Numerical solution of Volterra-Fredholm integral equation systems by operational matrices of integration based on Bernstein multi-scaling polynomials

This paper introduces a new method which is used to obtain numerical solutions of Volterra-Fredholm linear integral equation systems. In the proposed method, the functions and integrals of the system are approximated in terms of Bernstein multi-scale polynomials. Then, these approximations are applied to the Volterra-Fredholm integral equations system, which turns it into a linear system that is solvable by conventional methods. Then, in addition to convergence analysis, the accuracy and efficiency of the proposed method are evaluated by providing different examples and comparing their results with other similar methods.