non-standard finite difference method for numerical solution of second order linear fredholm integro-differential equations

in this article we have considered a non-standard finite difference method for the solution of second order  fredholm integro differential equation type initial value problems. the non-standard finite difference method and the composite trapezoidal quadrature method is used to transform the fredholm integro-differential equation into a system of equations. we have also developed a numerical method for the numerical approximation of the derivative of the solution of the problems. the numerical results in experiment on some  model problems show the simplicity and efficiency of the method. numerical results showed that the proposed method is convergent and at least second order of accurate.