Comparing cattaneo and fractional derivative models for heat transfer processes

We compare the model of heat transfer proposed by Cattaneo, Maxwell, and Vernotte with another one, formulated in terms of fractional differential equations, in one and two dimensions. These are only some of the numerous models that have been proposed in the literature over many decades to model heat transport and possibly heat waves, in place of the classical heat equation due to Fourier. These models are characterized by sound as well as by critical properties. In particular, we found that the Cattaneo model does not exhibit necessarily oscillations or negative values of the (absolute) temperature when the relaxation parameter, τ, drops below some value. On the other hand, the fractional derivative model may be affected by oscillations, depending on the specific initial profile. We also estimate the error made when the Cattaneo equation is adopted in place of the heat equation, and show that the approximation error is of order τ. Moreover, the solution of the Cattaneo equation converges uniformly to that of the heat equation as τ → 0+ in the full closed time interval [0; T] (for any given T > 0), while this does not occur for the time derivative, and the higher-order time derivatives blow up.