  In this paper, the crucial phenomenon of the expansion of the universe has
been discussed. For this purpose, we study the vacuum solutions of Bianchi
types $I$ and $V$ spacetimes in the framework of $f(R)$ gravity. In particular,
we find two exact solutions in each case by using the variation law of Hubble
parameter. These solutions correspond to two models of the universe. The first
solution gives a singular model while the second solution provides a
non-singular model. The physical behavior of these models is discussed.
Moreover, the function of the Ricci scalar is evaluated for both the models in
each case.
