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Session 1320

Clarification of Partial Differential Equation Solutions Using 2-D and 3-D Graphics and Animation Raymond G. Jacquot, Cameron H. G. Wright, Thomas V. Edgar, Robert F. Kubichek University of Wyoming

Abstract

The work discussed here demonstrates the use of two- and three-dimensional graphics and animation to clarify various solutions to partial differential equations describing a variety of dynamic physical problems. These graphical representations allow students to visualize the simultaneous variation of the dependent variable in space and time.

1. Introduction

There are a host of dynamic problems in the engineering disciplines that are described by partial differential equations. In the typical engineering math sequence the mathematics associated with their solutions often obscures the meaning and physical nature of the solutions. These problems arise in electromagnetics, vibrations, fluid dynamics, heat transfer and chemical mass and energy transport. The extensive graphics capabilities of MATLABTM make the illustration of these solutions a reasonable task. The idea of using graphics to illustrate the solutions to hyperbolic partial differential equations has been published but animation was not employed.1 Animations of bending vibrations in beams and longitudinal vibration of bars has been illustrated by Gramoll et al.2,3 The response of a plucked string employing implicit methods in time has been demonstrated previously but animation was not used there either.4 The ability to animate lumped parameter dynamic system behavior employing the handle graphics of MATLAB has been illustrated by Watkins et al.5 The response of a Bernoulli-Euler cantilever beam has been calculated and animated using central spatial differencing.6 A recent article illustrates the transport of pollutants employing web-based computer graphics.7

MATLAB is perhaps the most widely used general-purpose scientific and engineering software package in engineering education and engineering practice. It is thus appropriate to develop software for the purpose given here in that computing environment. The array computation ability and readily available graphics make such software development a reasonable task.

The authors have written MATLAB m-files to solve the following problems and illustrate their solutions. 1. Voltage waves on a sinusoidally driven lossless electrical transmission line. 2. The drawdown of well water in a pumped aquifer model. 3. Free and forced vibrations of a Bernoulli-Euler cantilever beam. 4. Conduction heat transfer in a slab with differing boundary temperatures.

Proceedings of the 2005 American Society for Engineering Education Annual Conference & Exposition Copyright  2005, American Society for Engineering Education

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Edgar, T., & Wright, C., & Kubichek, R., & Jacquot, R. (2005, June), Clarification Of Partial Differential Solutions Using 2 D And 3 D Graphics And Animation Paper presented at 2005 Annual Conference, Portland, Oregon. 10.18260/1-2--15308