Modified One-Leg $θ$-Methods for Linear Neutral Pantograph Equations with Multiple Delay Terms (Advances in Applied Mathematics and Mechanics, 2025)
In this paper, by combining the modified one-leg $θ$-methods with linear interpolation, a new class of one-leg $θ$-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 $θ$-methods. Under appropriate conditions, we prove, using the joint spectral radius method, that the proposed methods are asymptotically stable if and only if $0 < θ ≤ 1$. This indicates that the new methods overcome the limitation of the traditional $θ$-methods, which require $\frac{1}{2} ≤ θ ≤ 1$ to ensure asymptotic stability. Furthermore, several numerical experiments are provided to validate our theoretical results.