June 24, 2007
June 24, 2007
June 27, 2007
Computers in Education
12.58.1 - 12.58.13
A Library of MATLABTM Scripts for Illustration and Animation of Solutions to Partial Differential Equations
In the past three years the authors have developed a series of MATLABTM scripts that illustrate the solutions to partial differential equations commonly encountered in mathematics, engineering and physics courses. The objective of this paper is to create awareness, among teaching faculty, of the availability of this set of MATLABTM scripts to aid their teaching of physical phenomena governed by partial differential equations.
Over many years the authors have observed the difficulty students have with the solutions to partial differential equation problems and when they have completed such a solution they still cannot associate a physical interpretation with the resulting equation or equations. Since many students are graphical learners, we asked ourselves how the high quality and easy to use graphics available in MATLABTM might be exploited to help students better understand the solutions that they, their instructor or the textbook may have generated.
There has been considerable work done to exploit the use of computer graphics to clarify physical problems governed by partial differential equations. An early paper used MATLABTM to illustrate solutions to hyperbolic differential equations.1 Several papers at about the same time used computer animation to illustrate solutions for elastic wave propagation and beam vibration.2,3 The concept of using MATLABTM for the animation of lumped parameter dynamic systems was demonstrated by Watkins et al.4 Recently there have been a number of papers describing the graphical interpretation of partial differential equations. The transport of pollutants in groundwater has been described using web-based graphics5 and another paper reports a virtual laboratory for teaching quasistationary electromagnetics.6 Another recent paper discusses the solution of groundwater problems using a spreadsheet.7 Still another paper employs a spreadsheet to examine the topic of electromagnetic wave propagation.8 Two recent papers reported the use of animation to clarify a variety of partial differential equation solutions.9,10 There are a number of approaches to the animation of distributed parameter systems and one is the application of finite element software (ANSYSTM) to illustrate the vibration of beams and plates.11 A recent paper discusses the use of animation in MATLABTM to animate the solution to a variety of electrical transmission line problems.12 A very recent paper discusses how MATLABTM has been employed to illustrate the downwind transport of the chemical components of industrial stack emissions.13
Strategies for PDE Solution Presentation
There are a number of possible ways that graphical presentations may be employed to clarify responses of dynamic systems described by partial differential equations with one spatial variable and time as independent variables. The most obvious are:
• A plot of the solution as a function of time for several locations (location as a parameter). • A plot of the solution as a function of the spatial variable for several values of time (time as a parameter).
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