Modified One-Leg $\theta$-Methods for Linear Neutral Pantograph Equations with Multiple Delay Terms

Abstract

In this paper, by combining the modified one-leg $\theta$-methods with linear interpolation, a new class of one-leg $\theta$-methods for solving initial value problems of neutral pantograph equations with  multiple delay terms is presented. Our new approach, which is based on a geometric grid, exhibits better asymptotic stability compared to traditional $\theta$-methods. Under appropriate conditions, we prove, using the joint spectral radius method, that the proposed methods are asymptotically stable if and only if $0<\theta\le 1$. This indicates that the new methods overcome the limitation of the traditional $\theta$-methods, which require $\frac{1}{2}\le \theta\le1$ to ensure asymptotic stability. Furthermore, several numerical experiments are provided to validate our theoretical results.