example of solving the heat equation
Let a . Determine the temperature function on the plate, when the faces of the plate are .
The equation of the heat flow (http://planetmath.org/HeatEquation) in this case is
under the boundary conditions
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
and the boundary conditions
We separate the variables in (2):
This equation is not possible unless both sides are equal to a same negative , which implies for the solution
and for the solution
The two first boundary conditions give , , and since , we must have , i.e.
The fourth boundary condition now yields that ; thus and So (1) has infinitely many solutions
with and they all satisfy the boundary conditions except the third. Because of the linearity of (1), also the sum
of the functions (3) satisfy (1) and those boundary conditions, provided that this series converges. The third boundary condition requires that
The even (http://planetmath.org/EvenNumber) ’s here give 0 and the odd (http://planetmath.org/EvenNumber) give
Thus we obtain the solution
It can be shown that this series converges in the whole of the plate.
Visualization of the solution
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 . This phenomenon is actually inevitable given that the boundary conditions are actually discontinuous at the corners and .
More precisely, observe that when , the for reduces to the Fourier series
for the discontinuous function on :
That means the Fourier will necessarily be subject to the Gibbs phenomenon. Of course, the series also cannot converge absolutely; in other of the series decay too slowly in magnitude, adversely affecting the numerical solution.
http://gold-saucer.afraid.org/math/planetmath/ExampleOfSolvingTheHeatEquation/heat.pyPython program to compute and produce the two figures
|Title||example of solving the heat equation|
|Date of creation||2014-09-28 17:02:21|
|Last modified on||2014-09-28 17:02:21|
|Last modified by||pahio (2872)|
|Synonym||stationary example of heat equation|