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Article
Publication date: 17 May 2011

Manel Labidi and Khaled Omrani

The purpose of this paper is to implement variational iteration method (VIM) and homotopy perturbation method (HPM) to solve modified Camassa‐Holm (mCH) and modified…

Abstract

Purpose

The purpose of this paper is to implement variational iteration method (VIM) and homotopy perturbation method (HPM) to solve modified Camassa‐Holm (mCH) and modified Degasperis‐Procesi (mDP) equations.

Design/methodology/approach

Perturbation method is a traditional method depending on a small parameter which is difficult to be found for real‐life nonlinear problems. To overcome the difficulties and limitations of the above method, two new ones have recently been introduced by He, i.e. VIM and HPM. In this paper, mCH and mDP equations are solved through these methods.

Findings

To assess the accuracy of the solutions, the comparison of the obtained results with the exact solutions reveals that both methods are tremendously effective.

Originality/value

The paper shows that VIM and HPM can be implemented to solve mCH and mDP equations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 21 no. 4
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 23 March 2012

D.D. Ganji, M. Rahimi and M. Rahgoshay

The purpose of this paper is to determine the fin efficiency of convective straight fins with temperature dependent thermal conductivity by using Homotopy Perturbation Method.

Abstract

Purpose

The purpose of this paper is to determine the fin efficiency of convective straight fins with temperature dependent thermal conductivity by using Homotopy Perturbation Method.

Design/methodology/approach

Most engineering problems, especially heat transfer equations are in nonlinear form. Homotopy Perturbation Method (HPM) has been applied to solve a wide series of nonlinear differential equations. In this paper, HPM is used for obtaining the fin efficiency of convective straight fins with temperature‐dependent thermal conductivity. Comparison of the results with those of Homotopy Perturbation Method, exact solution, numerical results and Adomian's decomposition method (ADM) were been done by Cihat Arslanturk.

Findings

Results show that both Homotopy Perturbation Method and ADM applied to the nonlinear equations were capable of solving them with successive rapidly convergent approximations without any restrictive assumptions or transformations causing changes in the physical properties of the problem. Moreover, adding up the number of iterations leads to explicit solution for the problem. The results are just obtained with two iterations. This shows the accuracy and great potential of this method. Finally, it can be seen that, with increase of thermo‐geometric fin parameter (v), the fin efficiency increases too.

Originality/value

The results demonstrate good validity and great potential of the HPM for Heat Transfer equations in engineering problems.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 22 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 13 July 2010

A. Barari, B. Ganjavi, M. Ghanbari Jeloudar and G. Domairry

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…

Abstract

Purpose

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. This paper considers linear and nonlinear systems that are not only regarded as general boundary value problems, but also are used as mathematical models in viscoelastic and inelastic flows. The purpose of this paper is to present the application of the homotopyperturbation method (HPM) and variational iteration method (VIM) to solve some boundary value problems in structural engineering and fluid mechanics.

Design/methodology/approach

Two new but powerful analytical methods, namely, He's VIM and HPM, are introduced to solve some boundary value problems in structural engineering and fluid mechanics.

Findings

Analytical solutions often fit under classical perturbation methods. However, as with other analytical techniques, certain limitations restrict the wide application of perturbation methods, most important of which is the dependence of these methods on the existence of a small parameter in the equation. Disappointingly, the majority of nonlinear problems have no small parameter at all. Furthermore, the approximate solutions solved by the perturbation methods are valid, in most cases, only for the small values of the parameters. In the present study, two powerful analytical methods HPM and VIM have been employed to solve the linear and nonlinear elastic beam deformation problems. The results reveal that these new methods are very effective and simple and do not require a large computer memory and can also be used for solving linear and nonlinear boundary value problems.

Originality/value

The results revealed that the VIM and HPM are remarkably effective for solving boundary value problems. These methods are very promoting methods which can be wildly utilized for solving mathematical and engineering problems.

Details

Journal of Engineering, Design and Technology, vol. 8 no. 2
Type: Research Article
ISSN: 1726-0531

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Article
Publication date: 19 July 2011

Behrouz Raftari and Ahmet Yildirim

The purpose of this paper is to introduce a new version of the homotopy perturbation method (HPM) for solving the magnetohydrodynamic viscous flow due to a shrinking sheet.

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Abstract

Purpose

The purpose of this paper is to introduce a new version of the homotopy perturbation method (HPM) for solving the magnetohydrodynamic viscous flow due to a shrinking sheet.

Design/methodology/approach

Three terms from HPM solution are used.

Findings

The results show that this method is very effective and simple and can be applied to other nonlinear problems.

Research limitations/implications

Comparison between the HPM and homotopy analysis methods for the studied problem shows a remarkable agreement and reveals that the HPM needs less work.

Practical implications

It is suggested that this method should be called HPM with auxiliary parameters. This paper uses two auxiliary parameters, three or more auxiliary parameters could be used for accuracy consideration.

Originality/value

In this paper, a two‐parameter HPM is applied which is useful for finding an approximate analytical solution of MHD viscous flow due to a shrinking sheet.

Details

Engineering Computations, vol. 28 no. 5
Type: Research Article
ISSN: 0264-4401

Keywords

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Article
Publication date: 3 August 2012

M. Madani, Yasir Khan, Gh. Mahmodi, Naeem Faraz, Ahmet Yildirim and B. Nasernejad

The purpose of this paper is to present the problem of three‐dimensional flow of a fluid of constant density forced through the porous bottom of a circular porous slider…

Abstract

Purpose

The purpose of this paper is to present the problem of three‐dimensional flow of a fluid of constant density forced through the porous bottom of a circular porous slider moving laterally on a flat plate.

Design/methodology/approach

The transformed nonlinear ordinary differential equations are solved via the homotopy perturbation method (HPM) for small as well as moderately large Reynolds numbers. The convergence of the obtained HPM solution is carefully analyzed. Finally, the validity of results is verified by comparing with numerical methods and existing numerical results.

Findings

Close agreement of the two sets of results is observed, thus demonstrating the accuracy of the HPM approach for the particular problem considered.

Originality/value

Interesting conclusions which can be drawn from this study are that HPM is very effective and simple compared to the existing solution method, able to solve problems without using Padé approximants and can therefore be considered as a clear advantage over the N.M. Bujurke and Phan‐Thien techniques.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 22 no. 6
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 17 May 2011

Sefa Anıl Sezer, Ahmet Yıldırım and Syed Tauseef Mohyud‐Din

The purpose of this paper is to directly extend the homotopy perturbation method (HPM) that was developed for integer‐order differential equation, to derive explicit and…

Abstract

Purpose

The purpose of this paper is to directly extend the homotopy perturbation method (HPM) that was developed for integer‐order differential equation, to derive explicit and numerical solutions of the fractional KdV‐Burgers‐Kuramoto equation.

Design/methodology/approach

The authors used Maple Package to calculate the functions obtained from the HPM.

Findings

The fractional derivatives are described in the Caputo sense. HPM performs extremely well in terms of accuracy, efficiently, simplicity, stability and reliability.

Originality/value

The paper describes how the HPM has been successfully applied to find the solution of fractional KdV‐Burgers‐Kuramoto equation.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 21 no. 4
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 29 July 2014

Guanglu Zhou and Boying Wu

– The purpose of this paper is to present a general framework of Homotopy perturbation method (HPM) for analytic inverse heat source problems.

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123

Abstract

Purpose

The purpose of this paper is to present a general framework of Homotopy perturbation method (HPM) for analytic inverse heat source problems.

Design/methodology/approach

The proposed numerical technique is based on HPM to determine a heat source in the parabolic heat equation using the usual conditions. Then this shows the pertinent features of the technique in inverse problems.

Findings

Using this HPM, a rapid convergent sequence which tends to the exact solution of the problem can be obtained. And the HPM does not require the discretization of the inverse problems. So HPM is a powerful and efficient technique in finding exact and approximate solutions without dispersing the inverse problems.

Originality/value

The essential idea of this method is to introduce a homotopy parameter p which takes values from 0 to 1. When p=0, the system of equations usually reduces to a sufficiently simplified form, which normally admits a rather simple solution. As p is gradually increased to 1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 24 no. 6
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 30 March 2010

Ahmet Yıldırım

This paper aims to present a general framework of the homotopy perturbation method (HPM) for analytic treatment of fractional partial differential equations in fluid…

Abstract

Purpose

This paper aims to present a general framework of the homotopy perturbation method (HPM) for analytic treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense.

Design/methodology/approach

Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation and fractional Klein‐Gordon equation are investigated to show the pertinent features of the technique.

Findings

HPM is a powerful and efficient technique in finding exact and approximate solutions for fractional partial differential equations in fluid mechanics. The implementation of the noise terms, if they exist, is a powerful tool to accelerate the convergence of the solution. The results so obtained reinforce the conclusions made by many researchers that the efficiency of the HPM and related phenomena gives it much wider applicability.

Originality/value

The essential idea of this method is to introduce a homotopy parameter, say p, which takes values from 0 to 1. When p = 0, the system of equations usually reduces to a sufficiently simplied form, which normally admits a rather simple solution. As p is gradually increased to 1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 20 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 23 March 2012

Najeeb Alam Khan, Asmat Ara and Amir Mahmood

The purpose of this paper is to use the generalized differential transform method (GDTM) and homotopy perturbation method (HPM) for solving time‐fractional Burgers and…

Abstract

Purpose

The purpose of this paper is to use the generalized differential transform method (GDTM) and homotopy perturbation method (HPM) for solving time‐fractional Burgers and coupled Burgers equations. The fractional derivatives are described in the Caputo sense.

Design/methodology/approach

In these schemes, the solutions takes the form of a convergent series. In GDTM, the differential equation and related initial conditions are transformed into a recurrence relation that finally leads to the solution of a system of algebraic equations as coefficients of a power series solution. HPM requires a homotopy with an embedding parameter which is considered as a small parameter.

Findings

The paper extends the application and numerical comparison of the GDTM and HPM to obtain analytic and approximate solutions to the time‐fractional Burgers and coupled Burgers equations.

Research limitations/implications

Burgers and coupled Burgers equations with time‐fractional derivative used.

Practical implications

The implications include traffic flow, acoustic transmission, shocks, boundary layer, the steepening of the waves and fluids, thermal radiation, chemical reaction, gas dynamics and many other phenomena.

Originality/value

The numerical results demonstrate the significant features, efficiency and reliability of the two approaches. The results show that HPM is more promising, convenient, and computationally attractive than GDTM.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 22 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 12 April 2013

Mashallah Matinfar, Mostafa Eslami and Mohammad Saeidy

The purpose of this paper is to introduce a new homotopy perturbation method (NHPM) to solve Cauchy problem of unidimensional non‐linear diffusion equation.

Abstract

Purpose

The purpose of this paper is to introduce a new homotopy perturbation method (NHPM) to solve Cauchy problem of unidimensional non‐linear diffusion equation.

Design/methodology/approach

In this paper a modified version of HPM, which the authors call NHPM, has been presented; this technique performs much better than the HPM. HPM and NHPM start by considering a homotopy, and the solution of the problem under study is assumed to be as the summation of a power series in p, the difference between two methods starts from the form of initial approximation of the solution.

Findings

In this article, the authors have applied the NHPM for solving nonlinear Cauchy diffusion equation. In comparison with the homotopy perturbation method (HPM), in the present method, the authors achieve exact solutions while HPM does not lead to exact solutions. The authors believe that the new method is a promising technique in finding the exact solutions for a wide variety of mathematical problems.

Originality/value

The basic idea described in this paper is expected to be further employed to solve other functional equations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 23 no. 3
Type: Research Article
ISSN: 0961-5539

Keywords

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