Subdiffusions on Circular Branching Structures

Abstract Anomalous diffusions on classical structures, like the comb structure where the branches are perpendicular to the straight backbone, have been studied by many researchers. However, in many practical problems, the anomalous diffusions are associated with a region surrounded by a boundary, and the research about such a kind of branching structures is still deficient. In this article, a novel model, namely, a circular backbone with radial branches, is presented to describe the anomalous diffusion on more complex regions, which has the potential of modeling many problems in medical science and nature, for example, angiogenic vasculatures induced by tumor. Three types of diffusion in circular branching structures are discussed, including two-way diffusion, inward diffusion, and outward diffusion. A significant advantage of these models is the solvability: The Fokker–Planck equations could be solved by variable separation, the Laplace transformation and its numerical inversion. The angular initial time mean squared displacement (MSD) is obtained, and the results show that angular MSD is proportional to t 1 2 as t  → 0, indicating that these types of diffusion are subdiffusion. Moreover, a numerical simulation of inward model is studied and the results are in good agreement with our analytical solution. Finally, radial MSD is calculated numerically, which shows that the radial diffusion is normal diffusion in short time, and the sink boundary conditions have a different behavior from the case with the reflective boundary conditions.