Numerical Solution of the Nonlocal Diffusion Equation on the Real Line

Abstract

Numerical computation of a nonlocal diffusion equation on the real axis is considered in this paper. We first apply an extensively studied quadrature scheme to obtain a discrete nonlocal diffusion system on an unbounded domain. Then we derive an alternative formulation of the discrete problem based on the spectral analysis of the $z$-transform. This new formulation can be seen as a system defined on a bounded domain with an artificial boundary condition, and it allows us to reformulate the original infinite domain problem into an equivalent bounded domain problem. To numerically implement the exact artificial boundary condition, we apply the trapezoidal quadrature rule to approximate the contour integral induced by the inverse $z$-transform. Numerical examples are presented to demonstrate the effectiveness of our approach.