Exact solutions in two-dimensional metric $f(R)$-gravity

In this paper we consider the two-dimensional metric $f(R)$-gravity model for the metric tensor depending on two variable: time and one spacelike coordinate. We obtain exact analytical vacuum solutions for different forms of function $f(R)$ which are solutions of cosmological type. These solutions are expressed in terms of arbitrary functions, which, under certain conditions, can be chosen as new variables.