Project 1, 15 points
Math 314H, Spring 2000
Due: 2/18

Problem Description:  You are given a laterally insulated rod of a homogeneous 
material whose conductivity properties are unknown.  The rod is laid out on
the x-axis, 0 <= x <= 1.   A current is run through the rod, which results in a
heat source of 10 units of heat (per unit length) at each point along the rod. 
The rod is held at zero temperature at each end.  After a time the
temperatures in the rod settle down to a steady state.  A single measurement
is taken at x=0.3 which results in a temperature reading of approximately 11
units.   Based on this information, determine the best estimate you can for
the true value of the conductivity constant k of the material.  Also try
to guess a formula for the shape of the temperature function on the
interval [0,1] that results when this value of the conductivity is used.

Methodology: You should use the model that is presented on  pages 4-6 of the
text.  This will result in a linear system, which Maple can solve.   One way to
proceed is simply to use trial and error until you think you've hit on the
right value of k, that is, the one that gives a value of approximately 11
units at x =0.3.  Then plot the resulting approximate function doing a
dot-to-dot on the node values.  You should give some thought to step size h.

Output:  Return your results in the form of an annotated Maple notebook, which
should have the name of the team members at the top of the file and an
explanation of your solution in text cells interspersed between input cells,
that the user can happily click his/her way through.  This explanation
should be intelligible to your fellow students.  You will find it *very*
helpful to examine the Maple notebook LinAlgMaple5.ms  in my Public/MapleFiles
directory, especially the comments at the bottom of the notebook.

Comments:  This exercise introduces you to a very interesting area of
mathematics called "inverse theory."  The idea is, rather than proceeding
from problem (the governing equations for temperature values) to solution
(temperature values),  you are given the "solution", namely the measured
solution value at a point, and are to determine from this information the
"problem", that is, the conductivity coefficient that is needed to define the
governing equations.