

The numerical solution of a mixed linear integro delay differential-difference equation with piecewise interval is presented using the Chebyshev collocation method. The aim of this article is to present an efficient numerical procedure for solving a mixed linear integro delay differential difference equations. Our method depends mainly on a Chebyshev expansion approach. This method transforms a mixed linear integro delay differential-difference equations and the given conditions into a matrix equation which corresponds to a system of linear algebraic equation. The reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments and performed on the computer algebraic system …

Further Results On Fractional Calculus Of Saigo Operators, Praveen Agarwal

Applications and Applied Mathematics: An International Journal (AAM)

A significantly large number of earlier works on the subject of fractional calculus give interesting account of the theory and applications of fractional calculus operators in many different areas of mathematical analysis (such as ordinary and partial differential equations, integral equations, special functions, summation of series, et cetera). The main object of the present paper is to study and develop the Saigo operators. First, we establish two results that give the image of the product of multivariable H-function and a general class of polynomials in Saigo operators. On account of the general nature of the Saigo operators, multivariable H-function and …

An Approximate Analytical Algorithm For Solving The Multispecies Lotka-Volterra Equations, Abdolsaeed Alavi, Asghar Ghorbani

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, a new efficient method called the parametric iteration method (PIM) is applied to accurately solve the multispecies Lotka–Volterra equations (MLVEs). Some cases of MLVEs are highlighted in order to show the simplicity and efficiency of the method. The results obtained in this work demonstrate that the present algorithm is a powerful analytic tool for the solution of MLVEs.

A Constructive Proof Of Fundamental Theory For Fuzzy Variable Linear Programming Problems, A. Ebrahimnejad

Applications and Applied Mathematics: An International Journal (AAM)

Two existing methods for solving fuzzy variable linear programming problems based on ranking functions are the fuzzy primal simplex method proposed by Mahdavi-Amiri et al. (2009) and the fuzzy dual simplex method proposed by Mahdavi-Amiri and Nasseri (2007). In this paper, we prove that in the absence of degeneracy these fuzzy methods stop in a finite number of iterations. Moreover, we generalize the fundamental theorem of linear programming in a crisp environment to a fuzzy one. Finally, we illustrate our proof using a numerical example.

Modification Of Truncated Expansion Method For Solving Some Important Nonlinear Partial Differential Equations, N. Taghizadeh, M. Mirzazadeh

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, we implemented modification of truncated expansion method for the exact solutions of the Konopelchenko-Dubrovsky equation the (n+1)-dimensional combined sinhcosh- Gordon equation and the Maccari system. Modification of truncated expansion method is a powerful solution method for obtaining exact solutions of nonlinear evolution equations. This method presents a wider applicability for handling nonlinear wave equations.

On Stability Of Dynamic Equations On Time Scales Via Dichotomic Maps, Veysel F. Hatipoğlu, Zeynep F. Koçak, Deniz Uçar

Applications and Applied Mathematics: An International Journal (AAM)

Dichotomic maps are used to check the stability of ordinary differential equations and difference equations. In this paper, this method is extended to dynamic equations on time scales; the stability and asymptotic stability to the trivial solution of the first order system of dynamic equations are examined using dichotomic and strictly dichotomic maps. This method, in dynamic equations, also involves Lyapunov’s direct method.

Two Reliable Methods For Solving The Modified Improved Kadomtsev-Petviashvili Equation, N. Taghizadeh, S. R. Moosavi Noori

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, the tanh-coth method and the extended (G'/G)-expansion method are used to construct exact solutions of the nonlinear Modified Improved Kadomtsev-Petviashvili (MIKP) equation. These methods transform nonlinear partial differential equation to ordinary differential equation and can be applied to nonintegrable equation as well as integrable ones. It has been shown that the two methods are direct, effective and can be used for many other nonlinear evolution equations in mathematical physics.

Investigation Of Nonlinear Problems Of Heat Conduction In Tapered Cooling Fins Via Symbolic Programming, Hooman Fatoorehchi, Hossein Abolghasemi

Applications and Applied Mathematics: An International Journal (AAM)

In this paper, symbolic programming is employed to handle a mathematical model representing conduction in heat dissipating fins with triangular profiles. As the first part of the analysis, the Modified Adomian Decomposition Method (MADM) is converted into a piece of computer code in MATLAB to seek solution for the mentioned problem with constant thermal conductivity (a linear problem). The results show that the proposed solution converges to the analytical solution rapidly. Afterwards, the code is extended to calculate Adomian polynomials and implemented to the similar, but more generalized, problem involving a power law dependence of thermal conductivity on temperature. The …

Sensitivity Analysis In Magnetic Resonance Elastography And A Local Wavelength Reconstruction Based On Wave Direction, Christopher Gillam

All Dissertations

or the detection of early stage cancer. MRE utilizes interior data for its inverse problems, which greatly reduces the ill-posedness from which most traditional inverse problems suffer.
In this thesis, we first establish a sensitivity analysis for viscoelastic scalar medium with complex wave number and compare it with the purely elastic case. Also we estimate the smallest detectable inclusion for breast and liver, which is about twice larger than using the purely elastic model. We also found the existence of optimal frequency (50 Hz) that maximizes the detectability when the Voigt model is used.
Second, we propose a local wavelength …