Bernoulli operational matrix method for the numerical solution of nonlinear two-dimensional Volterra–Fredholm integral equations of Hammerstein type

Two-dimensional Volterra–Fredholm integral equations of Hammerstein type are studied. Using the Banach Fixed Point Theorem, the existence and uniqueness of a solution to these equations in the space $$L^\infty ([0,1]\times [0,1])$$ is proved. Then, the operational matrices of integration and product for two-variable Bernoulli polynomials are derived and utilized to reduce the solution of the considered problem to the solution of a system of nonlinear algebraic equations that can be solved by Newton’s method. The error analysis is given and some examples are provided to illustrate the efficiency and accuracy of the method.