Martha L. Abell's research while affiliated with Georgia Southern University and other places
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Publications (133)
Chapter 4 introduces operations on lists and tables. The examples used to illustrate the various commands in this chapter are taken from vector calculus, business, dynamical systems, and engineering applications.
Chapter 6 discusses several of Mathematica's functions that relate to ordinary and partial differential equations. The examples used to illustrate the various commands are similar to examples routinely done in a one- or two-semester differential equations course. For more detailed discussions regarding Mathematica and differential equations, see re...
Chapter 2 introduces the essential commands of Mathematica. Basic operations on numbers, expressions, functions, and two- and three-dimensional graphics are introduced and discussed. Mathematica's extensive image-processing capabilities are introduced.
Chapter 3 introduces Mathematica's calculus capabilities. The examples used to illustrate the various functions are similar to examples typically seen in a traditional calculus sequence. If you have trouble typing commands correctly, use the buttons on the Basic Math Input or Basic Math Assistant palettes to help you create templates in standard ma...
Chapter 5 discusses operations on matrices and vectors, including topics from linear algebra, linear programming, and vector calculus.
In this paper we modify a standard SIR model used to study the spread of some diseases by incorporating a disease that destroys the immunity that is conferred by having one of the other diseases or being vaccinated against the disease. A specific biological example of this occurs with measles. Studies of recent measles’ patients has shown that many...
In this study, we compare the effects of competitors in a chemostat when one of the competitors is lethal to the other. The first competitor (“the mutant”) is the desired organism because it provides a benefit, such as a substance that is harvested. However, when the mutant undergoes cell division the result may return to the original (“wild type”)...
In Chapter 4, we discussed several techniques for solving higher-order differential equations. In this chapter, we illustrate how some of these methods can be used to solve initial-value problems that model physical situations.
In previous chapters we have investigated solving the nth-order linear equation
In Chapter 7, we discuss several applications of systems of differential equations. These include standard linear applications that involve mechanical and electrical systems as well as diffusion and population problems. We also discuss several nonlinear applications including predator-prey interactions, food chains in the chemostat, and curvature,...
Chapter 10 introduces the separation of variables technique. The emphasis of the chapter is solving the one-dimensional heat equation, the one-dimensional wave equation, D’Alembert’s solution to the one-dimensional wave equation, Laplace’s equation, Laplace’s equation in a circular region, and the wave equation in a circular region.
When the space shuttle was launched from the Kennedy Space Center, its escape velocity could be determined by solving a first-order ordinary differential equation. The same can be said for finding the flow of electromagnetic forces, the temperature of a cup of coffee, the population of a species as well as numerous other applications. In this chapt...
The purpose of Differential Equations with Mathematica, Fourth Edition, is twofold. First, we introduce and discuss the topics covered in typical undergraduate and beginning graduate courses in ordinary and partial differential equations including topics such as Laplace transforms, Fourier series, eigenvalue problems, and boundary-value problems. S...
In previous chapters, we have seen that many physical situations can be modeled by either ordinary differential equations or systems of ordinary differential equations. However, to understand the motion of a string at a particular location and at a particular time, the temperature in a thin wire at a particular location and a particular time, or th...
Because of their importance in the study of systems of linear equations, we now review matrices and the operations associated with them.
In Chapters 2 and 3 we saw that first-order differential equations can be used to model a variety of physical situations. However, many physical situations need to be modeled by higher-order differential equations. In this chapter, we discuss several methods for solving higher-order linear differential equations.
An electrical circuit can be modeled with an ordinary differential equation with constant coefficients. This chapter discusses how a circuit involving loops can be described as a system of ordinary linear differential equations with constant coefficients. This derivation is based on several principles, such as Kirchhoff's current law and Kirchhoff'...
A differential equation is an equation that contains the derivative or differentials of one or more dependent variables in regard to one or more independent variables. If the equation contains only ordinary derivatives in regard to a single independent variable, then the equation is called an ordinary differential equation. This chapter discusses t...
We have focused our attention on solving ordinary differential equations that involve one dependent variable. However, many physical situations are modeled with more than one differential equation and involve more than independent variable. For example, if we want to determine the population of two interacting populations in a predator-prey relatio...
which describes the displacement of the simple modes. This equation can be rewritten in the form
We will devote a considerable amount of time in this course to developing explicit, implicit, numerical, and graphical solutions of differential equations. In this chapter, we discuss first-order ordinary differential equations (ODEs) and some methods used to construct explicit, implicit, numerical, and graphical solutions of them. Several of the e...
This chapter presents an introduction of Laplace transform and discusses several of its properties. In most cases, using the definition of the Laplace transform to calculate the Laplace transform of a function is a difficult and is a time consuming task. The definition of the Laplace transform is also difficult to apply in most of the cases. The ch...
This chapter describes how some of the techniques for solving higher-order differential equations methods can be used to solve initial-value problems that model physical situations. Suppose that a mass is attached to an elastic spring that is suspended from a rigid support such as a ceiling. The mass causes the spring to stretch to a distance from...
This text is for courses that are typically called (Introductory) Differential Equations, (Introductory) Partial Differential Equations, Applied Mathematics, and Fourier Series. Differential Equations is a text that follows a traditional approach and is appropriate for a first course in ordinary differential equations (including Laplace transforms)...
We compare the effects of interactive and noninteractive complementary nutrients on the growth of an organism in the chemostat. We also compare these two situations to the case when the nutrients are substitutable. In previous studies, complementary nutrients have been assumed to be noninteractive. However, more recent research indicates that some...
We incorporate basic genetics into an SIS endemic model of a disease that is spread primarily by individual–individual contact after it is introduced into the population. We illustrate that if a homozygote is immune to the disease or is resistant to the effects of the disease, the corresponding allele goes to fixation. On the other hand, if the het...
We incorporate basic genetics into an AIDS model. We illustrate that if a homozygote is immune to the disease or is resistant to the effects of the disease, the corresponding allele goes to fixation. On the other hand, if the heterozygote is immune to the disease or is resistant to the effects of the disease, polymorphism usually occurs.
Allelopathy is the chemical inhibition of one species by another. Bacteriocins, which are toxins produced by bacteria to inhibit the growth of closely related species, are a particular type of allelopathy that is of special interest because of the importance of bacteriocins in the food industry and in the development of vaccines. We form a model of...
Competing species use a variety of strategies to gain an advantage over a competitor. We show that a desirable auxotrophic mutant can sometimes gain a growth advantage over its parental (or, wild-type) organism by using an offensive inhibitory or lethal strategy against the parental organism that lower’s the parental organism’s growth rate. Our num...
Epistasis is the interaction between two or more genes to control a single phenotype. We model epistasis with a two-locus two-allele problem. The resulting model allows us to examine both population sizes as well as genotypic and phenotypic frequencies. In the context of an example, we show that if epistasis results in an undesirable phenotype, suc...
The theory of heterozygote advantage is often used to explain the genetic variation found in natural populations. If a large population randomly mates and the various genotypes have the same growth and death rates, the evolution of the genotypes follows Hardy–Weinberg proportions and polymorphism results. When other environmental stresses, like pre...
Many biological structures are products of repeated iteration functions. As such, they demonstrate characteristic, scale-invariant features. Fractal analysis of these features elucidates the mechanism of their formation. The objectives of this project were to determine whether human cranial sutures demonstrate self-similarity and measure their expo...
Objectives
Many biological structures are products of repeated iteration functions. As such, they demonstrate characteristic, scale-invariant features. Fractal analysis of these features elucidates the mechanism of their formation. The objectives of this project were to determine whether human cranial sutures demonstrate self-similarity and measure...
The Mo¨bius strip, torus, and Klein bottle are used to graphically and analytically illustrate the differences between orientable and non-orientable surfaces. An exercise/laboratory project using the non-orientable Boy surface is included.
Multiple comparison,methods (MCMs) are used to investigate dierences,between pairs of population means or, more generally, between subsets of population means using sam- ple data. Although several such methods are commonly,available in statistical software packages, users may be poorly informed about the appropriate method(s) to use and/or the corr...
The Landau–Lifshitz–Gilbert (LLG) equations describe the dynamics of ferromagnets. Using various assumptions, several exact solutions to this nonlinear system are determined in Refs. [Phys. Rev. Lett. 65 (1990) 787; J.F. Dillon, Domains and domain walls, in: G.T. Rado, H. Suhl (Eds.), A Treatise on Modern Theory and Materials. Vol. 3: Magnetism, Ac...
The Landau–Lifshitz–Gilbert (LLG) equations describe the dynamics of ferromagnets. The known exact solutions are reviewed and generalized. The results are visualized in several examples.
The daily closing values of the S&P 500 Index from January 1, 1926 through June 11, 1993, a total of 17,610 values, were entered into Mathematica, and the day-to-day percent changes were calculated. Using the Standard Mathematica Package Statistics ‵ContinuousDistributions‵ and the built-in function NonLinearFit, procedures were developed to find t...
A computer algebra system can be used to enhance numerous courses in applied mathematics. Not only can they alleviate the computational difficulties encountered, but they can offer the advantage of visualizing solutions. The solution of standard problems found in the study of ordinary and partial differential equations such as spring motion and the...
This chapter focuses on frequently used first-order ordinary differential equations and methods to construct their solutions. Differential equations are first encountered in the beginning integral calculus courses. Although the phrase differential equation is not frequently used at that point, the problem of finding a function whose derivative is a...
This chapter presents the applications of Laplace transforms. Laplace transforms are useful in solving the spring-mass systems. Although the method of Laplace transforms can be used to solve all problems based on applications of higher-order equations, this method is most useful in alleviating the difficulties associated with problems that involve...
This chapter presents an introduction to the essential commands of Mathematica. Basic operations on numbers, expressions, and functions are also discussed in the chapter. The basic arithmetic operations—such as addition, subtraction, multiplication, and division—are performed in the natural way with Mathematica. For a variety of reasons, however, n...
This chapter presents elementary operations on lists and tables. Lists are defined in a variety of ways. Lists may be completely typed in, or they may be created with either the Table or Array commands. Elements of lists can be numbers, ordered pairs, functions, and even other lists. Mathematica has built-in definitions of many commonly used specia...
This chapter describes the applications of first-order ordinary differential equations. Many interesting problems involving population can be solved with first-order differential equations. These include the determination of the number of cells in a bacteria culture, the number of citizens in a country, and the amount of radioactive substance remai...
This chapter discusses operations on matrices and vectors, including vector calculus and systems of equations. Several linear programming examples are presented in the chapter. Applications discussed include linear programming and vector calculus. Matrix algebra can be performed with Mathematica. In Mathematica, a matrix is a list of lists where ea...
This chapter presents Mathematica's built-in calculus commands. Mathematica is used to investigate limits graphically and numerically. Some limits involving rational functions are computed by factoring the numerator and denominator. Mathematica knows the familiar rules of differentiation—such as the product rule, quotient rule, and chain rule. It i...
Mathematica performs calculations when computing solutions of various differential equations. In some cases, it is used to find the exact solution of certain differential equations using the built-in command DSolve. Mathematica also contains the command NDSolve, which is used to obtain numerical solutions of other differential equations. This chapt...
This chapter focuses on the partial differential equations. Obtaining solutions of partial differential equations is connected with solving ordinary differential equations. The boundary value problems are solved in much the same way as initial-value problems. However, there is a difference between initial-value problems and boundary value problems...
This chapter presents an introduction to Mathematica, which is a system for doing mathematics on a computer. It combines symbolic manipulation, numerical mathematics, outstanding graphics, and a sophisticated programming language. Because of its versatility, Mathematica has established itself as the computer algebra system of choice for many comput...
In some cases, techniques used to solve ordinary differential equations with constant coefficients and typical applications of such differential equations can be used to solve differential equations with nonconstant coefficients. In other cases, different techniques should be used. Solving an arbitrary differential equation is a formidable, if not...
Many physical and mathematical situations are described by ordinary differential equations and others are described by partial differential equations. This chapter discusses that one way to solve some partial differential equations is the method of separation of variables. Partial differential equations cannot be studied without an introduction of...
This chapter provides an overview of Fourier series and its applications to partial differential equations. When the eigenfunctions are sines and cosines, the expansion is known as a Fourier series. To explain the convergence of the Fourier series, it defined as: a function is piecewise continuous on a finite interval if it is continuous at each po...
This chapter provides an overview of applications of power series. Power series solutions, because of their form, can be used to solve Cauchy–Euler equations. The chapter illustrates this method of solution. It shows that, in most cases, the result of the power series method only approximates an exact solution. It also examines a Cauchy–Euler probl...
Laplace transforms are useful in solving the spring-mass systems. Although the method of Laplace transforms can be used to solve problems of higher order equations, this method is most useful in alleviating the difficulties associated with problems of the type that involves piecewise-defined forcing functions. This chapter reviews the use of Laplac...
This chapter provides an overview of power series solutions of ordinary differential equations. It reviews the basic properties of power series and proofs of the major theorems. If the power series converges absolutely for all values of x, then the power series has infinite radius of convergence. A power series may be differentiated and integrated...
Mathematica is a system for doing mathematics on computer. It combines symbolic manipulation, numerical mathematics, outstanding graphics, and a sophisticated programming language. Because of its versatility, Mathematica has established itself as the computer algebra system of choice for many computer users. It is the most powerful and most widely...
This chapter discusses operations on matrices and vectors, including vector calculus and systems of equations. It presents several linear programming examples. Matrix algebra is performed with Mathematica. The chapter discusses the operations involved in matrix algebra and the method by which a matrix is entered. In Mathematica, a matrix is simply...
Becoming competent with Mathematica can take a serious investment of time. Messages that result from syntax errors are viewed light-heartedly. Ideally, instead of becoming frustrated, beginning Mathematica users will find it challenging and fun to locate the source of errors. In this process, it is natural that one will become more proficient with...
This chapter presents a list of built-in Mathematica objects starting with the alphabet D. It discusses these objects and illustrates their uses. D represents differentiation. This is accomplished in several forms. DampingFactor is a FindRoot option that can be used to control the convergence of Newton's method as each step in this root-finding met...
This chapter discusses the applications related to ordinary and partial differential equations. Mathematica can perform calculations necessary when computing solutions of various differential equations and, in some cases, can be used to find the exact solution of certain differential equations using the built-in command DSolve. In addition, Version...
This chapter discusses mathematical built-in calculus commands. The examples used to illustrate the various commands are similar to examples routinely done in first-year calculus courses. The Limit command is used along with Simplify to assist in determining the derivative of a function by using the definition of the derivative. The square brackets...
This chapter discusses a built-in Mathematica object starting with thealphabet X, that is, Xor. Xor represents the exclusive “or” function. Therefore, Xor [expression1, expression2, …] yields a value of False if there is an even number of expressions that are True and the rest are False. A value of True results if there is an odd number of True exp...
This chapter presents a list of built-in Mathematica objects starting with the alphabet U. It discusses these objects and illustrates their use. Some of these objects mentioned in the chapter are UnAlias, Underflow, Unevaluated, Uninstall, Union, and Unique. UnAlias is a system command with the attributes HoldAll and Protected. Underflow [ ] result...
This chapter discusses some of the more specialized packages available with Mathematica. It presents the packages within the numerical math folder in Version 2.0 are presented. The package Approximations. m contains useful commands for the approximation of functions with rational functions. In order to work with the approximating function obtained...
This chapter presents a list of built-in Mathematica objects starting with the alphabet Q. It discusses and illustrates the uses of QRDecomposition, Quartics, Quit, and Quotient. QRDecomposition [m] determines the QR decomposition of the matrix m. Quartics is an option used with functions, such as Roots and NSolve to indicate whether explicit solut...
This chapter presents a list of built-in Mathematica objects starting with the alphabet V. Some of these objects are WeierstrassP, WeierstrassPPrime, Which, While, With, and Word. With is used to define local constants. Word represents a Mathematica object made up of a sequence of characters delimited by word separators. These words are read by fun...
This chapter presents a list of built-in Mathematica objects starting with the alphabet P. Some of these objects listed in the chapter are PaddedForm, PageHeight, PageWidth, ParametricPlot, ParametricPlot3D, and Partition. The chapter discusses these objects and illustrates their uses. PaddedForm is used to allow for printing of the numbers that ha...
Citations
... The strong ties that this domain has with applied sciences, and the need for a concrete understanding of the model, make the interplay between visualization and conceptualization a core issue. For example, when solving a predator-prey problem for two species using the so-called Lotka-Volterra model, the graphs in the phase space provide great assistance in understanding the intertwined evolution of the populations (Abell and Braselton, 1996). In physics, chemistry, engineering, etc., ODEs play a central role and a purely theoretical development cannot be sufficient. ...
... 21 Poincaré map recently found its way to analyze cell development by Klimovskia et al, 22 where the investigators used Poincaré map to better delineate cell trajectory. We have used fractal analysis to study human cranial sutures and have recently demonstrated the interesting recursive nature of the monthly incidence of influenza infection in the U.S. 23,24 (Figure 4). The "attractor" within the phase space, and in this case, consists of many similar, but not identical triangular orbits by plotting X t , X t+Dt with Dt = 30 days. ...
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... all Hamiltonian operators compatible with a given Hamiltonian operator and the associated multiHamiltonian systems, see for example [10,14,17] and references therein. Another possible application of the results in question is e.g. the construction of new integrable systems in spirit of [9] in the new variables from Theorem 1 or Corollary 1 with the subsequent pullback to the original variables. ...
... Cette méthode présente toutefois un inconvénient. Le calcul offline des coefficients et la résolution online du ROM met en jeu les tenseurs 4. Dans le cas d'une inclusion circulaire ω ρ (ξ) = ρ ξ ξ et g ρ = ρ ρ sur l'interface solide-fluide Γ , ω ρ (ξ) = ξ et g ρ = 1 au bord périodique Γ de la cellule. 5. On pourra se reporterà [42], chapitre 6, pour une expression générale de gρ en fonction de ω ρ . 6 ...
... In classical algebra a point x satisfying x = S t (x) is called a fixed point or an invariant point and is distinguished from the stationary point. However in differential equations books (e.g. Conrad, 2003) and dynamical systems books (e.g. Wiggins, 2003) all these terms (and other ones, e.g., " equilibrium point " , " rest point " ) are used interchangeably. ...
... Let w − ∞ be the maxmin (lower value) of the continuous time differential game played using non anticipative strategies with delay. Then from Evans and Souganidis [13], extended in Cardaliaguet [6], Chapter 3, one obtains: ...
... For example, Hsu et al. [16], Wolkowicz and Lu [17] and Ellermeyer [18] studied the chemostat models in which two microorganisms feed on a single nutrient. However, some experiments show that the growth of a microorganism depends on a variety of nutrition factors such as carbon, nitrogen, energy, growth factors, inorganic salts, and water [19]. Recently, the growth of microorganism species in the chemostat on continuous multi-nutrient has been investigated by many researchers [20][21][22][23]. ...
... In this case, 68% of the variables should appear close to the average value +/− standard deviation (uncertainty range). Apart from the mean and the standard deviation, the normal distribution is also characterised by skewness and kurtosis coefficients [55]. As the value of the skewness coefficient increases, the analysed distribution diverges from normal. ...
... The new family of datasets envisioned for the interpretability of time series classification is based on five different and well-known chaotic dynamical systems, namely Chua [26], Duffing [27], Lorenz [28], Rikitake [29], and Rössler [30]. Each dynamical system (also referred to as attractor) constitutes a different class (or label) in the classification task, and is composed by three time series characterizing the nonlinear dynamics (composed by three state variables) of the attractor. ...
... There is a unique limit cycle that appears when κ < κ H (see Figure 9c-e) and, through a loop of heteroclinic orbits, the limit cycle vanishes when κ decreases to κ het = 0.14714 (see Figure 9f). The figures were drawn by Wolfram Mathematica [52]. ...
