Recently Liu applied the variational homotopy perturbation method for fractional initial
boundary value problems. This note concludes that the method is a modified variational iteration method using He’s polynomials.
A standard variational iteration algorithm for fractional differential equations is suggested.

1. Introduction

The variational iteration method [1, 2] has been shown to solve a large class of nonlinear differential problems effectively, easily, and accurately with the approximations converging rapidly to accurate solutions. In 1998, the method was first adopted to solve fractional differential equations [2]. Recently Liu applied the variational homotopy perturbation method for fractional initial boundary value problems [3]; however, the method is nothing but a modified variational iteration method.

2. Liu’s Work

Liu used the following example to elucidate the solution process [3]:
(2.1)∂αu∂tα-12x2∂2u∂x2=0.
The classical variational iteration algorithm reads [4]
(2.2)un+1(x,t)=un(x,t)-∫0t{∂αun(x,s)∂sα-12x2∂2un(x,s)∂x2}ds,
which is exactly the same as that in Liu’s work [3], where the nonlinear term is expanded into He’s polynomials [5]. So what Liu used is exactly the variational iteration method using He’s polynomials, which has been widely used for solving various nonlinear problems [6–8].

3. Conclusion

The so-called variational homotopy perturbation method is nothing but the variational iteration method using He’s polynomials. A standard variational iteration algorithm using He’s polynomials is suggested to follow Guo and Mei’s work [9], and the variational iteration algorithm using Adomian’s polynomials was given in [10].

Acknowledgment

The work is supported by PAPD (a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions).

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