## Homotopy perturbation method: a new nonlinear analytical technique.(English)Zbl 1030.34013

Summary: In this paper, a new perturbation method is proposed. In contrast to the traditional perturbation methods, this technique does not require a small parameter in an equation. In this method, according to the homotopy technique, a homotopy with an imbedding parameter $$p\in [0,1]$$ is constructed, and the imbedding parameter is considered as a “small parameter”, so the method is called the homotopy perturbation method, which can take the full advantages of the traditional perturbation methods and homotopy techniques. To illustrate its effectiveness and its convenience, a Duffing equation with high order of nonlinearity is used; the result reveals that its first order of approximation obtained by the proposed method is valid uniformly even for very large parameter, and is more accurate than the perturbation solutions.

### MSC:

 34A45 Theoretical approximation of solutions to ordinary differential equations 65L99 Numerical methods for ordinary differential equations 34E99 Asymptotic theory for ordinary differential equations

Duffing equation
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### References:

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