An Efficient Based Numerical Method for Accurate Integration of Volterra IntegroDifferential Equations

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This study presents a highly efficient New Numerical Method (NNM) for the accurate integration of second-kind Volterra Integro-Differential Equations (VIDEs). The New Numerical Method (NNM) is derived using a block algorithm based on a generalized linear multistep framework, enabling the construction of high-order schemes with multiple grid points. Analytical properties of the NNM, including order, error constants, consistency, zero-stability, convergence, and the region of absolute stability, are rigorously established to ensure reliability and robustness. Numerical simulations on representative VIDEs demonstrate the method’s superior accuracy and stability compared to existing techniques such as Adams–Bashforth–Moulton predictor-corrector methods, general linear methods, trigonometrically fitted schemes, and Haar wavelet methods. The results indicate that the NNM achieves exact or near-exact solutions across various step sizes, highlighting its potential as a powerful computational tool for solving complex integro-differential equations in science and engineering applications.

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