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I am trying to solve the one-dimensional transient heat equation with a specified flux in one end ($x=0$) and perfect insulation on the other ($x=L$):

$$\frac{\partial T(x,t)}{\partial x} = \alpha \frac{\partial^2 T(x,t)}{\partial x^2}$$ $$\frac{dT(0,t)}{dx}=W$$ $$\frac{dR(L,t)}{dx}=0$$

The initial temperature is constant throughout the system:

$$T(x,0)=1$$

By separation of variables I can get the general heat equation solution:

$$T(x,t) = Ae^{-\lambda \alpha t}\left[ B\sin(\sqrt{\lambda}x) + C\cos(\sqrt{\lambda}x)\right]$$

but I am stuck trying to apply the boundary conditions, specifically the constant flux. Every time I try to solve this I get "constants" that vary with time. It has been many moons since I have solved an equation like this, and I expected to get some complicated formula for $\lambda$, but unfortunately I can't even get that far.

I suppose the underlying question is, "Can the general heat equation solution above be solved to fit the initial and boundary conditions also specified above?".

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  • $\begingroup$ I've messed around with this a little bit and I'm not too sure it has a "pencil and paper" solution. But, it can be numerically simulated. $\endgroup$
    – K.defaoite
    Apr 27, 2022 at 17:31

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Separation of variables requires that the boundary conditions are homogeneous. So the first thing is to find a single inhomogeneous solution, not necessarily with $T(x,0)=0$, and subtract off. Maybe guess a solution of the form $T(x,t) = a x^2 + b t$.

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  • $\begingroup$ This answers my question. I did not know that I needed homogeneous BCs for separation of variables, oops! $\endgroup$ May 10, 2022 at 16:58

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