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\centerline{Mathematics 173 -- Applied Mathematics 1}
\centerline{Discussion 1 -- The Heat Equation}
\centerline{September 2, 1999}
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\noindent
{\it Background}
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Recall that last time, we gave a derivation of the Heat Equation
for a long, thin rod or wire:
$$c(x)\rho(x){\partial u \over \partial t} = {\partial \over \partial x}
\left(K_0 {\partial u\over \partial x}\right) + Q(x,t),\leqno(HE)$$
where:
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\item{$\bullet$} $u = u(x,t)$ is the {\it temperature} at location $x$
in the wire, at time $t$, 
\item{$\bullet$} $c(x)$ is the {\it specific heat} of the 
material in the wire at location $x$ (the amount of heat energy needed
to raise a unit mass one degree),
\item{$\bullet$} $\rho(x)$ is the {\it mass density} of the material 
at location $x$,
\item{$\bullet$} $K_0(x,t)$ is the {\it thermal conductivity} of the
material,
\item{$\bullet$} we assumed the lateral surface of the rod or wire
is insulated so not heat is lost or gained through that surface.
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\noindent
This is an example of a {\it partial differential equation} (PDE).  We
think of $c,\rho, K_0, Q$ as {\it known functions} (known once we know
the particular properties of the rod we are studying).  The temperature
function $u$ is the unknown, and (HE) gives a relation between partial
derivatives of $u$, and various known functions.

An important special case is the case where $c,\rho, K_0$ are all 
{\it constant}.  That is, the thermal properties do not
depend on the position within the wire, or on time -- we say the rod is 
{\it uniform}.  If we have a uniform rod, containing no heat sources (so 
$Q(x,t) = 0$ all $x,t$), then the heat equation reduces to 
$${\partial u \over \partial t} = k {\partial^2 u\over \partial x^2}\leqno(HE')$$
where $k = {K_0\over c\rho}$ is also constant.  
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\noindent
{\it Discussion Questions}
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\item{A)}  Show that each of the following functions is a {\it solution}
of the heat equation for a uniform rod with no heat sources -- that
its partial derivatives satisfy the relationship (HE').  Also
answer each of the questions connected with the specific solutions.
\itemitem{1)} $u_1(x,t) = a x + b$ ($a$,$b$ constant).  What is "special"
about this type of solution?  (What happens to the temperature as
$t$ changes?)
\itemitem{2)}  $u_2(x,t) = x(x^2 + 6kt)$.
\itemitem{3)}  $u_3(x,t) = e^{-kt} \sin(x)$ and $u_4(x,t) = e^{-kt} \cos(x)$.
Suppose the wire extends from $x = 0$ to $x = L = \pi$.  Is one of 
these solutions more reasonable physically than the other?  (Recall that $u = 0$
is the lowest possible temperature.)
\itemitem{4)}  
$$u_5(x,t) = {1\over \sqrt{t}} e^{-x^2\over 4kt}$$
Note that $u_5$ is not defined at $t = 0$.  Plot $u_5(x,t)$ for $k = 1$ and 
several different $t$ (at least $t = 1/4,1/2,1,2$) and describe
what is happening as $t$ increases.  What do you think the graph
of $\lim_{t\to 0^+} u_5(x,t)$ looks like?
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\item{B)}  One possible question we could ask is: Given a uniform rod
with known $c,\rho,K_0$, and no heat sources, if the
{\it initial temperature\/} $u(x,0)$ is known
for all $x$, does knowing that $u(x,t)$ satisfies the equation (HE') determine 
the value of $u(x,t)$ for all $t > 0$?
Think about this on physical grounds.  Should the answer be yes or no?
Take into account the following two different scenarios:
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\itemitem{} {\it scenario 1}:  The ends of the rod are maintained at
constant temperature $T = 0$.
\itemitem{} {\it scenario 2}:  The ends of the rod are {\it insulated}
so that no heat energy passes through the ends.  (Note the lateral
surface of the rod is insulated too.  In this scenario, there is no
transfer of heat energy between the rod and the ``outside world'' whatsoever).
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\item{} Describe what you think should happen in each of these scenarios if
$u(x,0) = f(x)$ is some known function.  Do you think you get the
same temperature function $u(x,t)$ for all $t > 0$ in these cases?
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\item{C)} Express the two scenarios above in the form of {\it equations}
involving $u(x,t)$ and its derivatives, assuming the ends of the
rod are $x = 0$ and $x = L$.  These are examples of what are known
as {\it boundary conditions}.  (Hint: For scenario 2, you probably want
to think about Fourier's Law.)
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\item{D)} Another form of boundary condition occurs if we assume that
the heat flux at one or both boundaries is {\it proportional to the 
difference between the temperature of the rod at that end and the 
ambient temperature surrounding that end of the rod.}  Express this
form of boundary condition as a pair of equations if we have
a rod in room maintained at a constant ambient temperature of $T_a$
degrees.
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\noindent
{\it Assignment}
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Group write-ups due in class on Thursday, September 9.
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