  The solutions of relativistic viscous hydrodynamics for longitudinal
expanding fireballs is investigated with the Navier-Stokes theory and
Israel-Stewart theory. The energy and Euler conservation equations for the
viscous fluid are derived in Rindler coordinates with the longitudinal
expansion effect is small. Under the perturbation assumption, an analytical
perturbation solution for the Navier-Stokes approximation and numerical
solutions for the Israel-Stewart approximation are presented. The temperature
evolution with both shear viscous effect and longitudinal acceleration effect
in the longitudinal expanding framework are presented and specifically
temperature profile shows symmetry Gaussian shape in the Rindler coordinates.
In addition, in the presence of the longitudinal acceleration expanding effect,
the results of the Israel-Stewart approximation are compared to the results
from Bjorken and Navier-Stokes approximation, and it gives a good description
than the Navier-Stokes theories results at the early stages of evolution.
