This problem is the unsteady version of the first example. Thus,
all of the boundary conditions shown in Fig. 1 apply.
In addition, we will specify that prior to time t=0 ,
the upper boundary is adiabatic so that the initial
temperature distribution within the box is given
by
We will use this
problem to illustrate superposition in
both the time and space domains and to
emphasize the
importance of physical reasoning during the solution
process.
As in the first example,
we homogenize the top surface boundary condition
by referencing temperatures to 
Due to the nonhomogeneities, we again assume that the problem
(i.e, the unsteady version of the problem shown in
the upper right
corner of Fig. 1) will have to be decomposed into two subproblems, each
containing one nonhomogeneous boundary.
A question arises as to the proper initial
condition for each subproblem. Physically, we note that prior to
the initiation of convective cooling on the top boundary,
this boundary is adiabatic. Thus, the initial temperature
distribution 
can only depend on x.
To prove this, note that prior to t= 0 , the temperature distribution
must be symmetric about the plane
i.e.,
Similarly, since 
and 
are adiabatic boundaries
then again by symmetry, so is 
Thus, by considering
smaller and smaller slices of the box, it is apparent that
across the width of the box. Thus, the initial temperature distribution
only depends on x and is given by
(This result should be verified by solving the steady
two-dimensional conduction equation.)
Based on the same argument, it
is apparent that the initial distribution in subproblem
2 is given by
Having specified the proper initial condition,
we now note that the physics of subproblem 1 (and 2)
indicate that a steady temperature distribution will eventually
appear. Thus, we guess a solution of the
form:
Since we have a non-zero initial condition, we anticipate
that the problem governing 
will have to be
homogeneous on all boundaries; this condition will allow us
to obtain orthogonal solutions in the x- and y-directions
which can then be used to satisfy the initial condition.
In other words, consistent with our
basic prescription, the problem defining
is limited to only one nonhomogeneity:
The problem for 
must account for
the nonhomogeneous condition on the left face and is given by:
The solution for was obtained in subproblem 1 of the first
example above and is given by:
where again
Turning to the problem on 
we assume a solution of the form
Thus, the governing equation (
) yields
Note that we have written the separated equation in this form since
we are seeking orthogonal solutions in the x- and y-directions.
Separating Y and P requires a second separation constant:
Based on the last two equations, it follows that
and
Using the boundary conditions defined above,
it is readily found that eigenvalues and eigenfunctions in the x and
y-direction are given by
and
where 
satisfies
The solution for is thus given by
where the coefficients 
are obtained
from the initial condition in ():
and where
Solution of Subproblem 2
As in the first example, subproblem 2 is essentially identical to
subproblem 1. Thus, the solution to subproblem 2 follows by making the
following substitutions:
The final solution to the problem is then the sum of the solutions to subproblem 1 and 2.