The Boundary Element Method Applied to the Solution of the Anomalous Diffusion Problem

Abstract

A Boundary Element Method formulation is developed for the solution of the two-dimensional anomalous diffusion equation. Initially, the Riemann–Liouville Fractional derivative is applied on both sides of the partial differential equation (PDE), thus transferring the fractional derivatives to the Laplacian. The boundary integral equation is then obtained through a Weighted Residual formulation that employs the fundamental solution of the steady-state problem as the weighting function. The integral contained in the Riemann–Liouville formula is evaluated assuming that both the variable of interest and its normal derivative are constant in each time interval. Five examples are presented and discussed, in which the results from the proposed formulation are compared with the analytical solution, where available, otherwise with those furnished by a Finite Difference Method formulation. The analysis shows that the new method is capable of producing accurate results for a variety of problems, but small time steps are needed to capture the large temporal gradients that arise in the solution to problems governed by PDEs containing the fractional derivative ∂αu/∂tα with α < 0.5. The use of the steady-state fundamental solution presents no hindrance to the ability of the new formulation to provide accurate solutions to time-dependent problems, and the method is shown to outperform a finite difference scheme in providing highly accurate solutions, even for problems dominated by conditions within the material and remote from the domain boundary.