<b>Bernoulli Polynomials for solving three-dimensional Volterra-Fredholm integral equations of the second kind</b><b></b>

In this work. our approach for solving three-dimensional linear Volterra-Fredholm integral equations (3D-VFIEs) based on Bernoulli polynomials. This approach was previously applied to solve two-dimensional Volterra-Fredholm integral equations. This method transforms the 3D-VFIEs into a system of linear algebraic equations. The Banach fixed-point theorem is employed to demonstrate the existence and uniqueness of the 3D-VFIE. It has demonstrated its efficiency and effectiveness in achieving accurate numerical results, outperforming other methods. The numerical method that we relied on in solving (3D-VFIEs) as explained in this paper, and the approximate solution that we reached it by comparing it to other solutions that were deduced by other methods showed us very close results for some methods such as the Lucas and Shifted Chebyshev polynomials methods and more efficient than other ones such as Haar Wavelet’s technique, Block-pulse functions and Modified block-pulse functions. Compared to other error rates, the error coefficient was the best. This is supported by examples of numerical solutions to linear integral equations, statistical tables, and figures, which provide the strongest evidence of the convergence of the exact and approximate solutions. The high accuracy of this method is verified through some numerical examples. The Maple 18 program outputs all of the results.