Exact solution and invariant for fractional Cattaneo anomalous diffusion of cells in two-dimensional comb framework

This paper presents an investigation on anomalous diffusion of cells in a two dimensional comb framework with effects of fractional Cattaneo flux. Formulated governing equation is an evolution equation with the coexisting characteristics of parabolic (diffusion) and hyperbolic (wave) for α in (0, 1). Exact solution is obtained by the special fractional integral transformations and a novel invariant is established, i.e., (the mean square displacement multiplied by the total number of cells along the x axis=0.5). Moreover, the characteristics of cells distribution, the total number and the mean square displacement of cells along the axis with different involved parameters, especially with the fractional parameter evolution, are shown graphically and analyzed in detail, which have never been reported before. For the cells distribution versus x, it turns from parabolic and hyperbolic with the decrease of t or the increase of α or ξ. It is monotone decreasing for the cells distribution versus α with different x, t and ξ. For the distribution versus t with different α and ξ or versus α with different t, it is monotone decreasing for the distribution of total number while monotone increasing for the distribution of mean square displacement. It is remarkable that the anomalous subdiffusion happens along the axis for arbitrary parameters which is different from the classical Cattaneo diffusion