Legendre collocation method for solving a class of the second order nonlinear differential equations with the mixed non-linear conditions

In this paper, a matrix method based on Legendre collocation points on interval is proposed for the approximate solution of some second order nonlinear ordinary differential equations with the mixed nonlinear conditions in terms of Legendre polynomials. The method, by means of collocation points, transforms the differential equation to a matrix equation which corresponds to a system of nonlinear algebraic equations with unknown Legendre coefficients. The numerical results show the effectiveness of the method for this type of equation. When this method is compared with the other usual techniques, results would be easier and have higher accuracy.