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THE VISUALIZATION OF BOUNDARY VALUE PROBLEMS

Abstract

In this paper the author will demonstrate how using Maple software, and physical models, in an introductory Boundary Value Problems course, helps students learn the concepts presented. By using Maple software and simple demonstrations done in class, the instructor and students were not only able to solve partial differential equations analytically, but were able to see how the solutions visually compared with the classroom demonstrations. Demonstrations will include the heat equation, the one dimensional wave equation, and the beating drum problem. The paper will also discuss why engineers and physicists can use only the first couple of terms of the solution to a boundary value problem. This course helps prepare engineering students to take courses such as Heat Transfer, Waves, Thermodynamics, and Electromagnetic Fields.

Introduction

When the author began teaching the Boundary Value Course it was decided to do a few things differently. The first change was to make the course more applied and less theoretical. To do this a new text was needed. While many texts were considered,including ones that were based on the use of a computer algebra system such as Maple , it was finally decided to pick a text that as one publisher stated "this text is the one the engineers would use." Thus, "Boundary Value Problems" by Powers was the text selected. For our purposes it has turned out to be a good choice.

The second change was to use Maple extensively in the course. The idea was that since we were going to rely heavily on the technique of separation of variables which leads to solutions obtained from Fourier Series, the students would be expected to actually calculate specific terms of the Fourier Series solution rather that simply writing down the general Fourier series.

The third change was the easiest to make. We simply advertised the course as a mathematics course that the engineering students might actually find useful.

This paper is going to deal mainly with the issues of using the Maple software and demonstrations and how they were used in the Boundary Values Course to help make the material "more interesting" and hopefully easier to understand. Our goal was to make sure

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Graves, E. (2010, June), The Visualization Of Boundary Value Problems Paper presented at 2010 Annual Conference & Exposition, Louisville, Kentucky. 10.18260/1-2--15731