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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…

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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

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Article
Publication date: 22 March 2013

Ahmet Bekir and Esin Aksoy

The purpose of this paper is to apply the exp‐function method to construct exact solutions of nonlinear wave equations. The proposed technique is tested on the (2+1) and…

Abstract

Purpose

The purpose of this paper is to apply the exp‐function method to construct exact solutions of nonlinear wave equations. The proposed technique is tested on the (2+1) and (3+1) dimensional extended shallow water wave equations. These equations play a very important role in mathematical physics and engineering sciences.

Design/methodology/approach

In this paper, the authors apply the exp‐function method to construct exact solutions of nonlinear wave equations.

Findings

In total, four forms of the extended shallow water wave equation have been studied, from the point of view of its exact solutions using computational method. Exp‐function method was employed to achieve the goal set for this work. The applied method will be used in further works to establish more entirely new solutions for other kinds of nonlinear wave equations. Finally, it is worthwhile to mention that the proposed method is straightforward, concise, and it is a promising and powerful new method for other nonlinear wave equations in mathematical physics.

Originality/value

The algorithm suggested in the paper is quite efficient and is practically well suited for use in these problems. The method is straightforward and concise, and its applications are promising.

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Article
Publication date: 2 November 2015

Zhijuan Jia, Mingsheng Hu, Qiaoling Chen and Suimin Jai

The fractional complex transform is used to convert the fractional differential equation to its differential partner and the exp-function method is to solve the resultant…

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164

Abstract

Purpose

The fractional complex transform is used to convert the fractional differential equation to its differential partner and the exp-function method is to solve the resultant equation. The exact solutions for the equation are successfully established. The paper aims to discuss these issues.

Design/methodology/approach

Use the chain rule of the local fractional derivative and the exp-function method.

Findings

Some new exact solutions for the fractional differential equation are successfully established, and the process of the solution is extremely simple and remarkably accessible.

Originality/value

The fractional complex transform is used to convert the fractional differential equation to its differential partner and the exp-function method is to solve the resultant equation.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 25 no. 8
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 7 April 2015

Xiao-rong Kang, Xian Daquan and Zhengde Dai

The purpose of this paper is to find new non-traveling wave solutions and study its localized structure of Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation.

Abstract

Purpose

The purpose of this paper is to find new non-traveling wave solutions and study its localized structure of Caudrey-Dodd-Gibbon-Kotera-Sawada (CDGKS) equation.

Design/methodology/approach

The authors apply the Lie group method twice and combine with the Exp-function method and Riccati equation mapping method to the (2+1)-dimensional CDGKS equation.

Findings

The authors have obtained some new non-traveling wave solutions with two arbitrary functions of time variable.

Research limitations/implications

As non-linear evolution equations is characterized by rich dynamical behavior, the authors just found some of them and others still to be found.

Originality/value

These results may help the authors to investigate some new localized structure and the interaction of waves in high-dimensional models. The new non-traveling wave solutions with two arbitrary functions of time variable are obtained for CDGKS equation using Lie group approach twice and combining with the Exp-function method and Riccati equation mapping method by the aid of Maple.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 25 no. 3
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 9 August 2011

Mehdi Dehghan, Jalil Manafian Heris and Abbas Saadatmandi

The purpose of this paper is to use He's Exp‐function method (EFM) to construct solitary and soliton solutions of the nonlinear evolution equation.

Abstract

Purpose

The purpose of this paper is to use He's Exp‐function method (EFM) to construct solitary and soliton solutions of the nonlinear evolution equation.

Design/methodology/approach

This technique is straightforward and simple to use and is a powerful method to overcome some difficulties in the nonlinear problems.

Findings

This method is developed for searching exact traveling wave solutions of the nonlinear partial differential equations. The EFM presents a wider applicability for handling nonlinear wave equations.

Originality/value

The paper shows that EFM, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving nonlinear evolution equations. Application of EFM to Fitzhugh‐Nagumo equation illustrates its effectiveness.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 21 no. 6
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 27 May 2020

Ji-Huan He, Fei-Yu Ji and Hamid Mohammad-Sedighi

The purpose of this paper is to demonstrate that the numerical method is not everything for nonlinear equations. Some properties cannot be revealed numerically; an example…

Abstract

Purpose

The purpose of this paper is to demonstrate that the numerical method is not everything for nonlinear equations. Some properties cannot be revealed numerically; an example is used to elucidate the fact.

Design/methodology/approach

A variational principle is established for the generalized KdV – Burgers equation by the semi-inverse method, and the equation is solved analytically by the exp-function method, and some exact solutions are obtained, including blowup solutions and discontinuous solutions. The solution morphologies are studied by illustrations using different scales.

Findings

Solitary solution is the basic property of nonlinear wave equations. This paper finds some new properties of the KdV–Burgers equation, which have not been reported in open literature and cannot be effectively elucidated by numerical methods. When the solitary solution or the blowup solution is observed on a much small scale, their discontinuous property is first found.

Originality/value

The variational principle can explain the blowup and discontinuous properties of a nonlinear wave equation, and the exp-function method is a good candidate to reveal the solution properties.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 31 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 5 January 2015

Ghodrat Ebadi, Aida Mojaver, Sachin Kumar and Anjan Biswas

The purpose of this paper is to discuss the integrability studies to the long-short wave equation that is studied in the context of shallow water waves. There are several…

Abstract

Purpose

The purpose of this paper is to discuss the integrability studies to the long-short wave equation that is studied in the context of shallow water waves. There are several integration tools that are applied to obtain the soliton and other solutions to the equation. The integration techniques are traveling waves, exp-function method, G′/G-expansion method and several others.

Design/methodology/approach

The design of the paper is structured with an introduction to the model. First the traveling wave hypothesis approach leads to the waves of permanent form. This eventually leads to the formulation of other approaches that conforms to the expected results.

Findings

The findings are a spectrum of solutions that lead to the clearer understanding of the physical phenomena of long-short waves. There are several constraint conditions that fall out naturally from the solutions. These poses the restrictions for the existence of the soliton solutions.

Originality/value

The results are new and are sharp with Lie symmetry analysis and other advanced integration techniques in place. These lead to the connection between these integration approaches.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 25 no. 1
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 7 November 2016

Xiao-rong Kang and Xian Daquan

The purpose of this paper is to find out some new rational non-traveling wave solutions and to study localized structures for (2+1)-dimensional Ablowitz-Kaup-Newell-Segur…

Abstract

Purpose

The purpose of this paper is to find out some new rational non-traveling wave solutions and to study localized structures for (2+1)-dimensional Ablowitz-Kaup-Newell-Segur (AKNS) equation.

Design/methodology/approach

Along with some special transformations, the Lie group method and the rational function method are applied to the (2+1)-dimensional AKNS equation.

Findings

Some new non-traveling wave solutions are obtained, including generalized rational solutions with two arbitrary functions of time variable.

Research limitations/implications

As a typical nonlinear evolution equation, some dynamical behaviors are also discussed.

Originality/value

With the help of the Lie group method, special transformations and the rational function method, new non-traveling wave solutions are derived for the AKNS equation by Maple software. These results are much useful for investigating some new localized structures and the interaction of waves in high-dimensional models, and enrich dynamical features of solutions for the higher dimensional systems.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 26 no. 8
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 6 March 2020

Ji-Huan He

This paper aims to review some effective methods for fully fourth-order nonlinear integral boundary value problems with fractal derivatives.

Abstract

Purpose

This paper aims to review some effective methods for fully fourth-order nonlinear integral boundary value problems with fractal derivatives.

Design/methodology/approach

Boundary value problems arise everywhere in engineering, hence two-scale thermodynamics and fractal calculus have been introduced. Some analytical methods are reviewed, mainly including the variational iteration method, the Ritz method, the homotopy perturbation method, the variational principle and the Taylor series method. An example is given to show the simple solution process and the high accuracy of the solution.

Findings

An elemental and heuristic explanation of fractal calculus is given, and the main solution process and merits of each reviewed method are elucidated. The fractal boundary value problem in a fractal space can be approximately converted into a classical one by the two-scale transform.

Originality/value

This paper can be served as a paradigm for various practical applications.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 30 no. 11
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 26 September 2019

Ji-Huan He

A three-dimensional (3D) unsteady potential flow might admit a variational principle. The purpose of this paper is to adopt a semi-inverse method to search for the

Abstract

Purpose

A three-dimensional (3D) unsteady potential flow might admit a variational principle. The purpose of this paper is to adopt a semi-inverse method to search for the variational formulation from the governing equations.

Design/methodology/approach

A suitable trial functional with a possible unknown function is constructed, and the identification of the unknown function is given in detail. The Lagrange multiplier method is used to establish a generalized variational principle, but in vain.

Findings

Some new variational principles are obtained, and the semi-inverse method can easily overcome the Lagrange crisis.

Practical implications

The semi-inverse method sheds a promising light on variational theory, and it can replace the Lagrange multiplier method for the establishment of a generalized variational principle. It can be used for the establishment of a variational principle for fractal and fractional calculus.

Originality/value

This paper establishes some new variational principles for the 3D unsteady flow and suggests an effective method to eliminate the Lagrange crisis.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 30 no. 3
Type: Research Article
ISSN: 0961-5539

Keywords

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