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Article
Publication date: 16 October 2009

Ahmet Yıldırım and Yves Cherruault

The purpose of this paper is to introduce an efficient method for solving susceptible‐infected‐removed (SIR) epidemic model. A SIR model that monitors the temporal dynamics of a…

Abstract

Purpose

The purpose of this paper is to introduce an efficient method for solving susceptible‐infected‐removed (SIR) epidemic model. A SIR model that monitors the temporal dynamics of a childhood disease in the presence of preventive vaccine. The qualitative analysis reveals the vaccination reproductive number for disease control and eradication. It introduces homotopy perturbation method (HPM) to overcome these problems.

Design/methodology/approach

The paper considers HPM to solve differential system which describes SIR epidemic model. 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 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. One of the most remarkable features of the HPM is that usually just few perturbation terms are sufficient for obtaining a reasonably accurate solution.

Findings

HPM is employed to compute an approximation to the solution of the non‐linear system of differential equations governing the problem.

Originality/value

The paper is of value in presenting, via some tables and figures, some numerical experiments which resulted from applying new methods on test problem.

Details

Kybernetes, vol. 38 no. 9
Type: Research Article
ISSN: 0368-492X

Keywords

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 mechanics…

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

Article
Publication date: 2 November 2010

Ahmet Yıldırım and Alev Kelleci

This paper aims to directly extend the homotopy perturbation method to study the coupled Burgers equations with time‐ and space‐fractional derivatives.

Abstract

Purpose

This paper aims to directly extend the homotopy perturbation method to study the coupled Burgers equations with time‐ and space‐fractional derivatives.

Design/methodology/approach

The realistic numerical solutions were obtained in a form of rapidly convergent series with easily computable components.

Findings

The figures show the effectiveness and good accuracy of the proposed method.

Originality/value

The paper obtains realistic numerical solutions in a form of rapidly convergent series with easily computable components. It shows the effectiveness and good accuracy of the proposed method.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 20 no. 8
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 29 March 2011

Syed Tauseef Mohyud‐Din, Ahmet Yildirim and Gülseren Demirli

The purpose of this paper is to consider analytical solution of wave system in Rn with coupling controllers by using the homotopy perturbation method (HPM).

216

Abstract

Purpose

The purpose of this paper is to consider analytical solution of wave system in Rn with coupling controllers by using the homotopy perturbation method (HPM).

Design/methodology/approach

HPM is applied to the system of linear partial differential equations, i.e. the system of waves in the two‐dimensional version of system equations (1) and (2). This problem is motivated by an analogous problem in ordinary differential equations for coupled oscillators and has potential application in isolating a vibrating object from the outside disturbances. For example, rubber or rubber‐like materials can be used to either absorb or shield a structure from vibration. As an approximation, these materials can be modeled as distributed springs.

Findings

In this paper, HPM was used to obtain analytical solution of wave system in with coupling controllers. The method provides the solutions in the form of a series with easily computable terms. Unlike other common methods for solving any physical problem, linear or nonlinear, that requires linearization, discretization, perturbation, or unjustified assumptions that may slightly change the physics of the problem, the HPM finds approximate analytical solutions by using the initial conditions only.

Originality/value

The method proposed in this paper is very reliable and efficient and is being used quite extensively for diversified nonlinear problems of a physical nature. The algorithm is being used for the first time on such problems.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 21 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 19 April 2011

Syed Tauseef Mohyud‐Din, Ahmet Yıldırım and Selin Sarıaydın

The purpose of this paper is to obtain soliton solution of the Kaup‐Kupershmidt (KK) equation with initial condition. The most important feature of this method is to obtain the…

Abstract

Purpose

The purpose of this paper is to obtain soliton solution of the Kaup‐Kupershmidt (KK) equation with initial condition. The most important feature of this method is to obtain the solution without direct transformation.

Design/methodology/approach

In this paper, the homotopy perturbation method (HPM) is used for obtaining soliton solution of the KK equation. The numerical solutions are compared with the known analytical solutions. The results of numerical examples are presented and only a few terms are required to obtain accurate solutions. Results derived from this method are shown graphically.

Findings

The authors obtained the one soliton solution for the KK equation by HPM. The numerical results showed that this method is very accurate. The HPM provides a reliable technique that requires less work if compared with the traditional techniques and the method does not also require unjustified assumptions, linearization, discretization or perturbation. The HPM is very easily applied to both differential equations and linear or nonlinear differential systems.

Originality/value

The paper describes how the authors obtained one soliton solution for the KK equation by HPM. The numerical results presented show that this method is very accurate.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 21 no. 3
Type: Research Article
ISSN: 0961-5539

Keywords

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 numerical…

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

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.

290

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

Article
Publication date: 30 March 2012

Mohammad Madani, Yasir Khan, Mahdi Fathizadeh and Ahmet Yildirim

The purpose of this paper is to report the effect of radiation on flow of a magneto‐micropolar fluid past a continuously moving plate with suction and blowing.

Abstract

Purpose

The purpose of this paper is to report the effect of radiation on flow of a magneto‐micropolar fluid past a continuously moving plate with suction and blowing.

Design/methodology/approach

The governing equations are transformed into dimensionless nonlinear ordinary differential equations by similarity transformation. Analytical technique, namely the homotopy perturbation method (HPM) combining with Padé approximants and finite difference method, are used to solve dimensionless non‐linear ordinary differential equations. The skin friction coefficient and local Nusselt numbers are also calculated. Beside this, the comparison of the analytical solution with numerical solution is illustrated by the graphs for different values of dimensionless pertinent parameters.

Findings

The authors have studied laminar magneto‐micropolar flow in the presence of radiation by using HPM‐Padé and finite difference methods. Results obtained by HPM‐Padé are in excellent agreement with the results of numerical solution.

Originality/value

The HPM‐Padé is used in a direct way without using linearization, discritization or restrictive assumption. The authors have attempted to show the capabilities and wide‐range applications of the HPM‐Padé in comparison with the finite difference solution of magneto‐micropolar flow in the presence of radiation problem.

Details

Engineering Computations, vol. 29 no. 3
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 13 April 2012

Syed Tauseef Mohyud‐Din, Yasir Khan, Naeem Faraz and Ahmet Yıldırım

The purpose of this paper is to apply exp‐function method to construct generalized solitary and periodic solutions of Fitzhugh‐Nagumo equation, which plays a very important role…

193

Abstract

Purpose

The purpose of this paper is to apply exp‐function method to construct generalized solitary and periodic solutions of Fitzhugh‐Nagumo equation, which plays a very important role in mathematical physics and engineering sciences.

Design/methodology/approach

The authors apply exp‐function method to construct generalized solitary and periodic solutions of Fitzhugh‐Nagumo equation.

Findings

Numerical results clearly indicate the reliability and efficiency of the proposed exp‐function method. The suggested algorithm is quite efficient and is practically well suited for use in these problems.

Originality/value

In this paper, the authors applied the exp‐function method to obtain solutions of the Fitzhugh‐Nagumo equation and show that the exp‐function method gives more realistic solutions without disturbing the basic physics of the physical problems.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 22 no. 3
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 4 May 2012

Ahmet Yıldırım and Hüseyin Koçak

The purpose of this paper is to implement the variational iteration method and the homotopy perturbation method to give a rational approximation solution of the foam drainage…

Abstract

Purpose

The purpose of this paper is to implement the variational iteration method and the homotopy perturbation method to give a rational approximation solution of the foam drainage equation with time‐ and space‐fractional derivatives.

Design/methodology/approach

The fractional derivatives are described in the Caputo sense. In these schemes, the solution takes the form of a convergent series with easily computable components.

Findings

Numerical examples are given to demonstrate the effectiveness of the present methods.

Originality/value

Results show that the proposed schemes are very effective and convenient for solving linear and nonlinear fractional differential equations with high accuracy.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 22 no. 4
Type: Research Article
ISSN: 0961-5539

Keywords

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