Cosmic acceleration and $f(R)$ theory: perturbed solution in a matter
  FLRW model
  In the present paper we consider $f(R)$ gravity theories in the metric
approach and we derive the equations of motion, focusing also on the boundary
conditions. In such a way we apply the general equations to a first order
perturbation expansion of the Lagrangian. We present a model able to fit
supernovae data without introducing dark energy.
