Spectral Bernoulli-Gauss Collocation Scheme for Pantograph-Type Volterra Integro-Differential Equations

In this work, we employ a spectral collocation approach to numerically solve pantograph-type Volterra integro-differential equations subject to given initial conditions. The scheme combines Bernoulli polynomials with Gauss quadrature for numerical integration. By leveraging this Bernoulli-Gauss framework, the original integro-differential problem is transformed into a solvable system of algebraic equations. Accurate approximations are achieved using only a modest number of collocation points. The convergence behavior of the method is illustrated through graphical analysis, revealing an exponential rate of convergence. To validate the proposed technique, we present several test problems whose numerical solutions are compared against exact results and those reported by alternative methods. These comparisons are summarized in tables and figures to highlight the accuracy and efficiency of the approach.