The One-Dimensional Heat Conduction Equation
Consider a thin bar of length
L
, of uniform cross-section and constructed of
homogeneous material.
Suppose that the side of the bar is perfectly
insulated so no heat transfer could occur through it (heat could possibly still
move into or out of the bar through the two ends of the bar).
Thus, the
movement of heat inside the bar could occur only in the
x
-direction.
Then,
the amount of heat content at any place inside the bar, 0 <
x
<
L
, and at any
time
t
> 0, is given by the temperature distribution function
u
(
x
,
t
).
It
satisfies the
homogeneous one-dimensional heat conduction equation
:
α
2
u
xx
=
u
t
Where the constant coefficient
α
2
is the
thermo diffusivity
of the bar, given
by
α
2
=
k
/
ρs
.
(
k
= thermal conductivity,
ρ
= density,
s
= specific heat, of the
material of the bar.)
Further, let us assume that both ends of the bar are kept constantly at 0
degree temperature (abstractly, by connecting them both to a heat reservoir
of the same temperature; more practically, say they are immersed in iced
water).
This assumption imposes explicit restriction on the bar’s ends, in
this case:
u
(0,
t
) = 0, and
u
(
L
,
t
) = 0.
t
> 0
Those two conditions are called the
boundary conditions
of this problem.
They literally specify the conditions present at the boundaries between the
bar and the outside.
In addition, there is an initial condition: the initial temperature distribution
within the bar,
u
(
x
,
0).
It is a snapshot of the temperature everywhere inside
the bar at
t
= 0.
Therefore, it is an (arbitrary) function of the spatial variable
x
only.
That is, the initial condition is
u
(
x
,
0) =
f
(
x
).