Abstract We analyze an anisotropic fractional diffusion equation that extends some known diffusion equations by considering a diffusion coefficient with spatial and time dependence, the presence of external forces and time fractional derivatives. We obtain new exact classes of solutions for a linear anisotropic fractional diffusion equation and investigate the time scaling behavior and an asymptotic solution for a nonlinear anisotropic fractional diffusion equation. We connect the asymptotic solution obtained with the distribution that emerges from the nonextensive statistics to the nonlinear case. We also verify different diffusive behavior, for instance, subdiffusion and superdiffusion, in each direction.