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The solution of $$\frac{\partial c}{\partial t} = D\frac{\partial^2c}{\partial x^2}$$ with boundary conditions $$\begin{cases} c(x,0) = C^\infty\\ c(0,t)=0\\ c(\infty,t)=C^\infty \end{cases}$$ is $$c(x,t) = C^\infty \operatorname{erfc}\Big(\frac{x}{2\sqrt{Dt}}\Big)$$ and then the flux is, $$J_{(x=0)} = D\frac{\partial c(x,t)}{\partial t} = \frac{C^\infty\sqrt{D}}{\sqrt{\pi t}}$$ what is the limit (maximum) of $J$?

When $t \to 0$ (at short times) then $J \to \infty$. It doesn't make sense.

doraemonpaul
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Kama
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    You are omitting lots of details. Not only does this make it difficult for us, the readers, to help you, it's also somewhat likely that the solution to your problem is precisely located in the details you've glossed over. (I'm specifically imagining the problem is that you did not find the full solution to your differential equation, and the solutions you missed include the specific solution you were actually looking for) –  Aug 24 '17 at 20:21
  • @Hurkyl I can include all the details, but they are not important because it is the simplest form of the solution to the diffusion equations. In any case, I add the boundary conditions for your reference. – Kama Aug 24 '17 at 20:27
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    Quick sanity check: wolfram alpha computing $\partial c(x,t)/\partial t$ gives a result that vanishes when you plug in $x=0$. Did you just compute the derivative wrong, or did I misunderstand the question or is there something else going on? –  Aug 24 '17 at 20:40

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In fact the solution of $\dfrac{\partial c}{\partial t}=D\dfrac{\partial^2c}{\partial x^2}$ with boundary conditions $\begin{cases}c(x,0)=C^\infty\\c(0,t)=0\\c(\infty,t)=C^\infty\end{cases}$ (e.g. according to Does solve PDE by combination of variables always cannot find the general solutions?) is $c(x,t)=C^\infty\text{erf}\left(\dfrac{x}{2\sqrt{Dt}}\right)$ , not $c(x,t)=C^\infty\text{erfc}\left(\dfrac{x}{2\sqrt{Dt}}\right)$ .

Now $D\dfrac{\partial c(x,t)}{\partial t}=-\dfrac{C^\infty\sqrt Dx}{2\sqrt{\pi t^3}}e^{-\frac{x^2}{4Dt}}$

$\left.D\dfrac{\partial c(x,t)}{\partial t}\right|_{x=0}=0$

doraemonpaul
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