New numerical algorithm for solving linear and nonlinear Volterra Integro-Differential equations of the Second Kind

Abstract

This study presents the development, analysis and numerical performance of a new numerical algorithm designed for solving Volterra integro-differential equations of the second kind. The method is derived using a power series approach with interpolation and collocation techniques, leading to a continuous and discrete block formulation suitable for solving linear and nonlinear Volterra integro-differential equations of the second kind. The fundamental properties of the algorithm such as order, error constant, consistency, zero-stability, convergence and absolute stability are rigorously examined and the method is shown to satisfy all necessary conditions for reliability and convergence. To assess its accuracy and efficiency, three numerical examples involving linear and nonlinear Volterra integro-differential equations with known exact solutions are solved. The results demonstrate that the developed algorithm consistently produces extremely small absolute errors, significantly outperforming several existing numerical methods. The algorithm proves to be highly accurate, stable and suitable for a wide range of volterra integro-differential problems.