Two-step integral collocation-variational iteration method for the solutions of integro-differential equations

In this paper, an algorithm based on integral collocation and variational iteration method for solving integro-differential equations is presented. In the rst instance, integro-differential equations are reduced to a system of integral equations after which we replaced all the derivatives in the new system of integral equations with their equivalent new derivatives. These new derivatives were obtained by approximating the nth order derivative with truncated Chebyshev series and then integrated n-times to obtain expressions for lower-order derivatives and the function itself. After the second iteration, the residual equation is formed which is collocated at the chosen collocation points and extra n equations are also obtained from the boundary conditions. Computational results are given for test examples to demonstrate the effectiveness, reliability, applicability and efficiency of the new method. It is shown that the solutions obtained from the method have very high degree of accuracy.