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Transient Heat Conduction Using Chebyshev Collocation

number of interior points
40
time
0.1
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
Consider the one-dimensional heat equation given by
1
α
u
t
=
2
u
2
x
, with
t0
and
0x1
.
This equation represents heat conduction in a rod. The boundary conditions are such that the temperature,
u
, is equal to 0 at both ends of the rod:
u(x=0,t)=0
and
u(x=1,t)=0
for
t0
.
Without loss of generality, one can take the thermal diffusivity,
α
, equal to
1
2
cm
/s
. The initial condition is given by
u(x,0)=1
for
0x1.
The temperature can be found using either NDSolve (solid colored curve) or the Chebyshev collocation technique (colored dots). Both methods give the same results, which are plotted in the same diagram at various values of time,
t
ranging from 0.01 to 0.1 with a span of 0.01. You can set the number of interior points used by the Chebyshev collocation method.
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