PDE program notes

**Introduction

     This folder contains a sequence of educational PDE solvers that I have
written, in an attempt to provide a user-friendly introduction to these
equations. They are called
     Heat Equation 1D,
     Heat Equation 2D,
     Wave Equation 1D,
     Wave Equation 2D.
The programs are updated versions of the ones I first placed in the Mathematics
Archives in 1995. They are "fat binaries", which means that they will run on both
68k and PowerPC Macintosh computers. The programs run at least on Quadra 610,
840, and 950, Centris 650, PowerMac 7100, 7200, and 7500. They will certainly not
run under System 6. I have not tested these programs on laptops. All four
programs are available by ftp from the Mathematics Archives at 
     archives.math.utk.edu 
in the directory 
     /software/Mac/diffEquations 
or from the author. 

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

**Intended Audience

     These programs are intended for use by beginning students. The purpose is to
develop familiarity with behavior of some of the solutions. The presentation is
entirely visual and does not require any programming of any kind. 
     You can use the programs without understanding the mathematics behind them,
but you must understand that the temperatures or wave shapes depend 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
and wave motion so that you can then understand the mathematics better. Normally
one would have studied ODEs for a while before jumping into equations like these,
but it is not strictly necessary.

**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.
     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.
     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 Wave Equation 1D and Wave Equation 2D

     Wave Equation 1D is supposed to help you visualise solutions to the wave
equation for the vibrations of a string. There is documentation included in the
program, under the Options menu. 

     Wave Equation 2D solves the two dimensional wave equation in a square
region. The solutions of this equation may be interpreted as the vibrations of a
square drum head, small amplitude waves in a square swimming pool, or as
idealised sound or electromagnetic waves in a square planar cavity. The program
does not assume any knowledge of any of these subjects on the part of the user.
The initial conditions in this case consist of a wave shape and a wave velocity,
since the wave equation is second order in time. 
     The program allows changes in the initial and boundary conditions, and even
allows the placement of barriers (or "break-waters") within the square, so that
you can design your own "harbors", so to speak. The wave can be viewed either in
perspective or in a plan view using color to represent height.