Legendre wavelets method for the solution of nonlinear problems in the calculus of variations

A numerical technique for solving the nonlinear problems of the calculus of variations is presented. Two nonlinear examples are considered. In the first example, the brachistochrone problem is formulated as a nonlinear optimal control problem, and in the second example, a higher-order nonlinear problem is given. An operational matrix of integration is introduced and is utilized to reduce the calculus of variations problems to the solution of algebraic equations. The method is general, easy to implement, and yields very accurate results.