Interpolated variational iteration method for initial value problems

• An interpolated version of the VIM called IVIM is presented.
• In the IVIM each term of the sequence is approximated by a piecewise linear function.
• Numerical results show that the IVIM is superior to the VIM.

In order to solve an initial value problem by the variational iteration method, a sequence of functions is produced that converges to the solution under suitable conditions. In the nonlinear case, the terms of the sequence become complicated after a few iterations, and thus computing a highly accurate solution is difficult or even impossible. In this study, we propose a new approach for one-dimensional initial value problems, which is based on approximating each term of the sequence by a piecewise linear function. Moreover, we prove the convergence of the method. Three illustrative examples are given to demonstrate the superior performance of the proposed method compared with the classical variational iteration method.