Neutron flux in multiplying medium: numerical and analytical solution
The geometry is a slab of length ‘a’, and the boundary condition are:
\Phi(\pm a/2) = 0
Here the balance equation:
L^2 {\partial^{2}\Phi(x,t) \over \partial x^{2}} + (k-1)\Phi(x,t) + S\delta(x)= l{\partial \Phi(x,t) \over \partial t}
The analytical soluton result to be:
\Phi = \sum_i A_0 cos(B_ix)e^{{(k_i-1)\over l_i} t}
where (B_i = \frac{\pi}{a} (2i+1), k_i = {k\over 1+L^2(B_i)^2}), l_i = {l\over 1+L^2(B_i)^2}. l depends on the material and k is equal to 1 for a critical situation (neutrons born = neutrons absorbed).
A_0 depends on the initial condition, for a delta source we have A_0 = {2\Phi(0)\over a}.
