Homotopy perturbation technique

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Abstract

The homotopy perturbation technique does not depend upon a small parameter in the equation. By the homotopy technique in topology, a homotopy is constructed with an imbedding parameter p∈[0,1], which is considered as a “small parameter”. Some examples are given. The approximations obtained by the proposed method are uniformly valid not only for small parameters, but also for very large parameters.

Introduction

In the last two decades with the rapid development of nonlinear science, there has appeared ever-increasing interest of scientists and engineers in the analytical techniques for nonlinear problems. The widely applied techniques are perturbation methods. But, like other nonlinear analytical techniques, perturbation methods have their own limitations. At first, almost all perturbation methods are based on an assumption that a small parameter must exist in the equation. This so-called small parameter assumption greatly restricts applications of perturbation techniques. As is well known, an overwhelming majority of nonlinear problems have no small parameters at all. Secondly, the determination of small parameters seems to be a special art requiring special techniques. An appropriate choice of small parameters leads to ideal results. However, an unsuitable choice of small parameters results in bad effects, sometimes seriously. Furthermore, the approximate solutions solved by the perturbation methods are valid, in most cases, only for the small values of the parameters. It is obvious that all these limitations come from the small parameter assumption.

In this paper, the author will first propose a new perturbation technique coupled with the homotopy technique. The proposed method, requiring no small parameters in the equations, can readily eliminate the limitations of the traditional perturbation techniques.

Section snippets

Basic idea of homotopy perturbation method

To illustrate the basic ideas of the new method, we consider the following nonlinear differential equationA(u)−f(r)=0,r∈Ωwith boundary conditionsB(u,u/n)=0,r∈Γ,where A is a general differential operator, B is a boundary operator, f(r) is a known analytic function, Γ is the boundary of the domain Ω.

The operator A can, generally speaking, be divided into two parts L and N, where L is linear, while N is nonlinear, Eq. (1), therefore, can be rewritten as followsL(u)+N(u)−f(r)=0.

By the homotopy

Example 1

At first, we will consider the Lighthill equation [3], for it is widely studied by the PLK method (Poincaré–Lightill–Kuo Method). The equation can be written as follows:(x+εy)dydx+y=0,y(1)=1.

We can readily construct a homotopy which satisfies(1−p)εYdYdx−εy0dy0dx+p(x+εY)dYdx+Y=0,p∈[0,1].

One may now try to obtain a solution of Eq. (10) in the formY(x)=Y0(x)+pY1(x)+p2Y2(x)+⋯,where the Yi(x),i=0,1,2,… are functions yet to be determined. The substitution of Eq. (11) into Eq. (10) yieldsεY0dY0dx−εy0dy

Conclusion

In this paper we have studied few problems with or without small parameters with the homotopy perturbation technique. The results show that:

  • 1.

    The proposed method does not require small parameters in the equations, so the limitations of the traditional perturbation methods can be eliminated.

  • 2.

    The initial approximation can be freely selected with possible unknown constants.

  • 3.

    The approximations obtained by this method are valid not only for small parameters, but also for very large parameters.

Acknowledgements

The work is supported by National Science Foundation of China and Shanghai Education Foundation for Young Scientists. The author wishes to thank an unknown referee for his valuable discussions, and part of his suggestion has been directly cited.

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