June 14, 2009
June 14, 2009
June 17, 2009
14.1044.1 - 14.1044.12
Scripts in MATLAB for Animation of the Solutions to Partial Differential Equations
Teaching of mathematics at all levels can be enhanced by the use of graphical techniques. That is particularly true for engineering mathematics, since well over half of engineering students are graphical learners and many presentations of mathematical concepts employ only symbolic presentations. Geometry and graphics were important parts of the Calculus Reform movement and can be used to advantage in the teaching of concepts of differentiation and integration as reflected in modern calculus texts.1 Differential equation pedagogy can be enhanced by use of integral curve and direction field sketches.2
In addition to their existence as interesting mathematical problems, partial differential equations occur in a variety of physical science problems such as vibration, acoustics, heat transfer, fluid dynamics, chemical, energy and momentum transport and electromagnetic wave propagation. In the past decade a number of authors have shown how computer graphics can enhance understanding of particular problems of partial differential equations.2-18 Watkins et al. demonstrated how the handle graphics of MATLABTM could be employed to animate the behavior of dynamic systems19. The purpose of this paper is to create an awareness of availability of a group of MATLAB scripts to aid in visualization of partial differential equation solutions.
Since partial differential equations involve two or more independent variables, a student with a correct solution in hand may still not be able to make sense of it, i.e. explain what is physically happening.
Here we will consider partial differential equations which have two independent variables, a spatial variable x, and a temporal variable t and this solution will be denoted as y(x,t). There are several ways of presenting the solution graphically which are: (i) plots of y(x,ti) as a function of x for selected values of t (freeze frame), (ii) plots of y(xj,t) as a function of t for selected values of x (measured values of y(x, t) at various locations xj), (iii) a plot of y(x,t) as a function of x and t in three dimensions, and (iv)using animation to plot and erase y(x,t) versus x for a series of closely spaced values of t such that a movie of y(x,t) is obtained. At the outset the authors thought that the third strategy would be easiest to understand, but that assumption turned out to be incorrect as the fourth strategy was by far the clearest.
MATLAB is perhaps the most widely used general-purpose scientific and engineering software package for engineering education and engineering practice. Generally it is not a symbolic processor such as Maple or Mathematica, however, MATLAB does have the Maple kernel imbedded in the Symbolic Toolbox. It is thus appropriate to develop software for the purpose
Jacquot, R., & Anderson, J., & Walrath, D. (2009, June), Scripts In Matlab For Animation Of The Solutions To Partial Differential Equations Paper presented at 2009 Annual Conference & Exposition, Austin, Texas. 10.18260/1-2--4614
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