On accuracy of numerical solution to boundary value problems on infinite domains with slow decay

Department of Mathematics and Statistics, Thompson Rivers University, Kamloops, British Columbia, Canada.

^* Corresponding author: cranberryana@gmail.com

Received: 29 November 2018
Accepted: 28 January 2019

Abstract

A numerical approach is developed to solve differential equations on an infinite domain, when the solution is known to possess a slowly decaying tail. An unorthodox boundary condition relying on the existence of an asymptotic relation for |y| ≫ 1 is implemented, followed by an optimisation procedure, allowing to obtain an accurate solution over a truncated finite domain. The method is applied to −(−Δ)^γ/2u − u + u^p = 0 in ℝ, a non-linear integro-differential equation containing the fractional Laplacian, and is easily expanded to asymmetric boundary conditions or domains of a higher dimension.