Along the whole positive -axis, we have an ideal heat-conducting rod, the surface of which is isolated. The initial temperature of the rod is 0 degrees. Determine the temperature function when at the time
(a) the head of the rod is set permanently to the constant temperature;
(b) through the head one directs a constant heat flux.
The heat equation in one dimension reads
In this we have
For solving (1), we first form its Laplace transform (see the table of Laplace transforms)
which is a simple ordinary linear differential equation
of order two. Here, is only a parametre, and the general solution of the equation is
(see this entry). Since
we must have . Thus the Laplace transform of the solution of (1) is in both cases (a) and (b)
For (a), the second boundary condition implies . But by (2) we must have , whence we infer that . Accordingly,
which corresponds to the solution function
of the heat equation (1).
For (b), the second boundary condition says that , and since (2) implies that , we can infer that now
which corresponds to