A Comparative Study of Bernoulli Collocation and Hermite-Galerkin Methods for Solving Two-Dimensional Nonlinear Volterra Integral Equations of the Second Kind (European Journal of Pure and Applied Mathematics, 2025)
This article is devoted to the presentation of two numerical methods which give the solution of a two-dimensional nonlinear Volterra integral equation of the second kind. The first method, Bernoulli collocation, depend on approximating the unknown function using Bernoulli polynomials, while applying the collocation technique at shifted Chebyshev points over the interval[0,1]. The second method, Hermite-Galerkin method, relies on constructing an operational matrices and applying the Galerkin projection, which we have a system of nonlinear algebraic equations from Volterra integral equation. Discussion on the existence and uniqueness of the solution is provided. Finally, the effect of that two numerical methods is described. To illustrate the previously described methods, several numerical examples are provided. Numerical results show that the Bernoulli collocation method consistently provides more accurate and efficient results than the Hermite–Galerkin method for the same number of collocation points. Comparisons with previously published approaches further demonstrate the superiority of the proposed methods in terms of convergence and stability.
Keywords: Non-linear two-dimensional Volterra integral equations, Bernouilli polynomials, shifted Chebyshev points, Hermite polynomials and Galerkin method