An Application of the Distributed-Order Time- and Space-Fractional Diffusion-Wave Equation for Studying Anomalous Transport in Comb Structures

Abstract

A comb structure consists of a one-dimensional backbone with lateral branches. These structures have widespread application in medicine and biology. Such a structure promotes an anomalous diffusion process along the backbone (x-direction), along with classical diffusion along the branches (y-direction). In this work, we propose a distributed-order time- and space-fractional diffusion-wave equation to model a comb structure in the more general setting. The distributed-order time- and space-fractional diffusion-wave equation is firstly formulated to study the anomalous diffusion in the comb model subject to an irregular convex domain with the motivation that the time-fractional derivative considers the memory characteristic and the space one with the variable diffusion coefficient possesses the nonlocal characteristic. The finite element method is applied to obtain the numerical solution. The stability and convergence of the numerical discretization scheme are derived and analyzed. Two numerical examples of relevance to the comb model are given to verify the correctness of the numerical method. Moreover, the influence of the involved parameters on the three-dimensional and axial projection drawing particle distribution subject to an elliptical domain are analyzed, and the physical meanings are interpreted in detail.