\input epsf
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\noindent {\bf Problem statement} If $u(x,t)$ is the temperature in a rod at time $t$ a distance $x$
from some fixed point, then to a good approximation $u(x,t)$ satisfies
the {\it Heat Equation}: 

$${{\partial u}\over{\partial t}} 
= D
{{\partial^2 u}\over{\partial t^2}}$$ 

\noindent where $D$ is a constant which describes the physical
properties of the rod. (To be more precise, the Heat Equation applies
when the surface of the rod is well insulated, so that heat can enter
or leave the rod only through its ends. The same equation also appears
in the analysis of many diffusion problems.)

\noindent For $t>0$, define $\displaystyle
u(x,t)={{e^{\bigl(\!{{-kx^2}\over t}\!\bigr)}}\over {\sqrt{t}}}$ where
$k$ is a constant. There is one value of $k$ for which this function
is a solution of the Heat Equation. Find the value of $k$ and verify
that the resulting function does solve the equation. The value of $k$
will be related to $D$.  

\smallskip

\noindent {\bf Comment} This is the most famous solution of the Heat
Equation and is called the {\it Fundamental Solution}.


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