Asian Journal of Pure and Applied Mathematics

This study introduces a robust numerical method that combines shifted Vieta-Lucas polynomials, the variational iteration approach (VIA), and collocation method (CM) to efficiently solve Volterra integro-differential equations. The method transforms the given equation into a solvable algebraic system by leveraging the orthogonal properties of shifted Vieta-Lucas polynomials as trial functions. Convergence is rigorously established via Banach's fixed-point theorem, confirming that the iterative scheme yields a unique, convergent Cauchy sequence. Validation on Linear second and third order problems demonstrates superior accuracy and faster convergence compared to traditional Legendre collocation methods, with absolute errors significantly reduced even at low polynomial degrees. Graphical results further confirm strong agreement with exact solutions, positioning the proposed technique as a powerful and mathematically sound alternative for integro-differential equations. All computations were done using Maple 18.