Exact solutions for a diffusion equation with a nonlinear external force

This work is devoted to investigate the solutions of the one-dimensional diffusion equation by taking the nonlinear external force F(x,t;ρ)=−k(t)x+K/x+κx|x|α−1η[ρ(x,t)]F(x,t;ρ)=−k(t)x+K/x+κx|x|α−1[ρ(x,t)]η into account. Our investigation is first performed by considering the case α=0α=0 and η=1η=1, which results in a Burgers like equation with a spatial and time dependent external force. After, we consider the case α≠0α≠0 and η=α+1η=α+1 and show that the solution found may be expressed in terms of the q-exponential functions present in the Tsallis formalism. In addition, we also discuss the stationary solution for α and η arbitraries.