Generalization of the Jarzynski equation for a system with variable temperature

Recently the Jarzynski equation have been under many investigations. In this equation a relation between the average exponential of the work done on a system under non-equilibrium condition and equilibrium free energy differences is established. In different papers the equation has already been extended to different conditions. In this paper we will extend it to a situation that a system is driven away from equilibrium by work performed on it but with a non-constant temperature. Then this generalized equation will be applied to one-dimensional classical and quantum harmonic oscillators.