New Iterative Method for Solving Nonlinear Volterra–Fredholm Integral Equations Using Bernoulli Polynomial

The work is aimed at proposing a novel iterative technique based on the combination of two powerful Taylor's and Bernoulli polynomials to solve nonlinear Volterra–Fredholm integral equations of the 2nd kind. The main contribution of this work is the development of a Bernoulli–Taylor iterative framework in which Bernoulli projection is combined with Taylor approximation within the nonlinear operator, leading to an explicit error estimate that simultaneously accounts for the Bernoulli approximation, Taylor truncation, and iterative errors. For this aim, an approximation of the unknown function by the Bernoulli polynomial with different degrees is proposed, and then the solution is obtained by taking the limit of the resulting solutions. The Taylor series expansion is taken for the other known functions in the integral equation. Also, theoretical analysis establishes the convergence and provides a general error estimate for the proposed method under the stated assumptions, whereas the numerical examples confirm its accuracy and effectiveness for the considered test problems, which illustrates that the proposed approach is valid not only for Volterra–Fredholm problems but also for Volterra and Fredholm nonlinear integral equations, and in each case, the solution converges to the exact solution. Finally, the numerical results are further validated through comparisons with existing methods reported in the literature, as well as by evaluating the maximum absolute error, demonstrating the high accuracy and effectiveness of the proposed method.