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

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Abstract

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.

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Acknowledgments

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

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

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    Wansheng Wang

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  1. Wansheng Wang
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Correspondence to Wansheng Wang.

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Communicated by: Zydrunas Gimbutas

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Wang, W. Optimal convergence orders of fully geometric mesh one-leg methods for neutral differential equations with vanishing variable delay. Adv Comput Math 45, 1631–1655 (2019). https://doi.org/10.1007/s10444-019-09688-8

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