Stability Analysis of $\Theta$-Methods for Nonlinear Neutral Functional Differential Equations

Abstract

In this paper the numerical solutions of nonlinear neutral differential equations with constant delay and with proportional delay, two typical examples of neutral functional differential equations, are investigated. The focus is on the stability of numerical solutions produced by $\theta$-methods. It is proved that if $\frac{1}{2}\leq\theta\leq 1$ the numerical solution to neutral differential equations with constant delay is globally stable and asymptotically stable under some conditions which are sufficient for stability and asymptotic stability of the exact solution. For neutral differential equations with proportional delay, two discretization approaches, the approach to transforming it into neutral equations with constant delay and the approach to directly discretizing it based on full-geometric mesh, are discussed. The integration on the first mesh interval $[0, T_0]$, which needs to be considered for both approaches, is also investigated. By taking into account the order of the methods, we construct two algorithms for the integration on $[0, T_0]$. It is also proved that under some assumptions the numerical solutions to the neutral equation with proportional delay produced by the two approaches are globally stable and asymptotically stable if $\frac{1}{2} < \theta\leq 1$.