Computational Fractional Dynamical Systems

9Variational Iterative Method

9.1 Introduction

In the previous chapters, we have discussed two semi‐analytical approaches, such as Adomian decomposition method (ADM) and homotopy perturbation method (HPM), to solve linear and nonlinear ordinary/partial/fractional differential equations. In this chapter, we will discuss the variational iterative method (VIM). VIM was first proposed by Chinese mathematician He (1998) and He and Wu (2007). He successfully used this technique for solving ordinary and partial differential equations. Subsequently, several researchers used this method for solving linear, nonlinear, homogeneous, and inhomogeneous differential equations (Geng et al. 2009; Odibat and Momani 2009; Odibat 2010; Khana et al. 2011). The principal benefit of this approach is its simplicity and ability to solve nonlinear equations. The approach is also valid in bounded and unbounded domains. It is based on the Lagrange multiplier method initiated by Inokuti et al. (1978). This approach is a modification of the general Lagrange multiplier approach to an iteration process, called functional correction (Momani and Odibat 2007). A substantial number of nonlinear problems are solved effectively by various researchers, generally with one or two iterations leading to good solutions. It is worth mentioning that almost all conventional perturbation methods are based on the assumption of small parameters. These small parameters are so sensitive that a slight change may influence the ...