The Green's function (GF) method is a powerful technique to describe non-equilibrium processes under inclusion of a broad class of boundary and initial conditions. Based on the Onsager theory of non-equilibrium the applicability of GFs is demonstrated to find the steady-state solution for the thermoelectric behaviour in one dimension. The spatial temperature profile is represented by the GF even if the Onsager transport coefficients are likewise coordinate dependent. The method can be directly used to study situations which are realized experimentally. The method can be extended to construct a systematic perturbation theory where the GF obeys a Dyson equation.