Fractional Variational Iteration Method versus Adomian’s Decomposition Method in Some Fractional Partial Differential Equations

Junqiang Song,Fengshun Lu,Xiaoqun Cao,Fukang Yin

doi:10.1155/2013/392567

Abstract

A comparative study is presented about the Adomian’s decomposition method (ADM), variational iteration method (VIM), and fractional variational iteration method (FVIM) in dealing with fractional partial differential equations (FPDEs). The study outlines the significant features of the ADM and FVIM methods. It is found that FVIM is identical to ADM in certain scenarios. Numerical results from three examples demonstrate that FVIM has similar efficiency, convenience, and accuracy like ADM. Moreover, the approximate series are also part of the exact solution while not requiring the evaluation of the Adomian’s polynomials.

Highlights

Fractional differential equations (FDEs), as a generalization of ordinary differential equations to an arbitrary order, have been proved to be a valuable tool in modelling many phenomena in the fields of physics, chemistry, engineering, aerodynamics, electrodynamics of complex medium, polymer rheology, and so forth [1,2,3,4,5,6,7,8,9]
Most of the nonlinear fractional differential equations do not have exact analytic solutions, so approximation and numerical techniques must be used. Some approximate methods such as Adomian’s decomposition method (ADM) [10,11,12,13], homotopy perturbation method (HPM) [14,15,16], variational iteration method (VIM) [17,18,19,20,21,22], homotopy analysis method (HAM) [23,24,25,26], fractional complex transform (FCT) [27,28,29,30,31], and wavelets method [32,33,34] have been given to find an analytical approximation to FDEs
The main goal of this work is to conduct a comparative study between fractional variational iteration method and the Adomian’s decomposition method

Summary

Introduction

Fractional differential equations (FDEs), as a generalization of ordinary differential equations to an arbitrary (noninteger) order, have been proved to be a valuable tool in modelling many phenomena in the fields of physics, chemistry, engineering, aerodynamics, electrodynamics of complex medium, polymer rheology, and so forth [1,2,3,4,5,6,7,8,9]. The prior work [46,47,48,49] has performed a comparative study of ADM and VIM and got two useful conclusions: on the one hand, ADM needs specific algorithms to evaluate the Adomian’s polynomials, Journal of Applied Mathematics while VIM handles linear and nonlinear problems in a similar manner without any additional requirement or restriction; on the other hand, Adomian’s decomposition method provides the components of the exact solution It has to be validated whether these conclusions are true for the scenario of FPDEs. In this paper, we consider the following fractional initial value problem: Dtαu (x, t) + N [u (x, t)] + L [u (x, t)] = g (x, t), t > 0, (1).

Preliminaries and Notations

VIM and FVIM

Adomian’s Decomposition Method

Applications and Results

Conclusion