Abstract and Applied Analysis

Volume 2012 (2012), Article ID 964974, 2 pages

doi:10.1155/2012/964974

Letter to the Editor

Comment on “Variational Iteration Method for Fractional Calculus Using He’s Polynomials”

Ji-Huan He

National Engineering Laboratory for Modern Silk, College of Textile and Engineering, Soochow University, 199 Ren-ai Road, Suzhou 215123, China

Received 17 November 2012; Accepted 1 December 2012

Copyright © 2012 Ji-Huan He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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]:

The classical variational iteration algorithm reads [4]

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).

References

J.-H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, vol. 20, no. 10, pp. 1141–1199, 2006.

J.-H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous media,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 57–68, 1998.

Y. Liu, “Variational homotopy perturbation method for solving fractional initial boundary value problems,” Abstract and Applied Analysis, vol. 2012, Article ID 727031, 10 pages, 2012.

J. H. He, “Asymptotic methods for solitary solutions and compactons,” Abstract and Applied Analysis, vol. 2012, Article ID 916793, 130 pages, 2012.

A. Ghorbani, “Beyond Adomian polynomials: He polynomials,” Chaos, Solitons and Fractals, vol. 39, no. 3, pp. 1486–1492, 2009.

M. A. Noor and S. T. Mohyud-Din, “Variational iteration method for solving higher-order nonlinear boundary value problems using He's polynomials,” The International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, pp. 141–156, 2008.

S. T. Mohyud-Din, “Solving heat and wave-like equations using He's polynomials,” Mathematical Problems in Engineering, vol. 2009, Article ID 427516, 12 pages, 2009.

M. A. Noor and S. T. Mohyud-Din, “Variational iteration method for fifth-order boundary value problems using He's polynomials,” Mathematical Problems in Engineering, vol. 2008, Article ID 954794, 12 pages, 2008.

S. Guo and L. Mei, “The fractional variational iteration method using He's polynomials,” Physics Letters A, vol. 375, no. 3, pp. 309–313, 2011.

J. Ji, J. Zhang, and Y. Dong, “The fractional variational iteration method improved with the Adomian series,” Applied Mathematics Letters, vol. 25, no. 12, pp. 2223–2226, 2012.