  Comptonization is a very important phenomenon in astrophysics. Kompaneets
equation describes up-Comptonization process in nonrelativistic astrophysics,
while it fails for down-Comptonization Scattering which is the most important
radiative transfer process in hard X-rays and $\gamma$-rays astronomy. In this
study, we explore both up-Comptonization and down-Comptonization processes. A
new relativistic corrections is introduced to the Kompaneets equation which is
valid in both nonrelativistic energy regime with the photon energy $h\nu <<
m_ec^2$ and the electron temperature $KT_e << m_ec^2$. Numerical comparisons
are presented which show excellent agreement between the Kompaneets equation
and the new equation.

High Energy Astrophysical Phenomena