Fractional and nonlinear diffusion equation: additional results

Abstract

We investigate the solutions of a generalized diffusion equation which extends some known equations such as the fractional diffusion equation and the porous medium equation. We start our study by considering the linear case and the nonlinear case afterward. The linear case is analyzed taking fractional time and spatial derivatives into account. In this context, we also discuss the modifications that emerge by considering a diffusion coefficient given by D(x)∝|x| −θ . For the nonlinear case accomplishing the fractional time derivative, we discuss scaling behavior of the time and the asymptotic for the solution of the nonlinear fractional diffusion equation. In this case, the connection between the asymptotic solution found here and the nonextensive Tsallis statistics is performed.