Advances in Computational Mathematics

, Volume 45, Issue 3, pp 1631–1655 | Cite as

Optimal convergence orders of fully geometric mesh one-leg methods for neutral differential equations with vanishing variable delay

  • Wansheng WangEmail author


The purpose of this paper is to obtain the error bounds of fully geometric mesh one-leg 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 \(h_{\max }\) 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.


Neutral functional differential equations Vanishing delay Fully geometric mesh one-leg methods Convergence orders Error estimates 

Mathematics Subject Classification (2010)

65L03 65L06 65L20 


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The authors would like to thank the anonymous referees for the valuable comments that lead to great improvements in the presentation of this paper; especially thank the referees for bringing us to interest in NFDEs with a vanishing delay and NFDEs with state-dependent delay.

Funding information

This work was supported by the Natural Science Foundation of China (Grant No. 11771060, 11371074).


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Authors and Affiliations

  1. 1.Department of MathematicsShanghai Normal UniversityShanghaiChina

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