Special Sessions on
Computer Enhanced Modelling

Table of Contents

• Schedule of Talks
• Titles and Abstracts of Invited Talks

Technology can be used to investigate a wide variety of applications of mathematics. At Georgia Southern University, students are required to complete at least one Applications Project each quarter in many of our courses. Through these assignments, students improve their problem-solving, analytical, and communication skills. The presentation will focus on applications of calculus and differential equations in such fields as biology, physics, chemistry, engineering, and economics. However, similar principles can be used to address applications in other areas.

Modern courses in differential equations are much more than the old catalogs of classifications and techniques. We'll discuss ways that students can investigate population dynamics and test both the appropriateness and validity of models. Students can use Maple for all four aspects: visualization, numeric, symbolic, and presentation. We'll close with a novel model of population growth and consider its dynamics.

The historical roots of the calculus of variations trace back to Isaac Newton's "Principia Mathematica" problem concerning the shape of a solid of revolution that experiences minimal resistance to rapid motion through a "rare medium" consisting of elastic particles. This presentation exploits modern computer algebra to explore the meaning and origin of Newton's analysis and solution.

We give some examples of computer projects, usable in the calculus sequence, that solve actual problems in acoustics and sound.

Ever since May's 1973 paper introduced chaos to the world of biological modeling there has been great excitement, but also controversy about the idea. While simple difference equation models do exhibit multiple periodicity and chaos, and the mathematics is undeniably elegant and fascinating, the parameter values under which this behavior appears have generally not been regarded as realistic in actual systems, nor, with the exception of Nicholson's blowflies, even in contrived laboratory systems. When apparently multiply periodic or chaotic behavior has been seen, other interpretations, for instance stochastic, have been proposed. We will discuss a number of articles that we have examined with a post-calculus class in Mathematical Biology dealing with these issues. We'll begin with the experiments of Nicholson and Bailey from the 1930's on host-parasitoid systems (the ``predator'' lays an egg under the skin of the ``prey''), and continuing to recent work of Dennis, Desharnais, Cushing and Costantino on flour beetles, in which both model and experiment yield 2-cycles and chaos. We will illustrate how rather straightforward Maple worksheets can be used to study not only trajectories of populations over time, but also analysis of the parameter space and bifurcation plots.

Some programs for the H-P 48 G calculator are presented which are designed to lessen the "drudgery" and enhance the comprehension. The graphs of solutions of boundary value problems (for example, isotherms) can be immediately obtained and greatly increase the understanding of the analytic solution.

With the use of technology, we can determine how varying conditions affect the range of a long jump. The presentation will include a brief description of the model and will focus on how factors such as air resistance affect the results. Exact and numerical solutions to the system of ordinary differential equations that models the physical situation will be presented.

MathView (formerly Theorist) offers symbolic, graphical, and numerical capabilities without requiring complex syntax or graphing calculator expertise. How MathView can be used to enhance the teaching and learning of college algebra, precalculus, and calculus will be demonstrated.

In Numerical Analysis, traditionally, algorithms are implemented for numerical solutions only. However, it will be much more interesting and useful if solutions also can be graphically displayed. Popular software such as Maple is not very successful for interactive graphic animation. A project for solving a first-order differential equation using a shareware, called Calculus Calculator (CC), will be presented. Although CC is not as powerful as Maple, it is much easier to learn, and more convenient for interactive graphics. In the project, an equation dx/dt = f(x,t), an initial condition x(a) = k, an interval [a,b] and a tolerance can be input; a numerical solution by Euler's method, Runge-Kutta method or Runge-Kutta-Fehlberg method can be displayed in a table; one or several solutions can be graphically displayed in different colors for comparison; the errors of solutions by different methods can be visually seen; in the adaptive method, the change of the length of each step can be visually seen in a computation process; the program can be run interactively and repeatedly for different purposes. Projects for other topics in CC also will be presented. Anyone can get a free copy of CC. Projects in Java will be discussed (or presented).