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Article
Publication date: 6 January 2012

Precious Sibanda, Sandile Motsa and Zodwa Makukula

The purpose of this paper is to study the steady laminar flow of a pressure driven third‐grade fluid with heat transfer in a horizontal channel. The study serves two…

Abstract

Purpose

The purpose of this paper is to study the steady laminar flow of a pressure driven third‐grade fluid with heat transfer in a horizontal channel. The study serves two purposes: to correct the inaccurate results presented in Siddiqui et al., where the homotopy perturbation method was used, and to demonstrate the computational efficiency and accuracy of the spectral‐homotopy analysis methods (SHAM and MSHAM) in solving problems that arise in fluid mechanics.

Design/methodology/approach

Exact and approximate analytical series solutions of the non‐linear equations that govern the flow of a steady laminar flow of a third grade fluid through a horizontal channel are constructed using the homotopy analysis method and two new modifications of this method. These solutions are compared to the full numerical results. A new method for calculating the optimum value of the embedded auxiliary parameter ∼ is proposed.

Findings

The “standard” HAM and the two modifications of the HAM (the SHAM and the MSHAM) lead to faster convergence when compared to the homotopy perturbation method. The paper shows that when the same initial approximation is used, the HAM and the SHAM give identical results. Nonetheless, the advantage of the SHAM is that it eliminates the restriction of searching for solutions to the nonlinear equations in terms of prescribed solution forms that conform to the rule of solution expression and the rule of coefficient ergodicity. In addition, an alternative and more efficient implementation of the SHAM (referred to as the MSHAM) converges much faster, and for all parameter values.

Research limitations/implications

The spectral modification of the homotopy analysis method is a new procedure that has been shown to work efficiently for fluid flow problems in bounded domains. It however remains to be generalized and verified for more complicated nonlinear problems.

Originality/value

The spectral‐HAM has already been proposed and implemented by the authors in a recent paper. This paper serves the purpose of verifying and demonstrating the utility of the new spectral modification of the HAM in solving problems that arise in fluid mechanics. The MSHAM is a further modification of the SHAM to speed up converge and to allow for convergence for a much wider range of system parameter values. The utility of these methods has not been tested and verified for systems of nonlinear equations. For this reason as much emphasis has been placed on proving the reliability and validity of the solution techniques as on the physics of the problem.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 22 no. 1
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 3 May 2013

A. Jabbari, H. Kheiri and A. Yildirim

The purpose of this paper is to obtain analytic solutions of the (1+1) and (2+1)‐dimensional dispersive long wave equations by the homotopy analysis and the homotopy Padé methods.

Abstract

Purpose

The purpose of this paper is to obtain analytic solutions of the (1+1) and (2+1)‐dimensional dispersive long wave equations by the homotopy analysis and the homotopy Padé methods.

Design/methodology/approach

The obtained approximation by using homotopy method contains an auxiliary parameter which is a simple way to control and adjust the convergence region and rate of solution series.

Findings

The approximation solutions by [m,m] homotopy Padé technique is often independent of auxiliary parameter ℏ and this technique accelerates the convergence of the related series.

Originality/value

In this paper, analytic solutions of the (1+1) and (2+1)‐dimensional dispersive long wave equations are obtained by the homotopy analysis and the homotopy Padé methods. The obtained approximation by using homotopy method contains an auxiliary parameter which is a simple way to control and adjust the convergence region and rate of solution series. The approximation solutions by [m,m] homotopy Padé technique are often independent of auxiliary parameter ℏ and this technique accelerates the convergence of the related series.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 23 no. 4
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 14 September 2012

Syed Tauseef Mohyud‐Din, Ahmet Yıldırım and Eda Yülüklü

The purpose of this paper is to present numerical solutions for the space‐ and time‐fractional Korteweg‐de Vries (KdV) equation using homotopy analysis method (HAM). The…

Abstract

Purpose

The purpose of this paper is to present numerical solutions for the space‐ and time‐fractional Korteweg‐de Vries (KdV) equation using homotopy analysis method (HAM). The space and time‐fractional derivatives are described in the Caputo sense. The paper witnesses the extension of HAM for fractional KdV equations.

Design/methodology/approach

This paper presents numerical solutions for the space‐ and time‐fractional KdV equation using HAM. The space and time‐fractional derivatives are described in the Caputo sense.

Findings

In this paper, the application of homotopy analysis method was extended to obtain explicit and numerical solutions of the time‐ and space‐fractional KdV equation with initial conditions. The homotopy analysis method was clearly a very efficient and powerful technique in finding the solutions of the proposed equations.

Originality/value

In this paper, the application of HAM was extended to obtain explicit and numerical solutions of the time‐ and space‐fractional KdV equation with initial conditions. The HAM was clearly very efficient and powerful technique in finding the solutions of the proposed equations. The obtained results demonstrate the reliability of the algorithm and its wider applicability to fractional nonlinear evolution equations. Finally, the recent appearance of nonlinear fractional differential equations as models in some fields such as the thermal diffusion in fractal media makes it necessary to investigate the method of solutions for such equations and the authors hope that this paper is a step in this direction.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 22 no. 7
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 25 February 2014

Hany Hassan and Mohammad Mehdi Rashidi

– The paper aims to find an accurate analytic solution (series solution) for the micropolar flow in a porous channel with mass injection for different values of Reynolds number.

Abstract

Purpose

The paper aims to find an accurate analytic solution (series solution) for the micropolar flow in a porous channel with mass injection for different values of Reynolds number.

Design/methodology/approach

In this paper, the homotopy analysis method (HAM) with different numbers of unknown convergence-control parameters has been used to derive accurate analytic solution for micropolar flow in a porous channel with mass injection. The possible optimal value of convergence-control parameter determined by minimizing the averaged residual error.

Findings

The results obtained from HAM solution with two parameters are compared with numerical results and that obtained from HAM solution with only one parameter. The results show that this method gives an analytical solution with high order of accuracy with a few iterations. As shown in this paper, by minimizing the averaged residual error, the authors can get the possible optimal value of the convergence-control parameters which may give the fastest convergent series.

Practical implications

The HAM with different numbers of unknown convergence-control parameters can be used to obtain analytic solutions for many problems in sciences and engineering.

Originality/value

This paper fulfils an identified need to evaluate the accurate analytic solution (series solution) of practical problem.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 24 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 18 April 2017

L. Ahmad Soltani, E. Shivanian and Reza Ezzati

The purpose of this paper is to present a new method based on the homotopy analysis method (HAM) with the aim of fast searching and calculating multiple solutions of…

Abstract

Purpose

The purpose of this paper is to present a new method based on the homotopy analysis method (HAM) with the aim of fast searching and calculating multiple solutions of nonlinear boundary value problems (NBVPs).

Design/methodology/approach

A major problem with the previously modified HAM, namely, predictor homotopy analysis method, which is used to predict multiplicity of solutions of NBVPs, is a time-consuming computation of high-order HAM-approximate solutions due to a symbolic variable namely “prescribed parameter”. The proposed new technique which is based on traditional shooting method, and the HAM cuts the dependency on the prescribed parameter.

Findings

To demonstrate the computational efficiency, the mentioned method is implemented on three important nonlinear exactly solvable differential equations, namely, the nonlinear MHD Jeffery–Hamel flow problem, the nonlinear boundary value problem arising in heat transfer and the strongly nonlinear Bratu problem.

Originality/value

The more high-order approximate solutions are computable, multiple solutions are easily searched and discovered and the more accurate solutions can be obtained depending on how nonhomogeneous boundary conditions are transcribed to the homogeneous boundary conditions.

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Article
Publication date: 22 March 2013

Behrouz Raftari, Heidar Khosravi and Ahmet Yildirim

The purpose of this paper is to obtain approximate analytical solution of the second order hyperbolic telegraph equation with initial conditions, by the homotopy analysis

Abstract

Purpose

The purpose of this paper is to obtain approximate analytical solution of the second order hyperbolic telegraph equation with initial conditions, by the homotopy analysis method (HAM).

Design/methodology/approach

The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of the series solutions.

Findings

Approximate analytical solution of the second order hyperbolic telegraph equation with initial conditions is obtained by the HAM. The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of the series solutions.

Originality/value

In this work, approximate analytical solution of the second order hyperbolic telegraph equation with initial conditions is obtained by the HAM. To show the efficiency of the present method, several examples are presented.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 23 no. 2
Type: Research Article
ISSN: 0961-5539

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Article
Publication date: 7 June 2013

F. Soltanian, Mehdi Dehghan and S.M. Karbassi

The main purpose of this paper is to find convenient methods to solve the differential‐algebraic equations which have great importance in various fields of science and engineering.

Abstract

Purpose

The main purpose of this paper is to find convenient methods to solve the differential‐algebraic equations which have great importance in various fields of science and engineering.

Design/methodology/approach

The paper applies a semi‐analytical approach, using both the homotopy analysis method (HAM) and the modified homotopy analysis method (MHAM) for finding the solution of linear and nonlinear DAEs.

Findings

The results show that the new modification can effectively reduce computational costs and accelerates the rapid convergence of the series solution.

Originality/value

Some high index DAEs are investigated to present a comparative study between the HAM and the MHAM.

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Article
Publication date: 26 October 2012

M.M. Rashidi, M. Keimanesh and S.C. Rajvanshi

The purpose of this study is to investigate the problem of pulsatile flow in a porous annulus for small Reynolds number.

Abstract

Purpose

The purpose of this study is to investigate the problem of pulsatile flow in a porous annulus for small Reynolds number.

Design/methodology/approach

The similarity transformation for the governing equations gives a system of nonlinear ordinary differential equations which are analytically solved by the homotopy analysis method (HAM). The analytic solutions of non‐linear differential equation are constructed in the series form. The convergence of the series solutions is carefully analyzed.

Findings

Graphical results are presented to investigate the influence of different parameters on the flow behavior. Comparison between the solutions obtained by the HAM and the numerical solution shows good agreement.

Originality/value

An analysis for study on flow of an incompressible viscous fluid in the region lying between two concentric porous cylinders, under the assumption that a periodic pressure gradient is imposed across the annulus and that there is a uniform small transfer across two walls is presented. The HAM solutions are obtained for the equations governing the fluid flow.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 22 no. 8
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 16 September 2013

Azizeh Jabbari, Hossein Kheiri and Ahmet Yildirim

– The purpose of this paper is to obtain analytic solutions of telegraph equation by the homotopy Padé method.

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Abstract

Purpose

The purpose of this paper is to obtain analytic solutions of telegraph equation by the homotopy Padé method.

Design/methodology/approach

The authors used Maple Package to calculate the solutions obtained from the homotopy Padé method.

Findings

The obtained approximation by using homotopy method contains an auxiliary parameter which is a simple way to control and adjust the convergence region and rate of solution series. The approximation solutions by [m, m] homotopy Padé technique are often independent of auxiliary parameter h and this technique accelerates the convergence of the related series. Finally, numerical results for some test problems with known solutions are presented and the numerical results are given to show the efficiency of the proposed techniques.

Originality/value

The paper is shown that homotopy Padé technique is a promising tool with accelerated convergence for complicated nonlinear differential equations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 23 no. 7
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 2 August 2013

Xindong Zhang, Leilei Wei, Bo Tang and Yinnian He

In this article, the authors aim to present the homotopy analysis method (HAM) for obtaining the approximate solutions of space‐time fractional differential equations with…

Abstract

Purpose

In this article, the authors aim to present the homotopy analysis method (HAM) for obtaining the approximate solutions of space‐time fractional differential equations with initial conditions.

Design/methodology/approach

The series solution is developed and the recurrence relations are given explicitly. The initial approximation can be determined by imposing the initial conditions.

Findings

The comparison of the HAM results with the exact solutions is made; the results reveal that the HAM is very effective and simple. The HAM contains the auxiliary parameter h, which provides a simple way to adjust and control the convergence region of series solution. Numerical examples demonstrate the effect of changing homotopy auxiliary parameter h on the convergence of the approximate solution. Also, they illustrate the effect of the fractional derivative orders a and b on the solution behavior.

Originality/value

The idea can be used to find the numerical solutions of other fractional differential equations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 23 no. 6
Type: Research Article
ISSN: 0961-5539

Keywords

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