Consider the temperature distribution in a perfectly insulated rod of length L , where one end, at x = 0, is maintained at a temperature...

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Consider the temperature distribution in a perfectly insulated rod of length
L
, where one end,
at
x
= 0, is maintained at a temperature of 0
0
C and the other end, at
x
=
L
, is insulated. This is
well modeled by the diff
usion equation,
where a is a constant. It is subjected to boundary conditions,
t
The method of separation of variables, with
Ө
(
x
,
t
) =
X
(
x
)
T
(
t
), is used to determine the
boundary
-
value problem satisfied by
X
(
x
) which is given by.
X
(
x
) =
sin(
βx
)
whe
re
β
is in
terms of
L
and
n
= 1,2,3,......
Calculate the value of
β
when
n =
2 and
L
= 0.81, giving your answer to 3 decimal places. You may
assume that
π = 3.142.
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