  In this study a spatio-temporal approach for the solution of the
time-dependent Boltzmann (transport) equation is derived. Finding the exact
solution using the Boltzmann equation for the general case is generally an open
problem and approximate methods are usually used. One of the most common
methods is the spherical harmonics method (the $P_N$ approximation), when the
exact transport equation is replaced with a closed set of equations for the
moments of the density, with some closure assumption. Unfortunately, the
classic $P_N$ closure yields poor results with low-order $N$ in highly
anisotropic problems. Specifically, the tails of the particle's positional
distribution as attained by the $P_N$ approximation, are inaccurate compared to
the true behavior. In this work we present a derivation of a linear closure
that even for low-order approximation yields a solution that is superior to the
classical $P_N$ approximation. This closure, is based on an asymptotic
derivation, both for space and time, of the exact Boltzmann equation in
infinite homogeneous media. We test this approximation with respect to the
one-dimensional benchmark of the full Green function in infinite media. The
convergence of the proposed approximation is also faster when compared to
(classic or modified) $P_N$ approximation.
