Extensions and solutions for nonlinear diffusion equations and random walks

We investigate a connection between random walks and nonlinear diffusion equations within the framework proposed by Einstein to explain the Brownian motion. We show here how to properly modify that framework in order to handle different physical scenarios. We obtain solutions for nonlinear diffusion equations that emerge from the random walk approach and analyse possible connections with a generalized thermostatistics formalism. Finally, we conclude that fractal and fractional derivatives may emerge in the context of nonlinear diffusion equations, depending on the choice of distribution functions related to the spreading of systems.