# Notes on the PDE programs

These programs are intended for use by beginning students. The purpose is to develop familiarity with behavior of some of the solutions, in hopes that the Mathematical treatment of these equations in whatever course the student is taking will then be easier to understand. The programs run at least on Quadra 610 and 950, Centris 650, and PowerMac 7100, except that the wave program does not run on my 950 for reasons I am unable to determine. They will certainly not run under System 6.

Please send comments to Bob Terrell, Math Dept, Cornell U., Ithaca NY, 14853, or email bterrell@math.cornell.edu

## Notes on Heat Equation 1D and Heat Equation 2D

These programs are supposed to help you visualise solutions to the heat equation in a bar or rectangle and the Laplace equation in a rectangle.

You can use the programs without understanding the mathematics behind them, but you must understand that the temperature depends on the initial and boundary conditions and that is what you are observing and modifying. In fact the purpose of the programs is to acquire some intuition about heat flow so that you can then understand the mathematics better.

There is documentation included in the programs, under the Options menu. For more information on the mathematics see any book on differential equations or Fourier series or engineering mathematics.

You will need a color Macintosh, and it is better if you have at least the standard 13" screen in order to read the documentation in the 2D program.

To see a relation between the two programs you can enter insulated boundaries on two opposite sides in the 2D program, and then you can solve essentially 1D problems like the 1D program does.

There is currently a bug in the 2D program which makes the central portion of the lower rectangle unresponsive to mouse clicks, but you can still enter initial and boundary conditions in that area by dragging over it.

Here are two fairly hard sample questions one might answer with this software:

• 1) Suppose you start with two checkerboards where the black squares are at 600 degrees and the white squares are at 0 degrees initially, and the only difference between the two checkerboards is that one is insulated around the edges and the other is maintained at 300 degrees around the edge. It is a fact that both will eventually reach an equilibrium temperature of 300 degrees, but the question is, which one approaches the equilibrium faster?
• 2) Suppose you begin with a metal rod which has a hot spot near one end, and 0 boundary conditions. What happens to the hot spot i.e., does the location of maximum temperature move or stay still as time goes by? If it moves, which direction does it move?

## Notes on the Wave Equation program

This program is supposed to help you visualise solutions to the wave equation for the vibrations of a string.

You can use the program without understanding the mathematics behind it, but you must understand that the string position depends on the initial and boundary conditions and that is what you are observing and modifying. In fact the purpose of the programs is to acquire some intuition about waves so that you can then understand the mathematics better.

There is documentation included in the program, under the Options menu. For more information on the mathematics see any book on differential equations or Fourier series or engineering mathematics.

Last Update: September 19, 1995