An Optimized Two-Step Hybrid Block Method with One Optimized and Three Free Parameter Hybrid Points for the Numerical Solution of Volterra Integral Equations of the Second Kind

An optimized two-step hybrid block approach with four hybrid points—consisting of one optimized hybrid point and three active free parameters for the numerical solution of second-kind Volterra integral equations is presented in this study. The suggested approach is developed by combining an appropriate power series with exponential fitting as the basis function to construct an effective numerical discrete scheme of the Volterra type that can approximate the solution of the Volterra integral equation of the second kind. While the unknown coefficients and the optimized hybrid point are found by applying suitable consistency requirements and equating the principal term of the local truncation error to zero, the remaining three off-grid points are integrated into the system as flexible free parameters. The continuous formulation of the approach is obtained using a collocation and interpolation procedure. The accuracy and computational efficiency of the final scheme are significantly enhanced by isolating and optimizing the key target off-grid point. Through rigorous Taylor series expansion of the three-step framework, the optimized value of this specific parameter is determined to be 47. The theoretical 400properties of the method, including consistency, convergence, zero-stability, gion of absolute stability, are established. Numerical experiments involving several test problems are conducted to assess the performance of the proposed method in comparison with existing numerical methods. The computed results, absolute errors, and error norms demonstrate that the proposed optimized three-step hybrid block method provides improved accuracy and competitive computational efficiency. The method is therefore shown to be a reliable and effective computational technique for solving Volterra integral equations of the second kind, particularly for problems requiring high accuracy and enhanced stability.