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A high accurate numerical framework for the solution of the vanishing-delay Volterra integro-differential equations via Legendre pseudo-spectral element approach

Author

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  • Yang, Yin
  • Yao, Pai
  • Tohidi, Emran

Abstract

In this study, we present a robust numerical framework for solving vanishing-delay Volterra integro-differential equations (VDVIDEs) using the Legendre pseudo-spectral element approach. Our first motivation stems from the limitations of traditional methods such as Runge–Kutta schemes when applied to linear vanishing-delay models. Building upon the robustness of pseudo-spectral element approaches, we extend the application to VDVIDEs and introduce a multi-step Legendre pseudo-spectral Galerkin method (MSLPSGM) to overcome issues related to long computational intervals and high oscillatory solutions. Our suggested approach provides upper bounds for estimating both the solution and its derivative, demonstrating a spectral rate of convergence as the length of sub-intervals decreases and the degree of approximations increases. Extensive numerical examples showcase the robustness and precision of the MSLPSGM in handling VDVIDEs with steep gradients and large computational intervals. The framework’s ability to maintain accuracy across various challenging scenarios underscores its potential as a powerful tool for solving complex integro-differential equations.

Suggested Citation

  • Yang, Yin & Yao, Pai & Tohidi, Emran, 2026. "A high accurate numerical framework for the solution of the vanishing-delay Volterra integro-differential equations via Legendre pseudo-spectral element approach," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 240(C), pages 403-422.
    • Handle: RePEc:eee:matcom:v:240:y:2026:i:c:p:403-422
    DOI: 10.1016/j.matcom.2025.07.030
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    References listed on IDEAS

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    1. Xiao-yong, Zhang, 2020. "A new strategy for the numerical solution of nonlinear Volterra integral equations with vanishing delays," Applied Mathematics and Computation, Elsevier, vol. 365(C).
    2. Yang, Yin & Yao, Pai & Tohidi, Emran, 2025. "Convergence analysis of an efficient multistep pseudo-spectral continuous Galerkin approach for solving Volterra integro-differential equations," Applied Mathematics and Computation, Elsevier, vol. 494(C).
    3. Yao, Guoqing & Tao, DongYa & Zhang, Chao, 2022. "A hybrid spectral method for the nonlinear Volterra integral equations with weakly singular kernel and vanishing delays," Applied Mathematics and Computation, Elsevier, vol. 417(C).
    4. Xiao-yong, Zhang, 2016. "A multistep Legendre pseudo-spectral method for Volterra integral equations," Applied Mathematics and Computation, Elsevier, vol. 274(C), pages 480-494.
    5. Guo, Yuling & Xu, Xiaoyu & Wang, Zicheng & Wang, Zhongqing, 2024. "An hp-version Legendre collocation method for the third-kind VIEs with nonlinear vanishing delay," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 223(C), pages 338-350.
    6. Bellen, Alfredo & Zennaro, Marino, 2003. "Numerical Methods for Delay Differential Equations," OUP Catalogue, Oxford University Press, number 9780198506546.
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