Application of Bernstein polynomials for solving Fredholm integro-differential-difference equations

In this paper, the Bernstein polynomials method is proposed for the numerical solution of Fredholm integro-differential-difference equation with variable coefficients and mixed conditions. This method is using a simple computational manner to obtain a quite acceptable approximate solution. The main characteristic behind this method lies in the fact that, on the one hand, the problem will be reduced to a system of algebraic equations. On the other hand, the efficiency and accuracy of the Bernstein polynomials method for solving these equations are high. The existence and uniqueness of the solution have been proved. Moreover, an estimation of the error bound for this method will be shown by preparing some theorems. Finally, some numerical experiments are presented to show the excellent behavior and high accuracy of this algorithm in comparison with some other well-known methods.