An investigation of space distributed-order models for simulating anomalous transport in a binary medium

Recent studies highlight that diffusion processes in highly heterogeneous, fractal-like media can exhibit anomalous transport phenomena, which motivates us to consider the use of generalised transport models based on fractional operators. In this work, we harness the properties of the distributed-order space fractional potential to provide a new perspective on dealing with boundary conditions for nonlocal operators on finite domains. Firstly, we consider a homogeneous space distributed-order model with Beta distribution weight. An a priori estimate based on the L2 norm is presented. Secondly, utilising finite Fourier and Laplace transform techniques, the analytical solution to the model is derived in terms of Kummer’s confluent hypergeometric function. Moreover, the finite volume method combined with Jacobi-Gauss quadrature is applied to derive the numerical solution, which demonstrates high accuracy even when the weight function shows near singularity. Finally, a one-dimensional two-layered problem involving the use of the fractional Laplacian operator and space distributed-order operator is developed and the correct form of boundary conditions to impose is analysed. Utilising the idea of ‘geometric reconstruction’, we introduce a ‘transition layer’ whereby the fractional operator index varies from fractional order to integer order across a fine layer at the boundary of the domain when transitioning from the complex internal structure to the external conditions exposed to the medium. An important observation is that for a fractional dominated case, the diffusion behaviour in the main layer is similar to fractional diffusion, while near the boundary the behaviour transitions to the case of classical diffusion.