© International Astronomical Union 2016.Stellar convection is customarily described by the mixing-length theory, which makes use of the mixing-length scale to express the convective flux, velocity, and temperature gradients of the convective elements and stellar medium. The mixing-length scale is taken to be proportional to the local pressure scale height, and the proportionality factor (the mixing-length parameter) must be determined by comparing the stellar models to some calibrator, usually the Sun. No strong arguments exist to suggest that the mixing-length parameter is the same in all stars and all evolutionary phases. Because of this, all stellar models in the literature are hampered by this basic uncertainty. In a recent paper (Pasetto et al. 2014) we presented a new theory that does not require the mixing length parameter. Our self-consistent analytical formulation of stellar convection determines all the properties of stellar convection as a function of the physical behavior of the convective elements themselves and the surrounding medium. The new theory of stellar convection is formulated starting from a conventional solution of the Navier-Stokes/Euler equations, i.e. the Bernoulli equation for a perfect fluid, but expressed in a non-inertial reference frame co-moving with the convective elements. In our formalism, the motion of stellar convective cells inside convective-unstable layers is fully determined by a new system of equations for convection in a non-local and time-dependent formalism. We obtained an analytical, non-local, time-dependent solution for the convective energy transport that does not depend on any free parameter. The predictions of the new theory are compared with those from the standard mixing-length paradigm with positive results for atmosphere models of the Sun and all the stars in the Hertzsprung-Russell diagram.