On the Reconstruction of the First Term in the Variational Iteration Method for Solving Differential Equations

Abstract

The well-known He's variational iteration method is an efficient and easy technique for finding the solution of various kinds of problems. In this method the solution of a nonlinear problem is found without any need to solve a linear or nonlinear system of equations. In the current work we present an idea for accelerating the convergence of the resulted sequence to the solution of the problem by choosing a suitable initial term. The efficiency of the method will be shown by testing the new algorithm on several examples.

Numerical Study of Second Painlevé Equation

“…Consequently, the solution is given by y = lim n→∞ y n . As said in [32], in some cases VIM converges to Taylors expansion of the method. In this case, an efficient after treatment technique has been proposed for extending the domain of the convergence.…”

“…The successive approximations y n (x) will be readily obtained upon using the obtained Lagrange multiplier and by using any selective function y 0 (x). In [32] an idea for accelerating the convergence of the resulted sequence to the solution of the problem by choosing a suitable initial term is presented. Consequently, the solution is given by y = lim n→∞ y n .…”

A numerical approach for a class of astrophysics equations using piecewise spectral-variational iteration method

“…VIM is an iterative integral-based scheme proposed by He (1999He ( , 2001He ( , 2002 which has been successfully used by numerous authors in a wide variety of linear and nonlinear models (Abbasbandy, 2007a;Abbasbandy, 2007b;Ganji et al, 2007;Miansari et al, 2008;Tatari and Dehghan, 2007;Saadatmandi and Dehghan, 2009a;Saadatmandi and Dehghan, 2009b;Tatari and Dehghan, 2009;Tatari and Dehghan, 2010;Dehghan and Salehi, 2010;Dehghan and Salehi, 2011). The idea of the method is based on constructing a correction functional by a general Lagrange multiplier, and the multiplier is chosen in such a way that its correction solution is improved with respect to the initial approximation or to the trial function.…”

Solution of a nonlinear time-delay model in biology via semi-analytical approaches

“…In [72] the variational iteration method was employed to solve the Kawahara equation that models the plasma waves, the capillary gravity water and describes water waves with surface tension. Authors of [64] presented an idea for accelerating the convergence of the resulted sequence to the solution of the problem by choosing a suitable initial term and examined the proposed idea for ordinary, partial differential, and integro-differential equations. In [65] the variational iteration method is used to compute the solution for the linear, variable coefficient, fractional derivative and multi-space telegraph equations, etc.…”