Revolutions in Differential Equations: Exploring ODEs with Modern Technology

by Michael J Kallaher

Overview

The articles in this volume provide an efficient way to learn about the direction in which the field of differential equations, and in particular its teaching, is going. The contributors are known not only for their research into the solution and understanding of differential equations, but also importantly for their experience as teachers of the subject. In particular, and this is a central theme of the book, the contributors have expert knowledge on the incorporation of modern technology into the teaching of ODE (ordinary differential equations) courses. Whatever their own field of expertise, this will be an essential purchase for anyone interested in the teaching of differential equations.

Product Details

ISBN-13:	9780883851609
Publisher:	Mathematical Association of America
Publication date:	08/01/1999
Series:	Notes Series , #50
Pages:	100
Product dimensions:	8.40(w) x 10.90(h) x 0.30(d)

Read an Excerpt

Modeling and Visualization in the Introductory ODE Course
Robert L. Borrelli
Courtney S. Coleman
Harvey Mudd College

Our increasingly technological society has welcomed computers with open arms. No longer the domain of the esoteric few, powerful and (relatively) inexpensive platforms are being marketed as a panacea for all that ails the civilized world. Software packages are being churned out by armies of programmers whose genius has produced code that not only addresses current societal needs, but also opens doors for more technological change. There are good reasons for this: Most science and engineering courses in the undergraduate curriculum have been around so long that their content and method of approach are pretty well defined and the supporting textbooks reflect this fact. It's in the nature of academia to be slow to change- we aren't trying to find a niche in a competitive market. Nevertheless, there are some very encouraging signs over the past decade that academia is responding to the challenge of using computers in the curriculum in an effective and creative way.
 Reform movements in calculus, linear algebra and differential equations are well under way, and all of them make good use of hands-on projects in connection with the modeling and visualization capabilities that technology provides. However these reform movements ultimately turn out in the twenty-first century, one thing is already clear; modeling and visualization will be in these courses in one way or another for a long time to come.
 What is modeling anyway? Scientists and engineers generally use the term to describe the process of translating a natural system info a form called a model that can be dealt with in a way that we have confidence in. the models that interest us here are mathematical models which involve ordinary differential equations.
 Curricular reform these days seems to be edging closer and closer to an interdisciplinary approach in which students take a more active role in their own course work. In mathematics departments this approach usually involves laboratory-based activities in which students play a hands-on role in converting "word problems" into mathematical models and "solving" them. We will describe modeling activities in which are suitable for a course in differential equations.

1. The First Course in Differential Equations
Trends in the undergraduate math curriculum have been influenced by a variety of factors. First and foremost is the ready availability of powerful platforms and excellent software for both numerical and symbolic computations. The reform movements in all disciplines have been affected by these technology twins, but the undergraduate mathematics curriculum has arguably been affected the most. The primary use of technology in instruction is to visualize mathematical concepts, solutions of equations, etc., in the context of a laboratory environment. The first course in ODEs seems to be an ideal place to use computers for modeling and visualization.
 We have put in more years than we care to remember in redesigning the first course in ODEs at Harvey Mudd College to bring in modeling and visualization as an essential component of the course. When we first embarked on this path in the late '70's, we couldn't find any solvers which were both robust and easy-to-use, or any affordable platforms on which the solvers could run. So our first task was to design our own solver package that cold be used reliably on available platforms in a dynamical systems lab environment. With the collaboration of four of our students (Ned Freed, Dan Newman, Kevin Carosso and Tony Leneis), we finally after considerable effort, produced an ODE solver package which was suitable for our needs, and the National Science Foundation helped us to set up a lab dedicated to this course. At last we were in business, or so we thought.
 As is often the case at universities and colleges with engineering programs, our first course in ODEs is a required 3-credit-hour sophomore course with no time set aside for laboratory instruction. The traditional syllabus for the course was designed to serve the needs of our client disciplines. There was little flexibility in the course syllabus and no chance the college would give us an additional credit-hour to set up a companion laboratory course, so what to do? Well, first we made our solver absolutely transparent to use (no mean feat) so that students could use it on their own from the get-go with very little instruction required. Our solver is network accessible, and so we included drivers for most graphics platforms. That way students could access the solver from their dorm rooms (or anywhere really) and use their favorite platforms as a front end; X-windows, PCs, MACs, etc. Printing on an y academic laser printer could also be done over the network. Graphs are automatically imprinted with the user name, the date, and the time so that they can be retrieved at any time. Next, we created a collection of computer experiments (later published as DE Lab Workbook) which were more-or-less self-contained and designed to go along with a standard ODE course. We assigned one (or more) experiments per week from this collection that the students could do as "homework".  In fact, each experiment replaces one of the homework sets that otherwise would have been done that week. This slight syllabus change was necessary because students would be quick to see that we were cramming a four-credit hour course into three-credit hours.
 Students use any solver they like for laboratory assignments. At the first lecture we distribute a hand-out describing all the currently available  ODE solvers with instructions on access from the various colleges computer labs. This arrangement works well for a required course that anyone in the department may be assigned to teach; individual instructions are free to select any textbook/solver combination.
 In contrast to the reform calculus experience, there has been little doubt about the direction in which the introductory ODE course should change; the only details to be worked out concern the resources which support different modes of instruction. From 1992 to 1997 the NSF-funded Consortium for ODE Experiments (CODEE) has published a newsletter with information about incorporating hands-on projects ("experiments") in an ODE course. Experiments always involve a modeling component and are designed to address some questions in the modeling environment.

2. ODE Solvers
There are a number of excellent ODE software packages that support modeling and visualization in a first course in ODEs. Some are commercially distributed, and some are shareware. Some are built to run as a component of a large multi-purpose package, and some are stand-alone ODE packages. The stand-alone ODE packages are usually designed to run on only one platform, whereas the multi-purpose packages run on all platforms.

 Here are brief descriptions of three solver packages that differ in their approach but are designed to provide a lab experience to go along with an ODE course.

Interactive Differential Equations (IDE): A collection of interactive tools designed to explore a single concept or application in an ODE course, such as the logistic equation, direction fields, oscillators, numerical methods, the phase plane, eigenvectors. Laplace transforms, series solutions, chaos, bifurcations, and many other topics, each with linked animations of the systems being modeled. An important feature of the package is that the tools demonstrate visually the connections between real world phenomena and the mathematical models that describe them.
 There are 97 interactive illustrations, called tools, arranged into 31 lab collections. The intuitive point-and-click interface allows the user to interact with the tools by setting initial conditions and using sliders to very parameters for both linear and nonlinear models. Each set of tools is accompanied by a workbook lab consisting of background,, instructions, and experiments, with space for writing answers. An instructor's manual is available with sample answers filled in.
 These tools were developed by a team of mathematics faculty including John Cantwell, jean Marie McDill, Steven Strogatz, and Beverly West; the software designer, who originated the package was Hubert Hohn. The package is now available for Windows 95 as well as for the Macintosh.
 
 
Internet Differential Equations Activities (IDEA):
This product can be thought of as an interactive virtual lab book for differential equations at the undergraduate level. IDEA has the basic goal of developing and disseminating software for numerical explorations of mathematical models using differential equations. These materials are available over the Web and can be used by anyone with connections to the Internet. The IDEA developers, Tom LoFaro and Kevin cooper,  have created tools to assist instructors in the development of their own Web materials and/or contribute to the IDEA site. Currently, most explorations are based on biology, chemistry, and ecology. The IDEA approach provides resources that can give students an appreciate for research projects involving differential equations not found in traditional texts.
 

ODE Architect: With NSF/DUE support, the CODEE consortium John Wiley & Sons and the software house Intellipro are producing an interactive multimedia ODE package which is built over a robust solver engine designed by Larry Shampine. Scheduled for release in 1998 the CD-ROM runs on a PC under Windows 3.2 or better and is accompanied by a lab workbook or experiments. With video, sound, animation, and dynamic graphics, this interactive package provides motivation for modeling, analysis, visualization, discovery, and interpretation in any ODE lab environment. In addition to the solver tool there are 13 interactive modeling modules on ODEs and dynamical systems, as well as a library of ODEs with their dynamically generated solution curves and orbits. Each module leads the user through a model building process via several exploration screens, and ends up with questions. These questions take the user to the solver tools and to the accompanying lab workbook where the user is asked to carry out graphics-based experiments to explain what is going on. Users can enter their own ODEs and explore dynamical systems with 2D or 3D graphics or numerical tables by seeing what happens when data and parameters change. Graphs are editable and aces can be scaled and labeled, equidistant-in-tiume orbital points marked. Graphs of solutions and orbits can also be colorized, animated, displayed in various line styles, overlayed with graphs of functions, and graphed together with solution curves of other ODEs. All of this is possible with no programming or special commands to remember. ODE Architect has a report writing feature; graphs can be cut and pasted into reports.
 
3. Laboratory Experiments for an ODE Course
We can use technology to examine dynamical processes and their ODE models that would have been inaccessible only a few years ago. Hands-on experience in the setting of an ODE laboratory is at the heart of this approach. The "laboratory" might be a room with computers or individuals at home or in a dormitory working with their own computer, or even lecture demos. Whatever the mode, here are some central ideas that come up in the dynamical systems laboratory.

Modeling
* Derivative as a rate of change
* Balance law: net rate of change=rate in-rate out
* Compartmental models
* Newton's force laws
* Circuit laws for voltages and currents
* Chemical law of Mass Action

Visualization
* Derivative as slope of a curve
* The art of making graphs that tell a story
* Effective use of computer graphics; choosing appropriate displays
* How to interpret graphs; extracting information from graphs
* Change scales, time span, viewpoint to get the most informative graph

Solution Behavior
* Do we use theory, formulas or computer simulation to study solution behavior?
* Do solutions tend to an equilibrium state or periodic solution with advancing time? Or do they become chaotic?
* How sensitive are solutions to changes in data and system parameters?
* Is analysis of the sign of a rate function useful to see how solutions behave?
* What happens to long-term solution behavior as a parameter changes?
* If the modeling system is nonlinear, can it be approximated by a linear system, or is the behavior due to the nonlinearities?

Computer Techniques
* Should the variable be scaled before computing?
* Scale out system parameters not relevant to the sensitivity study
* Does your solve handle on-off functions of engineering, or can you work around it?
* Is the solution behavior generated by your numerical solver really there, or is it an artifact of the numerics? Has your solver overlooked an important aspect of solution behavior because its internal settings are inappropriate?

Well that's a long list of things to keep in mind, but the numerical experiments that follow touch on many of these points. Don't look for the standard experiments and models that can be found in many places. Our aim is to present challenging models that can now be handled by students in an introductory ODE course if they have access to a decent numerical solver.

Table of Contents

1. Modeling and visualization in the introductory ODE course R. L. Borelli and C. S. Coleman; 2. Differential equations in the information age W. E. Boyce; 3. A geometric approach to ODEs M. Branton and M. Hale; 4. Differential equations on the internet K. D. Cooper and T. LoFaro; 5. Data as an essential part of a course on differential equations D. O. Lomen; 6. Qualitative study of differential equations V. S. Manoranjan; 7. Teaching methods in ODE courses L. F. Shampine and Ian Gladwell; 8. Technology in differential equations courses: my experiences, student reactions B. H. West.

Editorial Reviews

"A useful resource for any instructor who teaches a course that incorporates DEs"

Telegraphic Review

"The text serves as an excellent introduction to instructors wishing to incorporate computer based methods into their teaching..... an extremely interesting, and informative book. The articles are, well written and provide explicit examples as well as many useful and practical ideas from several innovative and capable practitioners in this rapidly evolving area of undergraduate education."
—Crux Mathematicorum

Alan Law

"The revolution is worth joining, and this volume is worth having, for anyone who teaches differential equations..the volume serves its purpose well by presenting a number of ideas for using technology in teaching differential equations."

MAA Online

Eight contributions provide undergraduate mathematics instructors with ideas on how technology can be incorporated into differential equations courses. Topics include the use of laboratories in ODE courses, dynamical systems, and Internet resources. Lacks subject index. Annotation c. Book News, Inc., Portland, OR (booknews.com)

Booknews