On the Self-Similar, Wright-Function Exact Solution for Early-Time, Anomalous Diffusion in Random Networks: Comparison with Numerical Results

Abstract

Recently, Zhang and Padrino in (Int J Multiph Flow 92:70–81, 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 the finite volume method 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 $$x t^{-1/4}$$ , describing sub-diffusion, instead of $$x t^{-1/2}$$ as in ordinary diffusion. They argue that the sub-diffusive trend is the result of the time required 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 given in the literature for the same boundary and initial conditions in terms of the Wright function (Gorenflo et al. in J Comput Appl Math 118(1):175–191, 2000). This solution, in turn, is written in terms of generalized hypergeometric functions, readily available in calculus software packages. Comparing predictions from the exact solution with Zhang and Padrino’s numerical results leads to excellent agreement, serving as validation of their numerical approach.