Partial Differential Equations PowerTool

by Dr. Jim Herod

Conclusion: Partial Differential Equations, Analysis, and Maple

      Since the time when Maple IV was created, I have tried to push Maple up the curriculum. Before retiring from Georgia Tech, I participated in a project to create worksheets for every course in the first two years of undergraduate mathematics. At the same time, high schools in metropolitan Atlanta began to include Maple as a part of the preparation of their students for college mathematics. I was also asked to assist in creating materials for those pre-college courses.

      But it has always been my perspective that Maple is more than a teaching tool. It is a working tool. If Maple were used through the calculus and in an ordinary differential equations class and then were put aside, then its real power would be lost. Maple is a tool that should be readily at hand for making models and for analyzing the implications of the models in any field on the campus.

      Toward accomplishing that goal, Maple must move across the campus and up the curriculum.

      E. Yeargers, R. Shonkwiler, and I took Maple across the campus in one direction through our cross listed, team taught course in Mathematical Biology. The result of that endeavor is included in the text published by Birkhauser and titled An Introduction to the Mathematics of Biology with Computer Algebra Models . The idea of carrying computer algebra systems into the curriculum of the Life Sciences is wide spread among those who have seen the power of Maple as a working tool. These efforts must be continued.

      Using Maple in a study of undergraduate ordinary differential equations is common. There are many authors who have worked to make materials available for doing this. Writings in partial differential equations using Maple are less common. Yet, every engineering, physics, or chemistry student uses partial differential equations to model physical phenomena in their area. Undergraduate courses in partial differential equations might take an entire academic year. Major topics would be methods of Fourier Series, characteristics, Green's functions, Laplace transforms, and numerical analysis. These would be introduced while examining linear systems. Yet, the world is non-linear, and we need a year of graduate partial differential equations to begin that topic.

     The first question facing any writer deciding to make notes for partial differential equations is what to include. The reader can get a glimpse of my decisions by looking over the titles of the Sections. The omissions stand out. Where are Green's functions? How can we discuss the method of characteristics and not illustrate shocks arising in first order systems? Where are numerical methods for three dimensional diffusion equations. One colleague reading the notes pointed out the omission of time dependent boundary conditions for the diffusion equation using the methods of separation of variables.

      The list goes on. Thinking of the list, the problem changes: where is the stopping place in constructing a set of lecture notes for a first look at using Maple in an undergraduate partial differential equations course.

      I stop here.