AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 964974 10.1155/2012/964974 964974 Letter to the Editor Comment on “Variational Iteration Method for Fractional Calculus Using He’s Polynomials” He Ji-Huan 1 National Engineering Laboratory for Modern Silk College of Textile and Engineering Soochow University 199 Ren-ai Road Suzhou 215123 China scu.edu.tw 2012 6 12 2012 2012 17 11 2012 01 12 2012 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.

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 . Recently Liu applied the variational homotopy perturbation method for fractional initial boundary value problems ; 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 : (2.1)αutα-12x22ux2=0. The classical variational iteration algorithm reads  (2.2)un+1(x,t)=un(x,t)-0t{αun(x,s)sα-12x22un(x,s)x2}ds, which is exactly the same as that in Liu’s work , where the nonlinear term is expanded into He’s polynomials . So what Liu used is exactly the variational iteration method using He’s polynomials, which has been widely used for solving various nonlinear problems .

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 , and the variational iteration algorithm using Adomian’s polynomials was given in .

Acknowledgment

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

He J.-H. Some asymptotic methods for strongly nonlinear equations International Journal of Modern Physics B 2006 20 10 1141 1199 10.1142/S0217979206033796 2220565 ZBL1102.34039 He J.-H. Approximate analytical solution for seepage flow with fractional derivatives in porous media Computer Methods in Applied Mechanics and Engineering 1998 167 1-2 57 68 10.1016/S0045-7825(98)00108-X 1665221 ZBL0942.76077 Liu Y. Variational homotopy perturbation method for solving fractional initial boundary value problems Abstract and Applied Analysis 2012 2012 10 727031 2926899 ZBL1246.65191 He J. H. Asymptotic methods for solitary solutions and compactons Abstract and Applied Analysis 2012 2012 130 916793 10.1155/2012/916793 Ghorbani A. Beyond Adomian polynomials: He polynomials Chaos, Solitons and Fractals 2009 39 3 1486 1492 10.1016/j.chaos.2007.06.034 2512946 ZBL1197.65061 Noor M. A. Mohyud-Din S. T. Variational iteration method for solving higher-order nonlinear boundary value problems using He's polynomials The International Journal of Nonlinear Sciences and Numerical Simulation 2008 9 141 156 Mohyud-Din S. T. Solving heat and wave-like equations using He's polynomials Mathematical Problems in Engineering 2009 2009 12 427516 10.1155/2009/427516 2539709 ZBL1181.80014 Noor M. A. Mohyud-Din S. T. Variational iteration method for fifth-order boundary value problems using He's polynomials Mathematical Problems in Engineering 2008 2008 12 954794 10.1155/2008/954794 2402983 ZBL1151.65334 Guo S. Mei L. The fractional variational iteration method using He's polynomials Physics Letters A 2011 375 3 309 313 10.1016/j.physleta.2010.11.047 2748835 ZBL1241.35216 Ji J. Zhang J. Dong Y. The fractional variational iteration method improved with the Adomian series Applied Mathematics Letters 2012 25 12 2223 2226 10.1016/j.aml.2012.06.007