The slowing down of neutrons in the moderator of a nuclear reactor can be described by the slowing down density \(q\) (neutrons \(\mathrm{m}^{-3} \mathrm{~s}^{-1}\) ) in accordance with the so-called Fermi age equation, \(\frac{\partial q}{\partial \tau}=\frac{\partial^{2} q}{\partial x^{2}}\), where \(\tau\) is the "age" of the neutron (in this theory, the unit of \(\tau\) is \(\mathrm{m}^{2}\) ). This equation can be solved assuming a plane source of neutrons \(S\) (neutrons \(\mathrm{m}^{-2} \mathrm{~s}^{-1}\) ) in the \(y z\) plane at the origin of the coordinate system, where \(x\) is the distance from the plane source. The boundary conditions for this equation consider symmetry at \(x=0\) and require that: (i) \(q\) remains finite such that \(\left.\frac{\partial q}{\partial x}ight|_{x=0}=0, \tau>0\), and (ii) \(q\) is finite as \(x ightarrow \infty, \tau>0\).

(a) Using a Fourier cosine transform on defining \(Q(w, \tau)=F_{c}\{q(x, \tau)\}\), show that this problem yields the transformed solution: \(Q(w, \tau)=A_{1} e^{-w^{2} \tau}\), where \(A_{1}\) is an arbitrary constant.

(b) Taking the inverse Fourier cosine transform, show that the solution for \(q(x, \tau)\) is \(q(x, \tau)=\frac{A}{\sqrt{\tau}} e^{-x^{2} /(4 \tau)}\), where \(A\) is a constant.

(c) Using a source condition that follows from the conservation of mass \(S=\) \(\int_{-\infty}^{\infty} q(x, \tau) d x\), evaluate the constant \(A\). This source condition replaces the initial condition for \(\tau=0\).