Optimal convergence orders of fully geometric mesh one-leg methods for neutral differential equations with vanishing var
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Optimal convergence orders of fully geometric mesh one-leg methods for neutral differential equations with vanishing variable delay Wansheng Wang1 Received: 14 June 2018 / Accepted: 21 March 2019 / Published online: 10 April 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract The purpose of this paper is to obtain the error bounds of fully geometric mesh oneleg methods for solving the nonlinear neutral functional differential equation with a vanishing delay. For this purpose, we consider Gq -algebraically stable one-leg methods which include the midpoint rule as a special case. The error of the first-step integration implemented by the midpoint rule on [0, T0 ] is first estimated. The optimal convergence orders of the fully geometric mesh one-leg methods with respect to T0 and the mesh diameter hmax are then analyzed and provided for such equation. Numerical studies reported for several test cases confirm our theoretical results and illustrate the effectiveness of the proposed method. Keywords Neutral functional differential equations · Vanishing delay · Fully geometric mesh one-leg methods · Convergence orders · Error estimates Mathematics Subject Classification (2010) 65L03 · 65L06 · 65L20
1 Introduction In this paper, we focus on the error estimates of fully geometric mesh (FGM) one-leg methods (OLMs) for the numerical solution of the initial-value problems for neutral functional differential equation (NFDEs) with a vanishing variable delay, which is an equation of the form y (t) = f (t, y(t), y(η(t)), y (η(t))), t ∈ JT := [0, T ], (1.1) y(0) = φ, Communicated by: Zydrunas Gimbutas Wansheng Wang
[email protected] 1
Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China
1632
Wansheng Wang
where f : JT × Cd × Cd × Cd → ×Cd is a given mapping, 0 ≤ η(t) ≤ λt, t ∈ JT , with λ ∈ (0, 1), T > 0 is a real constant, and φ ∈ Cd is a vector. The so-called generalized pantograph equation (GPE), NFDEs with a proportional delay η(t) = λt, is a special case of (1.1). The equations of the form (1.1) have found applications in many areas of science (see, for example, [5, 18]). The stability and convergence of numerical methods for NFDEs have drawn much attention in the last decades. The linear stability of the trapezoidal rule for NFDEs with vanishing delays was studied in [8]. The variable step and variable order algorithms for nonlinear NFDEs were systematically investigated by Jackiewicz and his coworkers (see, e.g., [16, 17]). The expected pth-order convergence of one-leg methods on a uniform mesh has been shown for neutral differential equations with a variable delay in [26]. However, little attention was paid to the error estimates of numerical methods for neutral equation (1.1) on a geometric mesh. Since the use of geometric mesh can reduce the computational cost, permit the derivation of the classical optimal, or of quasi-optimal, orders of superconvergence of numerical methods for smooth problems at the mesh points, and remedy the order reduction for numer
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