Interpolated variational iteration method for initial value problems
For researchers solving one-dimensional initial value problems, this method offers a practical improvement over the variational iteration method by reducing computational complexity.
The paper proposes an interpolated variational iteration method that approximates each iteration term by a piecewise linear function to overcome complexity in nonlinear initial value problems, demonstrating superiority over the classical method through three examples.
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.