2.1. The Continuous Neutron Transport Equation

The combination of terms describing the creation and destruction of neutrons results in the neutron transport equation. In effect, the neutron transport equation (NTE) models the behavior of advecting neutrons as they interact with the surrounding medium [LM93]. The NTE combines the time-dependent, collision, and streaming terms with the scattering, fission, and external sources, with the \(d\bx\), \(d\bsOmega\), and \(dE\) terms being divided out. Rearranging, this results in the form [LM93]

(1)\[\begin{split}\begin{aligned} \left( \frac{1}{v(E)}\frac{\partial}{\partial t} + \bsOmega \cdot \nabla + \sig{t}(\bx, E) \right) & \Psi(\bx,\bsOmega,E,t) = \\ & \hspace{-50pt} \int_{0}^{\infty}dE' \; \sig{s} (\bx, E' \rightarrow E) \Phi(\bx, E', t) + Q(\bx,\bsOmega, E, t) \\ & \hspace{-50pt} + \chi(\bx, E) \int_{0}^{\infty} dE' \; \nu(x, E') \sig{f}(\bx, E') \Phi (\bx, E', t), \end{aligned}\end{split}\]

with the scalar flux represented as \(\Phi (\bx, E, t)\) and obtained by integrating the angular flux over angle. The angular flux depends on a spatial coordinate \(\bx \in D \subset \bbR^3\), angular coordinate \(\bsOmega \in \bbS^2\), energy \(E > 0\), and time \(t > 0\).

The initial condition for Equation (1) is

\[\Psi(\bx, \bsOmega, E, t=0) = f( \bx, \bsOmega, E) \quad \text{for} \quad \bx \in D, \quad \bsOmega \in \bbS^2, \quad E > 0,\]

where \(f\) is given. The incoming boundary data is prescribed as

\[\Psi(\bx, \bsOmega, E, t) = b( \bx, \bsOmega, E, t) \quad \text{for} \quad \bx \in \partial D, \quad \bn(\bx) \cdot \bsOmega < 0, \quad E>0, \quad t >0,\]

where \(\bn (\bx)\) is the unit outward normal at \(\bx \in \partial D\), the boundary of \(D\).