  In order to solve an initial value problem by the variational iteration
method, a sequence of functions is produced which converges to the solution
under some suitable conditions. In the nonlinear case, after a few iterations
the terms of the sequence become complicated, and therefore, computing a highly
accurate solution would be difficult or even impossible. In this paper, for
one-dimensional initial value problems, we propose a new approach which is
based on approximating each term of the sequence by a piecewise linear
function. Moreover, the convergence of the method is proved. Three illustrative
examples are given to show the superiority of the proposed method over the
classical variational iteration method.
