Stretched Gaussian Asymptotic Behavior for Fractional Giona–Roman Equation on Biased Heterogeneous Fractal Structure in External Force Fields

We introduce a biased heterogeneous fractional Giona-Roman equation (BHFGRE) on biased heterogeneous fractal structure media describing systems involving external force fields. The BHFGRE is shown to obey generalized Einstein relation, and its stationary solution is the generalized Boltzmann distribution. It is proved that the asymptotic shape of its solution is a stretched Gaussian and that its solution can be expressed in the form of a function of a dimensionless similarity variable for the case of constant potentials and generic potentials.