Discretization of fractional boundary value problems using split operator local extension problems

Abstract

Partial differential equations which include nonlocal operators have recently become a major focus of mathematical and computational research. The efficient computational implementation of a nonlocal operator remains an important question. In this article, we introduce an operator splitting approach to the discretization of nonlocal boundary value problems. It has been shown that the fractional Laplacian in Rd is identical to equivalent local extension problem in Rd+1 with a nonlinear Neumann boundary condition. We present an operator splitting approach to the extended problem that enables a consistent stable implementation. Computational experiments are presented using finite differences and meshless methods.