Computational method based on Bernstein operational matrices for nonlinear Volterra–Fredholm–Hammerstein integral equations

In this paper, we present a method to solve nonlinear Volterra–Fredholm–Hammerstein integral equations in terms of Bernstein polynomials. Properties of these polynomials and operational matrix of integration together with the product operational matrix are first presented. These properties are then utilized to transform the integral equation to a matrix equation which corresponds to a system of nonlinear algebraic equations with unknown Bernstein coefficients. The method is computationally very simple and attractive and numerical examples illustrate the efficiency and accuracy of the method.

► The Volterra–Fredholm–Hammerstein integral equation has been solved by Bernstein polynomials operational matrices. ► The dual matrix, operational matrix of integration and coefficient matrix of Bernstein polynomials are presented. ► The Volterra–Fredholm–Hammerstein integral equation converted to nonlinear algebraic equations. ► The method is computationally very simple and attractive. ► The numerical examples and the compared results by other methods results show the efficiently of proposed method.