Exact solution of a radiative transfer problem by the Laplace transform and Wiener-Hopf technique (Q1318304)

The authors consider the following transfer equation \[ \mu {\partial I \over \partial t} (t,\mu)=I(t,\mu)-{1 \over 2} \int^ 1_{-1} p(\mu,\mu ') I(t,\mu ')d \mu ' \tag{1} \] where \(p(\mu,\mu ')=1+ {1 \over 8} (3 \mu^ 2-1)\) \((3(\mu ' )^ 2-1)\) and \(I\) has to satisfy the boundary conditions (2) \(I(0,\mu)=0\), \(\forall \mu \in [-1,0)\), \(I (t,\mu)e^{-t/ \mu} \to 0\) as \(t \to+\infty\), \(\forall \mu \in [-1,1] \backslash \{0\}\). Using Wiener-Hopf techniques related to the Laplace transform, the authors determine an explicit representation formula (depending upon an arbitrary constant \(F)\) for the solutions to problem (1)--(2).