On the Picard–Lindelof method for nonlinear second-order differential equations

We consider a second-order nonlinear ordinary differential equation which satisfies a Lipschitz-continuity condition and show that the method of variation of parameters allows us to write the solution as a Volterra integral equation that includes the initial values of the solution and its first-order derivative. We also show that this integral formulation can be written as an iterative method which converges uniformly to the unique solution of the problem. By making use of an integral identity, the initial conditions can be eliminated from the integral equation and a new integral formulation that depends on the second-order derivative of the solution is obtained. It is shown that this integral formulation is exactly the same as that of the variational iteration method but does not require a variational principle; neither does it require constrained variations for nonlinear terms. We also show that the two integral formulations can be written in differential form and correspond to a two-level iterative method. By writing the second-order ordinary differential equation as a two-equation system of first-order ordinary differential equations, three different iterative procedures are developed, but only one of them coincides with that derived from the two integral equations for the solution.