Abstract and Applied Analysis

Volume 2012, Article ID 857612, 7 pages

http://dx.doi.org/10.1155/2012/857612

## Homotopy Perturbation Method with an Auxiliary Term

^{1}National Engineering Laboratory for Modern Silk, College of Textile and Engineering, Soochow University, 199 Ren-ai Road, Suzhou 215123, China^{2}Modern Textile Institute, Donghua University, Shanghai 200051, China

Received 2 October 2011; Accepted 17 November 2011

Academic Editor: Muhammad Aslam Noor

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

The two most important steps in application of the homotopy perturbation method are to construct a suitable homotopy equation and to choose a suitable initial guess. The homotopy equation should be such constructed that when the homotopy parameter is zero, it can approximately describe the solution property, and the initial solution can be chosen with an unknown parameter, which is determined after one or two iterations. This paper suggests an alternative approach to construction of the homotopy equation with an auxiliary term; Dufing equation is used as an example to illustrate the solution procedure.

#### 1. Introduction

The homotopy perturbation method [1–7] has been worked out over a number of years by numerous authors, and it has matured into a relatively fledged theory thanks to the efforts of many researchers, see Figure 1. For a relatively comprehensive survey on the concepts, theory, and applications of the homotopy perturbation method, the reader is referred to review articles [8–11].

The homotopy perturbation method has been shown to solve a large class of nonlinear differential problems effectively, easily, and accurately; generally one iteration is enough for engineering applications with acceptable accuracy, making the method accessible to nonmathematicians.

In case of higher-order approximates needed, we can use parameter-expansion technology [12–14]; in this paper, we suggest an alternative approach by adding a suitable term in the homotopy equation.

#### 2. Homotopy Perturbation Method

Consider a general nonlinear equation where and are, respectively, the linear operator and nonlinear operator.

The first step for the method is to construct a homotopy equation in the form [3–5] where is a linear operator with a possible unknown constant and can approximately describe the solution property. The embedding parameter monotonically increases from zero to unit as the trivial problem is continuously deformed to the original one .

For example, consider a nonlinear oscillator For an oscillator, we can use sine or cosine function. We assume that the approximate solution of (2.3) is where is the frequency to be determined later. We, therefore, can choose Accordingly, we can construct a homotopy equation in the form When , we have which describes the basic solution property of the original nonlinear equation, (2.3).

When , (2.6) becomes the original one. So the solution procedure is to deform from the initial solution, (2.4), to the real one. Due to one unknown parameter in the initial solution, only one iteration is enough. For detailed solution procedure, refer to [5].

If a higher-order approximate solution is searched for, we can construct a homotopy equation in the form We expand the solution and the coefficient, zero, of the linear term into a series in : where the unknown constant, , is determined in the ()th iteration. The solution procedure is given in [5].

#### 3. Homotopy Equation with an Auxiliary Term

In this paper, we suggest an alternative approach to construction of homotopy equation, which is where is an auxiliary parameter. When , (3.1) turns out to be that of the classical one, expressed in (2.2). The auxiliary term, , vanishes completely when or ; so the auxiliary term will affect neither the initial solution () nor the real solution (). The homotopy perturbation method with an auxiliary term was first considered by Noor [15].

To illustrate the solution procedure, we consider a nonlinear oscillator in the form where and are positive constants.

Equation (3.2) admits a periodic solution, and the linearized equation of (3.2) is where is the frequency of (3.2).

We construct the following homotopy equation with an auxiliary term: Assume that the solution can be expressed in a power series in as shown in (2.9).

Substituting (2.9) into (3.4), and processing as the standard perturbation method, we have with initial conditions Solving (3.5), we have

Substituting into (3.6) results into

Eliminating the secular term needs A special solution of (3.10) is If only a first-order approximate solution is enough, we just set ; this results in

The accuracy reaches 7.6% even for the case .

The solution procedure continues by submitting into (3.7); after some simple calculation, we obtain No secular term in requires Solving (3.11) and (3.15) simultaneously, we obtain and the approximate solution is , where is given in (3.16).

In order to compare with the perturbation solution and the exact solution, we set . In case , (3.16) agrees with that obtained by classical perturbation method; when , we have The exact period reads where .

In case , we have Comparing between (3.17) and (3.20), we find that the accuracy reaches 5.5%, while accuracy of the first-order approximate frequency is 7.6%.

If a higher-order approximate solution is needed, we rewrite the homotopy equation in the form The coefficient, 1, in the auxiliary term, is also expanded in a series in in the form where is identified in the ()th iteration. The solution procedure is similar to that illustrated above.

#### 4. Discussions and Conclusions

Generally the homotopy equation can be constructed in the form where and are functions of , satisfying and , and can be generally expressed in the form For example, for the Blasius equation where the superscript denotes derivative with respect to , we can construct a homotopy equation in the form where and are unknown constants to be determined.

The operator can be also a nonlinear one, for example, if we want to search for a solitonary solution, we can choose .

The homotopy equation can be easily constructed, and the solution procedure is simple. This paper can be considered a standard homotopy perturbation algorithm and can be used as a paradigm for many other applications.

#### 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, “Homotopy perturbation technique,”
*Computer Methods in Applied Mechanics and Engineering*, vol. 178, no. 3-4, pp. 257–262, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J.-H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linear problems,”
*International Journal of Non-Linear Mechanics*, vol. 35, no. 1, pp. 37–43, 2000. View at Publisher · View at Google Scholar - J.-H. He, “An elementary introduction to the homotopy perturbation method,”
*Computers & Mathematics with Applications*, vol. 57, no. 3, pp. 410–412, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J.-H. He, “Recent development of the homotopy perturbation method,”
*Topological Methods in Nonlinear Analysis*, vol. 31, no. 2, pp. 205–209, 2008. View at Google Scholar · View at Zentralblatt MATH - J.-H. He, “New interpretation of homotopy perturbation method,”
*International Journal of Modern Physics B*, vol. 20, no. 18, pp. 2561–2568, 2006. View at Publisher · View at Google Scholar - M. Rentoul and P. D. Ariel, “Extended homotopy perturbation method and the flow past a nonlinearly stretching sheet,”
*Nonlinear Science Letters A*, vol. 2, pp. 17–30, 2011. View at Google Scholar - M. A. Abdou, “Zakharov-Kuznetsov equation by the homotopy analysis method and Hirota's Bilinear method,”
*Nonlinear Science Letters B*, vol. 1, pp. 99–110, 2011. View at Google Scholar - J.-H. He, “Some asymptotic methods for strongly nonlinear equations,”
*International Journal of Modern Physics B*, vol. 20, no. 10, pp. 1141–1199, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - J. H. He, “An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering,”
*International Journal of Modern Physics B*, vol. 22, no. 21, pp. 3487–3578, 2008. View at Publisher · View at Google Scholar · View at Scopus - D. D. Ganji and S. H. H. Kachapi, “Analysis of nonlinear equations in fluids,”
*Progress in Nonlinear Science*, vol. 2, pp. 1–293, 2011. View at Google Scholar - D. D. Ganji and S. H. H. Kachapi, “Analytical and numerical methods in engineering and applied sciences,”
*Progress in Nonlinear Science*, vol. 3, pp. 1–579, 2011. View at Google Scholar - S. Q. Wang and J. H. He, “Nonlinear oscillator with discontinuity by parameter-expansion method,”
*Chaos, Solitons and Fractals*, vol. 35, no. 4, pp. 688–691, 2008. View at Publisher · View at Google Scholar · View at Scopus - L. Xu, “Application of He's parameter-expansion method to an oscillation of a mass attached to a stretched elastic wire,”
*Physics Letters A*, vol. 368, no. 3-4, pp. 259–262, 2007. View at Publisher · View at Google Scholar · View at Scopus - D. H. Shou and J. H. He, “Application of parameter-expanding method to strongly nonlinear oscillators,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 8, no. 1, pp. 121–124, 2007. View at Google Scholar · View at Scopus - M. A. Noor, “Some iterative methods for solving nonlinear equations using homotopy perturbation method,”
*International Journal of Computer Mathematics*, vol. 87, no. 1–3, pp. 141–149, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH