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

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.