We use homotopy perturbation method (HPM) to handle the foam drainage equation. Foaming occurs in many distillation and absorption processes. The drainage of liquid foams involves the interplay of gravity, surface tension, and viscous forces. The concept of He's homotopy perturbation method is introduced briefly for applying this method for problem solving. The results of HPM as an analytical solution are then compared with those derived from Adomian's decomposition method (ADM) and the variational iteration method (VIM). The results reveal that the HPM is very effective and convenient in predicting the solution of such problems, and it is predicted that HPM can find a wide application in new engineering problems.

Most scientific problems and physical phenomena occur nonlinearly. Except in a limited number of these problems, finding the exact analytical solutions of such problems are rather difficult. Therefore, there have been attempts to develop new techniques for obtaining analytical solutions which reasonably approximate the exact solutions [

The homotopy perturbation method was established by He [

Foams are of great importance in many technological processes and applications, and their properties are subject of intensive studies from both practical and scientific points of view [

Schematic illustration of the interdependence of drainage, coarsening, and rheology of foams [

The pursuit of analytical solutions for foam drainage equation is of intrinsic scientific interest. To the best of the authors’ knowledge, there is no paper that has solved the nonlinear foam drainage equation by HPM. In this paper, the basic idea of HPM is described, and then, it is applied to study the following nonlinear foam drainage equation [

To explain this method, let us consider the following function:

In this section, we obtain an analytical solution of (

In this section, we present the results with HPM to show the efficiency of the method, described in the previous section for solving (

The solution of (

The solution of (

The solution of (

The absolute error of HPM at

The effect of

It is important to see the difference between the results obtained by ADM [

Comparison between absolute errors of ADM, VIM, and HPM for

−10 | 0 | 0 | |

−8 | 0 | ||

−6 | 0 | ||

−4 | 0 | ||

−2 | 0.000236656 | 0 | |

−1 | 0.00523834 | 0 | |

0 | 0 |

Comparison between absolute errors of ADM, VIM, and HPM for

−10 | 0 | ||

−8 | 0 | ||

−6 | 0 | ||

−4 | |||

−2 | 0.00197296 | 0.000002123 | |

−1 | 0.0485679 | — | 0.000050433 |

0 | 0.00051592 | 0.000014557 |

Comparison between absolute errors of ADM, VIM, and HPM for

−10 | 0 | ||

−8 | 0 | ||

−6 | |||

−4 | 0.0000146727 | ||

−2 | 0.0064218 | 0.001716375 | |

−1 | 0.094494 | — | 0.043767689 |

0 | 3.71367 | 0.126259125 |

In this work, He’s homotopy perturbation method has been successfully utilized to derive approximate explicit analytical solution for the nonlinear foam drainage equation. The results show that this perturbation scheme provides excellent approximations to the solution of this nonlinear equation with high accuracy and avoids linearization and physically unrealistic assumptions. This new method is extremely simple, easy to apply, needs less computation, and accelerates the convergence to the solutions. The results obtained here are compared with results of exact solution. The current work illustrates that the HPM is indeed a powerful analytical technique for most types of nonlinear problems and several such problems in scientific studies and engineering may be solved by this method.

The authors gratefully acknowledge the support of the Department of Mechanical Engineering and the talented office of Semnan University for funding the current research grant.