# Nonlinear Diffusion Equation with Reaction Terms: Analytical and Numerical Results

E.K. Lenzi, M.A. Ribeiro, M.E.K. Fuziki, M.K. Lenzi, H.V. Ribeiro, Applied Mathematics and Computation 330, 254--265 (2018)

### Abstract

We investigate a process obtained from a combination of nonlinear diffusion equations with reaction terms connected to a reversible process, i.e., 1 -> 2, of two species. This feature implies that the species 1 reacts producing the species 2, and vice-versa. A particular case emerging from this scenario is represented by 1 -> 2 (or 2 ->1), characterizing an irreversible process where one species produces the other. The results show that in the asymptotic limit of small and long times the behavior of the species is essentially governed by the diffusive term terms. For intermediate times, the behavior of the system depends on the reaction terms, particularly on the rates related to the reaction terms. In the presence of external forces, significant changes occur in the asymptotic limits. For these cases, we relate the solutions with the q-exponential function of the Tsallis statistic to highlight the compact or long-tailed behavior of the solutions and to establish a connection with the Tsallis thermo-statistic. We also extend the results to the spatial fractional differential operator by considering long-tailed distributions for the probability density function.