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WOLFRAM|DEMONSTRATIONS PROJECT

Heat Conduction in a Rod

​
collocation points
20
t
0.05
General
:Exp[-712.585] is too small to represent as a normalized machine number; precision may be lost.
General
:Exp[-750.583] is too small to represent as a normalized machine number; precision may be lost.
General
:Exp[-789.568] is too small to represent as a normalized machine number; precision may be lost.
General
:Further output of General::munfl will be suppressed during this calculation.
Consider the problem of unsteady-state heat conduction in a rod, as governed by the heat equation:
∂T
∂t
=
2
∂
T
∂
2
x
.
The initial and boundary conditions are:
t=0
,
T(x,0)=2x
for
0≤x≤1/2
and
T(x,0)=2(1-x)
for
1/2≤x≤1
,
x=0
,
T(0,t)=0
,
x=1
,
T(1,t)=0
,
where
T
is the temperature,
t
is time, and
x
is the position.
This problem has an analytical solution in the form of a Fourier series after separation of variables:
T(x,t)=
8
2
π
∞
∑
n=1
1
2
n
sin
nπ
2
sin(nπx)
-
2
n
2
π
t
e
.
This Demonstration plots the solution
T(x,t)
. The numerical solution obtained using Chebyshev orthogonal collocation is given by the red dots. The analytical solution is given by the blue curve. Excellent agreement between the two solutions is observed. You can vary the value of
t
as well as the number of Chebyshev collocation points,
N+1
.
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