Analysis of IVPs and BVPs on Semi-Infinite Domains via Collocation Methods

Abstract

We study the numerical solutions to semi-infinite-domain two-point boundary value problems and initial value problems. A smooth, strictly monotonic transformation is used to map the semi-infinite domain $x\in [0,\infty )$ onto a half-open interval $t\in [-1,1)$. The resulting finite-domain two-point boundary value problem is transcribed to a system of algebraic equations using Chebyshev-Gauss (CG) collocation, while the resulting initial value problem over a finite domain is transcribed to a system of algebraic equations using Chebyshev-Gauss-Radau (CGR) collocation. In numerical experiments, the tuning of the map $\varphi :[-1,+1)\to [0,+\infty )$ and its effects on the quality of the discrete approximation are analyzed.