On the self-similar, early-time, anomalous diffusion in random networks - Approach by fractional calculus

In a recent work, Zhang and Padrino (2017) derived an equation for diffusion in random networks consisting of junction pockets and connecting channels by applying the ensemble average method to the mass conservation principle. The resulting integro-differential equation was solved numerically using finite volumes for the test case of one-dimensional diffusion in the half-line. For early time, they found that the numerical predictions of pocket mass density depend on the similarity variable xt −1/4, representing sub-diffusion. They argue that the sub-diffusive behavior is a consequence of the time needed to establish a linear concentration profile inside a channel. By theoretical analysis of the diffusion equation for small time, they confirmed this finding. Nevertheless, they did not present an exact solution for the small time limit to compare with. Here, starting with their small-time leading order diffusion equation in (x,t) space, we use elements of fractional calculus to cast it into a form for which an analytical solution has been obtained in the literature for the same boundary and initial conditions in terms of the Fox H-function (Schneider and Wyss, 1989). For ease of computation, we express the solution in terms of the Meijer G-function. We compare the exact solution with Zhang and Padrino's numerical predictions, resulting in excellent agreement, thereby validating their numerical approach.