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data. Initial conditions can be entered by clicking in a graph window or via the keyboard. Graph scales can be set automatically or manually. Numerical values of solutions can be viewed in tabular form. Parameter-sensitive analysis is made easy with a built-in parameter-sweep tool. You can do parameter and initial-value sweeps to see the effects of data changes on orbits and solution curves. Graphs are editable and you can scale and label axes, mark equidistant-in-time orbital points, color the graphs, change line styles, overlay graphs of functions and solution curves for different ODEs—all with no programming or special commands to remember.
The solvers in the ODE Architect are state-of-the-art numerical solvers based on those developed by Dr. L.F. Shampine and Dr. I. Gladwell at Southern Methodist University. For a delightfully readable account on using numerical ODE solvers in teaching ODEs, please refer to their paper:
Shampine, L.F., and Gladwell, I., “Teaching Numerical Methods in
in the book Revolutions in Differential Equations, edited by Michael J. Kalla-her in the MAA Notes series.
Module 1, “Modeling with the ODE Architect”, is an on-line tutorial for many of the features of the Tool. The Architect also has help facilities and the multimedia side is self-documenting.
The ODE Library has dozens of pre-programmed, editable, and interactive ODE files covering a wide range of topics from mathematics, physics, chemistry, population biology, and epidemiology. There are also many ODEs to illustrate points such as data compression, ODEs with singular coefficients, bifurcations, limit cycles, and so on. Each Library file has explanatory text along with the equations and includes an illustrative graph or graphs. The Library files are organized into folders by topic and they have descriptive titles to facilitate browsing. These files also provide a marvelous way to learn how to use the tool.
ODE Architect was developed with partial support from the NSF/DUE,1 by the Consortium for ODE Experiments (C-ODE-E), Intellipro, Inc., and John Wiley & Sons, Inc. C-ODE-E saw to the mathematical side of things, Intellipro rendered C-ODE-Es work into an attractive multimedia software package, and John Wiley coordinated the efforts of both teams.
As in any project like this, we owe a debt of gratitude to many people: reviewers, beta testers, students, programmers, and designers. Specifically, we want to thank the other members of the C-ODE-E Evaluation Committee, Barbara Holland (John Wiley & Sons), Philippe Marchal (Intellipro), Michael Moody (Harvey Mudd College), and Beverly West (Cornell University). Without the many hours of hard work they put in on this project, it could not have been done.
We especially want to thank Professor L.F. Shampine for providing the excellent solver codes (developed by himself and his colleague, Ian Gladwell), and for his continuing support of the project. Thanks to Mark DeMichele at Intellipro, who wrote the code for the Architect and the implementation of the Shampine/Gladstone solver codes, and who put up with our constant “advice”. Another very special thanks to David Richards for designing and implementing the LATEX macros for the Companion book and for his patience and meticulous attention to detail during the many revisions. We also very much appreciate the reviewers, editors, and evaluators Susan Gerstein, Zaven Karian, Mario Martelli, Lang Moore, Douglas Quinney, David Cook, and Robert Styer for their many helpful comments and suggestions.
Finally, a big “thank you” to our students Tiffany Arnal, Claire Lau-nay, Nathan Jakubiak, John Lu, Joel Miller, Justin Radick, Paul SanGiorgio, Treasa Sweek, and many others who read chapters, tested modules, and commented freely (even favorably) on what they experienced.
Robert L. Borrelli Courtney S. Coleman Claremont, CA
1The work on ODE Architect and its Companion book was supported in part by the National Science Foundation under Grant Numbers DMI-9509135 and DUE-9450742. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
Modules/Chapters 1-3 are all introductory modules for first-order ODEs and simple systems of ODEs. Any of these modules/chapters can be used at the beginning of an ODE course, or at appropriate places in elementary calculus courses.
Modules/Chapters 4-9 involve higher-order ODEs and systems and their applications. Once students understand how to deal with two-dimensional systems graphically, any of these modules/chapters is easily accessible.