Study via Approximate Symmetries of the Emergence of Secondary Flux in the Bevilacqua-Gale\~ao Model
The Bevilacqua-Gale\~ao Model of Anomalous Diffusion introduces two fluxes: a primary flux that follows Fick's law of diffusion, representing the fraction of particles undergoing classical diffusion, and a secondary flux modeled by a fourth-order differential term, which accounts for retention phenomena. We investigate the ``analytic emergence'' of this secondary flux by treating the model as a (singular) perturbation of the heat equation, which describes the classical diffusion. Rather than applying traditional perturbation methods, or a straightforward Fourier transformation, we employ the powerful framework of enhanced modern group analysis to study this problem. Specifically, by utilizing approximate symmetries we derive an analytic expression for the emergent secondary diffusion as a perturbation of the classical diffusion process, the approximate fundamental solution and the approximate solution of the related Cauchy problem.