The nonlinear fractional diffusion equation ∂tρ=r1−d∂rμ′{rd−1D(r,t;ρ)∂rμρν}−r1−d∂r{rd−1F(r,t)ρ}+α¯(t)ρ is studied by considering the diffusion coefficient D(r,t;ρ)=D(t)r−θργ and the external force F(r,t)=−k1(t)r+kαrα. In addition, a rich class of diffusive processes, including normal and anomalous ones, is obtained from the study present in this work.

We investigate the solutions of a generalized diffusion-like equation by considering a spatial and time fractional derivative and the presence of non-local terms, which can be related to reaction or adsorption–desorption processes. We use the Green function approach to obtain solutions and evaluate the spreading of the system to show a rich class of behaviors. We also connect the results obtained with the anomalous diffusion processes.

The area of networks is very interdisciplinary and exhibits many applications in several fields of science. Nevertheless, there are few studies focusing on geographically located $d$-dimensional networks. In this paper, we study the scaling properties of a wide class of $d$-dimensional geographically located networks which grow with preferential attachment involving Euclidean distances through $r_{ij}^{-{\alpha }_A}$ (${\alpha }_A\ge 0$). We have numerically analyzed the time evolution of the connectivity of sites, the average shortest path, the degree distribution entropy, and the average clustering coefficient for $d=1,2,3,4$ and typical values of ${\alpha }_A$. Remarkably enough, virtually all the curves can be made to collapse as functions of the scaled variable ${\alpha }_A/d$. These observations confirm the exist- ence of three regimes. The first one occurs in the interval ${\alpha }_A/d\in [0,1]$; it is non-Boltzmannian with very-long-range interactions in the sense that the degree distribution is a $q$ exponential with $q$ constant and above unity. The critical value ${\alpha }_A/d=1$ that emerges in many of these properties is replaced by ${\alpha }_A/d=1/2$ for the $\beta$ exponent which characterizes the time evolution of the connectivity of sites. The second regime is still non-Boltzmannian, now with moderately-long-range interactions, and reflects in an index $q$ monotonically decreasing with ${\alpha }_A/d$ increasing from its critical value to a characteristic value ${\alpha }_A/d\simeq 5$. Finally, the third regime is Boltzmannian-like (with $q\simeq 1$) and corresponds to short-range interactions.