Comparison between variational iteration method and Gegenbauer–Galerkin method for solving two dimensional nonlinear Volterra integral equations of the second kind

This paper intends to introduce two numerical techniques—the variational iteration method and the Gegenbauer–Galerkin method—for obtaining solutions to two dimensional nonlinear Volterra integral equations of the second kind. The first method, the variational iteration method, generates a convergent sequence of functions by approximating the exact solution with many iterations. The second method, the Gegenbauer–Galerkin method, transforms the Volterra integral equation into a system of nonlinear algebraic equations. This paper also discusses the existence and uniqueness of solutions. To demonstrate the effectiveness of these methods, several numerical examples are provided. Numerical results show that the variational iteration method consistently provides more accurate and efficient results than the Gegenbauer–Galerkin method.