This might as well be a mathematics question since I am only looking for a way to solve a basic model of the neutrondistribution in an infinite plane subcritical reactor.
So the model for the neutrondistribution looks like this:
$\frac{d^{2}\phi}{{dx}^{2}}-\gamma^{2}\phi+\frac{Q}{D}\delta(x)=0$
The source of the neutrons is located at $x=0$, the net flow of neutrons is $Q$ per second and per unit area (perpendicular to the $x$-axis).
Now, we seperate this into two regions, namely $x<0$ and $x>0$. For the region $x>0$ we obtain the second order differential equation $\frac{d^{2}\phi}{{dx}^{2}}-\gamma^{2}\phi = 0$ which can be solved directly, yielding $\phi(x) =A\exp{(\gamma x)}+C\exp{(-\gamma x)}$. Naturally as $x\rightarrow \infty$ then $\phi \rightarrow 0$, so $A=0$. We can yield a similar result for the other result, namely $\phi(x)=D\exp{(\gamma x)}$, $x<0$. At the boundary we have that these two functions (let's name them $\phi_1$ and $\phi_2$ for $x<0$ and $x>0$, respectively), have the same value: $\phi_1(0)=\phi_2(0)$. This yields $D=C$.
Now, the last boundary condition is tricky. My teacher tells me that that the last final BC that sets the value of $D$ comes from integrating the first equation:
$\lim_{\epsilon \to 0}\int_{-\epsilon}^{\epsilon}[\frac{d^{2}\phi}{{dx}^{2}}-\gamma^{2}\phi+\frac{Q}{D}\delta(x)]$
I am not sure how this gives me anything, since the integral can be seperated into the different regions where we know that the differential equation equals zero. I got a (reasonable) result by using Fick's law instead, with
$\lim_{x \to 0}(-D\frac{d\phi}{dx})=Q/2$
How do I solve this problem going the way my teacher hinted about? This is supposedly the more mathematical way.
Best regards SimpleP.
Okay, so I got an answer (finally) to this question. I am going to write the solution below.
$\lim_{\epsilon\rightarrow 0}\int_{-\epsilon}^{\epsilon}[\frac{d^2\phi}{dx^2}-\gamma^2\phi+\frac{Q}{D}\delta(x)]dx=\lim_{\epsilon\rightarrow 0}(\int_{-\epsilon}^{\epsilon}\frac{d^2\phi}{dx^2}dx-\int_{-\epsilon}^{\epsilon}\gamma^2\phi dx+\int_{-\epsilon}^{\epsilon}\frac{Q}{D}\delta(x)dx)$.
Now the first trem integrates to $\lim_{\epsilon\rightarrow0}[\phi'(\epsilon)-\phi'(-\epsilon)]=\phi'(0+)-\phi'(0-)$
The second term becomes $\lim_{\epsilon\rightarrow0}(-\int_{-\epsilon}^{\epsilon}\gamma^2\phi dx)=\{Continuous\:and\:finite\:at\:boundary\}=0$
The third term just becomes, per definition $\lim_{\epsilon\rightarrow0}\int_{-\epsilon}^{\epsilon}\frac{Q}{D}\delta(x)dx=\frac{Q}{D}$
You put this back into the equation and you obtain Fick's law through purely mathematical means. Best regards.