The Fractional Laplacian: An adaptive finite difference approach in one and two dimensions, with applications.

Abstract

The fractional Laplacian can be viewed as a generalisation of the ordinary Laplacian and likewise can be used to model systems in sectors such as engineering, hydrology and biology. Super diffusion, a consequence of Lévy flights, can be described using the fractional Laplacian, − (−∆)^α/2 , of order α ∈ (0, 2). The fractional Laplacian has numerous definitions and here we will use the Riesz potential definition in one and two dimensions (which is equal to the Riesz derivative in one dimension). Methods exist to approximate the fractional Laplacian on fixed uniform grids including the L1 and L2 methods derived from the Caputo fractional derivative definitions. In this thesis, the new L1(NU) and L1(NU)-2D methods for α ∈ (0, 1), and the L2(NU) and L2(NU)-2D methods for α ∈ (1, 2) will be presented to approximate the fractional Laplacian on non-uniform adaptive finite difference meshes in one and two dimensions. The L1(NU) and L1(NU)-2D methods for α ∈ (0, 1) display some unstable behaviours and therefore require a higher number of grid points, and less adaptive grids to work well. The L2(NU) and L2(NU)-2D methods for α ∈ (1, 2) are successfully applied to space-fractional heat, advection-diffusion and Fisher’s PDEs where the solutions and adaptive grids behave as expected.