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Old   November 12, 2013, 11:37
Post 1 dimensional heat conduction
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We have one dimensional steady heat conduction with heat generation term. I use this to find temperature distribution within a rod. Now, i will require two boundary conditions in space co-ordinates. I specify, one end as insulated and other end with a specified heat flux. Physically this is not correct. However, mathematically the differential equation requires two boundary conditions. So, can we conclude that, we cannot specify two Neumann boundary conditions in this case?
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Old   November 13, 2013, 03:30
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Rami Ben-Zvi
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Hi dreamz,

Let us try to naively solve your energy eqaution
q,x = -(kT,x),x = S
assuming for simplicity k and S are constant.
q = -kT,x = Sx + A
-kT = 0.5*S*x^2 + Ax + B
To determine the integration constants A and B, use your BCs:
q(0) = 0 => A=0
q(L) = Q = SL (where Q is the prescribed value at x=L).

Now you have two problems:
(1) A contradiction: if both S and Q are given, you cannot satisfy the last equation.
(2) You cannot determine B.

Conclusion: the problem is ill-posed.
On the other hand, if your BCs were both Dirichlet or one Dirichlet and one Neumann, then the problem is well-posed, as you may see.
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1 dimensional, differential equation, heat conduction, neumann bcs

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