Vieta-Lucas Polynomial Computational Tecnique for Volterra Integro-Differential Equations

Abstract

In this study, we introduce a computational technique for tackling Volterra Integro-Differential Equations (VIDEs) using shifted Vieta-Lucas polynomials as the foundational basis functions. The approach involves adopting an approximative solution strategy through the utilization of Vieta-Lucas polynomials. These polynomials are then integrated into the pertinent VIDEs. Subsequently, the resulting equation is subjected to collocation at evenly spaced intervals, generating a system of linear algebraic equations with unspecified Vieta-Lucas coefficients. To solve this system, we employ a matrix inversion method to deduce the unknown constants. Once these constants are determined, they are incorporated into the earlier assumed approximate solution, thus yielding the sought-after approximated solution. To validate the accuracy and efficiency of this technique, we conducted numerical experiments. The obtained results underscore the outstanding performance of our method in comparison to outcomes found in existing literature. The precision and effectiveness of the approach are further illustrated through the utilization of tables.