Heat Conduction Equation
المؤلف:
المرجع الالكتروني للمعلوماتيه
المصدر:
المرجع الالكتروني للمعلوماتيه
الجزء والصفحة:
...
13-7-2018
1972
Heat Conduction Equation
A partial differential diffusion equation of the form
 |
(1)
|
Physically, the equation commonly arises in situations where 
is the thermal diffusivity and 
the temperature.
The one-dimensional heat conduction equation is
 |
(2)
|
This can be solved by separation of variables using
 |
(3)
|
Then
 |
(4)
|
Dividing both sides by 
gives
 |
(5)
|
where each side must be equal to a constant. Anticipating the exponential solution in 
, we have picked a negative separation constant so that the solution remains finite at all times and 
has units of length. The 
solution is
 |
(6)
|
and the 
solution is
 |
(7)
|
The general solution is then
If we are given the boundary conditions
 |
(11)
|
and
 |
(12)
|
then applying (11) to (10) gives
 |
(13)
|
and applying (12) to (10) gives
 |
(14)
|
so (10) becomes
 |
(15)
|
Since the general solution can have any 
,
 |
(16)
|
Now, if we are given an initial condition 
, we have
 |
(17)
|
Multiplying both sides by 
and integrating from 0 to 
gives
 |
(18)
|
Using the orthogonality of 
and 
,
so
 |
(22)
|
If the boundary conditions are replaced by the requirement that the derivative of the temperature be zero at the edges, then (◇) and (◇) are replaced by
 |
(23)
|
 |
(24)
|
Following the same procedure as before, a similar answer is found, but with sine replaced by cosine:
 |
(25)
|
where
 |
(26)
|
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