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Education 3)  Technology in the Teaching and Learning of Differential Equations

Research and educational tools for differential equations in Maple

Maple's ODE and PDE solvers, nowadays powerful tools for finding exact solutions, were developed as part of a computational environment for differential equations (DEs). In this talk, an overview of this computational environment, focusing on its research and educational tools, is given. Finally, the present status of Maple's ODE and PDE solvers, as well as their possible evolution as artificial intelligence systems, are discussed.

ROBERT CORLESS, University of Western Ontario
Applications of the NODES package for solving IVPs for ODE in Maple: modules for classroom use

The package NODES (for Numerical Ordinary Differential Equations) developed in Maple by Larry Shampine and myself is quite convenient for classroom explorations within Maple. This talk will explore several application modules, ranging from simple introductory examples through IVP that require custom Maple tools to generate the input for NODES. Other tools, useful for assessment of the quality of the computed solution and its relevance to the problem at hand, are also discussed.

CHRIS ESSEX, Department of Applied Mathematics, University of Western Ontario, Ontario
Numerical monsters

When certain seemingly innocent computations performed with computers are not merely wrong, but are dramatically wrong, they provide a powerful reason to be cautious about computations using computers. Furthermore, should these calculations be such that they can provide a general insight into how and when computer calculations do go wrong, beyond just dismissing error as ``noise'', a natural lifelong appreciation for the subject of numerical analysis can be engendered and users exposed to the calculations will become masters of their tools. The presentation can be regarded as a gallery of simple calculations that can be depended on to produce such insightful numerical mayhem, even across different machines and across very different software. While forms of these numerical ``monsters'' are not unknown, even experienced computer users can find them capable of surprises in the potent forms presented.

KEITH GEDDES, University of Waterloo, Waterloo, Ontario
Generating numerical ODE formulas via a symbolic calculus of divided differences

Divided differences of vector functions can be applied to generate numerical update formulas for ODEs, as proposed by W. Kahan. A computer algebra system renders the symbolic manipulations practical to perform. The technique is presented here using the Maple system. Some examples illustrate the potential of this hybrid symbolic-numeric approach to solving initial-value problems. We discuss this in an educational context.

To be announced

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