Fractional potential: A new perspective on the fractional Laplacian problem on bounded domains

Abstract

In this work, the fractional Laplacian operator is studied based on its spectral definition for a bounded, one-dimensional domain. We first note that both the analytical and numerical solutions of the fractional diffusion equation subject to Neumann boundary conditions do not exist due to the need to raise the zero eigenvalue to a negative fractional index. To overcome this issue, we use the concept of regularisation and modify the Neumann boundary conditions to Robin type by introducing regularisation parameters that ensure the smallest eigenvalue of both the Sturm–Liouville system and the matrix representation of the Laplacian operator is nonzero. We then observe that these regularisation parameters can have a significant impact on the solution behaviour of the fractional diffusion model. This finding motivates the use of what we call the ‘fractional potential’ to define a new fractional model that is far less sensitive to the regularisation parameter and makes imposing fractional Neumann boundary conditions also feasible. Next, we compare the results of this new fractional model with a fractional Riesz problem subject to Neumann boundary conditions on a bounded symmetric domain. An interesting finding is that the steady-state solution for the fractional Laplacian problem leads to a constant function, while the fractional Riesz operator results in a function exhibiting boundary singularities. Finally, we propose a three-layered fractional model based on the Laplacian operator to naturally impose the traditional boundary conditions by using the idea of ‘geometric reconstruction’. We derive semi-analytical and numerical solutions for this model. An important outcome is that by choosing an extremely small scale for the boundary transition layer on both sides of the domain, this three-layered model can be used to approximate the fractional Laplacian problem on a bounded domain. The proposed model captures the fractional diffusion behaviour in the inner layer well while describing the classical diffusion behaviour near the boundary. We show that this model is ideal for modelling anomalous diffusion in a binary medium exhibiting distinct heterogeneity.