Application of sigmoidal operational matrix of integration and collocation method to numerically solve Hammerstein-type functional integral equations

In this research, we investigate a functional Volterra-Hammerstein integral equation, providing conditions that ensure the existence and uniqueness of a solution in the space of square integrable functions. We obtain the operational matrix of integration of sigmoidal polynomials and employ it with the collocation method to approximate the solution. This method reduces the considered functional integral equation into a system of nonlinear algebraic equations, solvable through standard numerical methods such as Newton's method or Picard iteration method. A significant feature of this approach is that it avoids the need for integration to discretize the integral operator. We also provide an error estimation and include illustrative examples to demonstrate the method's efficiency and applicability.