Interpolated variational iteration method for initial value problems

TL;DR

This paper introduces an interpolated variational iteration method that simplifies the computation of solutions to initial value problems by approximating sequence terms with piecewise linear functions, improving efficiency and convergence.

Contribution

The paper proposes a novel approach using piecewise linear approximation within the variational iteration method for better computational efficiency and convergence in solving initial value problems.

Findings

The method converges under suitable conditions.

It outperforms the classical variational iteration method in examples.

The approach simplifies complex iterative sequences.

Abstract

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.