A direct approach based on spline for the solution of problems in calculus of variations

A method based on sextic spline method is developed for the numerical solution of brachistochrone problem. The nonlinear Brachistochrone problem is a well-known problem in calculus of variations. We used a direct approach using sexic B-spline connected with Lagrange multiplier technique to approximate the solution. The integrals have been solved by a p point Gauss-Legendre quadrature. Obtained orders of convergence show that the method is optimal. We have compared the numerical results with some existed methods which show the efficiency of our proposed method computationally.