Consider a rod of length m, laterally insulated (heat only flows inside the rod). Initially the temperature is
(5.48) |
The left and right ends are both attached to a thermostat, and the temperature at the left side is fixed at a temperature of and the right end at . There is also a heater attached to the rod that adds a constant heat of to the rod. The differential equation describing this is inhomogeneous
Since the inhomogeneity is time-independent we write
(5.50) |
where will be determined so as to make satisfy a homogeneous equation. Substituting this form, we find
(5.51) |
To make the equation for homogeneous we require
(5.52) |
which has the solution
(5.53) |
At the same time we let carry the boundary conditions, , , and thus
(5.54) |
The function satisfies
This is a problem of a type that we have seen before. By separation of variables we find
(5.56) |
The initial condition gives
(5.57) |
from which we find
(5.58) |
And thus
(5.59) |
Note: as , . As can be seen in Fig. 5.2 this approach is quite rapid – we have chosen in that figure, and summed over the first 60 solutions.