Let a thin square-formed plate of heat conducting homogeneous material be in the
-plane with sides on the
-axis (isolated), on the line
(held at the constant temperature
), and on the vertical lines
and
(both held at the constant temperature
). Determine the temperature
function
on the plate, when the faces of the plate are isolated.
The equation of the heat flow in this stationary case is
![]() |
(1) |
We first try to separate the variables, i.e. seek the solution of (1) of the form
Then we get
and thus (1) gets the form
![]() |
(2) |
We separate the variables in (2):
This equation is not possible unless both sides are equal to a same negative constant
and for
The two first boundary conditions give
Therefore
The fourth boundary condition now gives that
![]() |
(3) |
of the functions (3) satisfy (1) and those boundary conditions, provided that this series converges. The third boundary condition requires that
on the interval
The even
Thus we obtain the solution
It can be shown that this series converges in the whole square of the plate.
Remark. The function
has been approximated in the plot by computing a partial sum of the true infinite-series solution. However,
there is substantial numerical error in the approximate solution near
, evident in the small oscillations observed in the surface plot, that should not be there in theory.
This phenomenon is actually inevitable given that the boundary conditions are actually discontinuous at the corners
and
.
More precisely, observe that when
, the formula for
reduces to the Fourier series
for the discontinuous function on
That means the Fourier expansion will necessarily be subject to the Gibbs phenomenon. Of course, the series also cannot converge absolutely; in other words, the terms of the series decay too slowly in magnitude, adversely affecting the numerical solution.
As of this snapshot date, this entry was owned by bloftin.