Computational method for Volterra integro-differential equations of the second kind

Abstract

This study presents a computational method (CM) for solving second-kind Volterra integro-differential equations (VIDEs), which are widely used in engineering, physics, biology, and control systems. The method combines interpolation and collocation techniques to produce a continuous one-step computational scheme that accurately approximates the solution while maintaining stability and convergence properties. Consistency, zero-stability, order of accuracy, convergence, and the region of absolute stability are examined to confirm the suitability of the method for both linear and nonlinear problems. Numerical experiments with selected VIDEs compare the CM with the fifth--order Adams–Bashforth–Moulton predictor--corrector method, the two-point three-step block method, the trigonometrically fitted block method, the Haar wavelet method, and trapezoidal schemes. Across the tested examples, the CM provides accurate, stable, and efficient approximations, making it a useful tool for solving complex integro-differential systems in practical applications.