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Application of the Bernstein Polynomials for Solving the Nonlinear Fredholm Integro-Differential Equations

In this article an efficient numerical method for finding solution of the nonlinear Fredholm integro-differential equations on base of Bernstein polynomials basis would be presented. For this purpose at the beginning we express briefly some properties of Bernstein polynomials and after that with respect to relation between Bernstein and Legendre polynomials, operational matrices of integration and product of Bernstein Polynomials and also dual operational matrix of Bernstein basis vector, all will be presented. Then with approximate approach the solution of integro-differential equation with C T �(x) form (in which C is the unknown coefficients vector and�(x) is the Bernstein basis vector) and it’s usage of presented matrices, mentioned equation and it’s initial conditions will be converted to an equivalent matrix equation. Coefficients vector C is the solution of this matrix equation. At the end with presentation of five numerical examples the method will be evaluated.