Search results

1 – 10 of 98
To view the access options for this content please click here
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

To view the access options for this content please click here
Article
Publication date: 15 August 2019

Mehdi Dehghan, Mostafa Abbaszadeh, Amirreza Khodadadian and Clemens Heitzinger

The current paper aims to develop a reduced order discontinuous Galerkin method for solving the generalized Swift–Hohenberg equation with application in biological science…

Abstract

Purpose

The current paper aims to develop a reduced order discontinuous Galerkin method for solving the generalized Swift–Hohenberg equation with application in biological science and mechanical engineering. The generalized Swift–Hohenberg equation is a fourth-order PDE; thus, this paper uses the local discontinuous Galerkin (LDG) method for it.

Design/methodology/approach

At first, the spatial direction has been discretized by the LDG technique, as this process results in a nonlinear system of equations based on the time variable. Thus, to achieve more accurate outcomes, this paper uses an exponential time differencing scheme for solving the obtained system of ordinary differential equations. Finally, to decrease the used CPU time, this study combines the proper orthogonal decomposition approach with the LDG method and obtains a reduced order LDG method. The circular and rectangular computational domains have been selected to solve the generalized Swift–Hohenberg equation. Furthermore, the energy stability for the semi-discrete LDG scheme has been discussed.

Findings

The results show that the new numerical procedure has not only suitable and acceptable accuracy but also less computational cost compared to the local DG without the proper orthogonal decomposition (POD) approach.

Originality/value

The local DG technique is an efficient numerical procedure for solving models in the fluid flow. The current paper combines the POD approach and the local LDG technique to solve the generalized Swift–Hohenberg equation with application in the fluid mechanics. In the new technique, the computational cost and the used CPU time of the local DG have been reduced.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 29 no. 8
Type: Research Article
ISSN: 0961-5539

Keywords

To view the access options for this content please click here
Article
Publication date: 19 December 2018

Hui Wang and Tian-Tian Zhang

The purpose of this paper is to study stability analysis, solition solutions and Gaussian solitons of the generalized nonlinear Schrödinger equation with higher order…

Abstract

Purpose

The purpose of this paper is to study stability analysis, solition solutions and Gaussian solitons of the generalized nonlinear Schrödinger equation with higher order terms, which can be used to describe the propagation properties of optical soliton solutions.

Design/methodology/approach

The authors apply the ansatz method and the Hamiltonian system technique to find its bright, dark and Gaussian wave solitons and analyze its modulation instability analysis and stability analysis solution.

Findings

The results imply that the generalized nonlinear Schrödinger equation has bright, dark and Gaussian wave solitons. Meanwhile, the authors provide the graphical analysis of such solutions to better understand their dynamical behavior. Some constraint conditions are provided which can guarantee the existence of solitons. The authors analyze its modulation instability analysis and stability analysis solution.

Originality/value

These results may help us to further study the local structure and the interaction of solutions in generalized nonlinear Schrödinger -type equations. The authors hope that the results provided in this work can help enrich the dynamic behavior of the generalized nonlinear Schrödinger--type equations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 29 no. 3
Type: Research Article
ISSN: 0961-5539

Keywords

To view the access options for this content please click here
Article
Publication date: 3 December 2020

Na Liu

The purpose of this paper is to study the homoclinic breather waves, rogue waves and multi-soliton waves of the (2 + 1)-dimensional Mel’nikov equation, which describes an…

Abstract

Purpose

The purpose of this paper is to study the homoclinic breather waves, rogue waves and multi-soliton waves of the (2 + 1)-dimensional Mel’nikov equation, which describes an interaction of long waves with short wave packets.

Design/methodology/approach

The author applies the Hirota’s bilinear method, extended homoclinic test approach and parameter limit method to construct the homoclinic breather waves and rogue waves of the (2 + 1)-dimensional Mel’nikov equation. Moreover, multi-soliton waves are constructed by using the three-wave method.

Findings

The results imply that the (2 + 1)-dimensional Mel’nikov equation has breather waves, rogue waves and multi-soliton waves. Moreover, the dynamic properties of such solutions are displayed vividly by figures.

Research limitations/implications

This paper presents efficient methods to find breather waves, rogue waves and multi-soliton waves for nonlinear evolution equations.

Originality/value

The outcome suggests that the extreme behavior of the homoclinic breather waves yields the rogue waves. Moreover, the multi-soliton waves are constructed, including the new breather two-solitary and two-soliton solutions. Meanwhile, the dynamics of these solutions will greatly enrich the diversity of the dynamics of the (2 + 1)-dimensional Mel’nikov equation.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 31 no. 5
Type: Research Article
ISSN: 0961-5539

Keywords

To view the access options for this content please click here
Article
Publication date: 27 November 2018

Lakhveer Kaur and Abdul-Majid Wazwaz

The purpose of this paper is to explore new reduced form of the (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili (BKP) equation by considering its bilinear…

Abstract

Purpose

The purpose of this paper is to explore new reduced form of the (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili (BKP) equation by considering its bilinear equations, derived from connection between the Hirota’s transformation and Bell polynomials.

Design/methodology/approach

Based on the bilinear form of new reduced form of the (3 + 1)-dimensional generalized BKP equation, lump solutions with sufficient and necessary conditions to guarantee analyticity and rational localization of the solutions are discovered. Also, extended homoclinic approach is applied to considered equation for finding solitary wave solutions.

Findings

A class of the bright-dark lump waves are fabricated for studying different attributes of (3 + 1)-dimensional generalized BKP equation and some new exact solutions including kinky periodic solitary wave solutions and line breathers periodic are also obtained by Following the extended homoclinic approach.

Research limitations/implications

The paper presents that the implemented methods have emerged as a promising and robust mathematical tool to manage (3 + 1)-dimensional generalized BKP equation by using the Hirota’s bilinear equation.

Practical implications

By considering important characteristics of lump and solitary wave solutions, one can understand the shapes, amplitudes and velocities of solitons after the collision with another soliton.

Social implications

The analysis of these higher-dimensional nonlinear wave equations is not only of fundamental interest but also has important practical implications in many areas of mathematical physics and ocean engineering.

Originality/value

To the best of the authors’ knowledge, the acquired solutions given in various cases have not been reported for new reduced form of the (3 + 1)-dimensional generalized BKP equation in the literature. These obtained solutions are advantageous for researchers to know objective laws and grab the indispensable features of the development of the mathematical physics.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 29 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

To view the access options for this content please click here
Article
Publication date: 1 February 2021

Shou-Fu Tian, Xiao-Fei Wang, Tian-Tian Zhang and Wang-Hua Qiu

The purpose of this paper is to study the stability analysis and optical solitary wave solutions of a (2 + 1)-dimensional nonlinear Schrödinger equation, which are derived…

Abstract

Purpose

The purpose of this paper is to study the stability analysis and optical solitary wave solutions of a (2 + 1)-dimensional nonlinear Schrödinger equation, which are derived from a multicomponent plasma with nonextensive distribution.

Design Methodology Approach

Based on the ansatz and sub-equation theories, the authors use a direct method to find stability analysis and optical solitary wave solutions of the (2 + 1)-dimensional equation.

Findings

By considering the ansatz method, the authors successfully construct the bright and dark soliton solutions of the equation. The sub-equation method is also extended to find its complexitons solutions. Moreover, the explicit power series solution is also derived with its convergence analysis. Finally, the influences of each parameter on these solutions are discussed via graphical analysis.

Originality Value

The dynamics of these solutions are analyzed to enrich the diversity of the dynamics of high-dimensional nonlinear Schrödinger equation type nonlinear wave fields.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 31 no. 5
Type: Research Article
ISSN: 0961-5539

Keywords

To view the access options for this content please click here
Article
Publication date: 1 August 2016

Abdul-Majid Wazwaz

The purpose of this paper is to concern with a reliable treatment of the (2+1)-dimensional and the (3+1)-dimensional logarithmic Boussinesq equations (BEs). The author…

Abstract

Purpose

The purpose of this paper is to concern with a reliable treatment of the (2+1)-dimensional and the (3+1)-dimensional logarithmic Boussinesq equations (BEs). The author uses the sense of the Gaussian solitary waves to determine these gaussons. The study confirms that models characterized by logarithmic nonlinearity give gaussons solitons of distinct physical structures.

Design/methodology/approach

The proposed technique, as presented in this work has been shown to be very efficient for solving nonlinear equations with logarithmic nonlinearity.

Findings

The (2+1) and the (3+1)-dimensional BEs were examined as well. The examined models feature interesting results in propagation of waves and fluid flow.

Research limitations/implications

The paper presents a new efficient algorithm for the higher dimensional logarithmic BEs.

Practical implications

The work shows the effect of logarithmic nonlinearity compared to other nonlinearities where standard solitons appear in the last case.

Social implications

The work will benefit audience who are willing to examine the effect of logarithmic nonlinearity.

Originality/value

The paper presents a new efficient algorithm for the higher dimensional logarithmic BEs.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 26 no. 6
Type: Research Article
ISSN: 0961-5539

Keywords

To view the access options for this content please click here
Article
Publication date: 6 August 2021

Yasir Khan

Nizhnik–Novikov–Veselov system (NNVS) is a well-known isotropic extension of the Lax (1 + 1) dimensional Korteweg-deVries equation that is also used as a paradigm for an…

Abstract

Purpose

Nizhnik–Novikov–Veselov system (NNVS) is a well-known isotropic extension of the Lax (1 + 1) dimensional Korteweg-deVries equation that is also used as a paradigm for an incompressible fluid. The purpose of this paper is to present a fractal model of the NNVS based on the Hausdorff fractal derivative fundamental concept.

Design/methodology/approach

A two-scale transformation is used to convert the proposed fractal model into regular NNVS. The variational strategy of well-known Chinese scientist Prof. Ji Huan He is used to generate bright and exponential soliton solutions for the proposed fractal system.

Findings

The NNV fractal model and its variational principle are introduced in this paper. Solitons are created with a variety of restriction interactions that must all be applied equally. Finally, the three-dimensional diagrams are displayed using an appropriate range of physical parameters. The results of the solitary solutions demonstrated that the suggested method is very accurate and effective. The proposed methodology is extremely useful and nearly preferable for use in such problems.

Practical implications

The research study of the soliton theory has already played a pioneering role in modern nonlinear science. It is widely used in many natural sciences, including communication, biology, chemistry and mathematics, as well as almost all branches of physics, including nonlinear optics, plasma physics, fluid dynamics, condensed matter physics and field theory, among others. As a result, while constructing possible soliton solutions to a nonlinear NNV model arising from the field of an incompressible fluid is a popular topic, solving nonlinear fluid mechanics problems is significantly more difficult than solving linear ones.

Originality/value

To the best of the authors’ knowledge, for the first time in the literature, this study presents Prof. Ji Huan He's variational algorithm for finding and studying solitary solutions of the fractal NNV model. The reported solutions are novel and present a valuable addition to the literature in soliton theory.

Details

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

Keywords

To view the access options for this content please click here
Article
Publication date: 11 October 2018

Lian-Li Feng and Tian-Tian Zhang

The purpose of this paper is to find homoclinic breather waves, rogue waves and soliton waves for a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation…

Abstract

Purpose

The purpose of this paper is to find homoclinic breather waves, rogue waves and soliton waves for a (3 + 1)-dimensional generalized Kadomtsev–Petviashvili (gKP) equation, which can be used to describe the propagation of weakly nonlinear dispersive long waves on the surface of a fluid.

Design/methodology/approach

The authors apply the extended Bell polynomial approach, Hirota’s bilinear method and the homoclinic test technique to find the rogue waves, homoclinic breather waves and soliton waves of the (3 + 1)-dimensional gKP equation.

Findings

The results imply that the gKP equation admits rogue waves, homoclinic breather waves and soliton waves. Moreover, the authors also find that rogue waves can come from the extreme behavior of the breather solitary wave. The authors analyze the propagation and interaction properties of these solutions to better understand the dynamic behavior of these solutions.

Originality/value

These results may help us to further study the local structure and the interaction of waves in KP-type equations. It is hoped that the results can help enrich the dynamic behavior of such equations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 29 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

To view the access options for this content please click here
Article
Publication date: 12 May 2021

Abdul-Majid Wazwaz

This study aims to develop two integrable shallow water wave equations, of higher-dimensions, and with constant and time-dependent coefficients, respectively. The author…

Abstract

Purpose

This study aims to develop two integrable shallow water wave equations, of higher-dimensions, and with constant and time-dependent coefficients, respectively. The author derives multiple soliton solutions and a class of lump solutions which are rationally localized in all directions in space.

Design/methodology/approach

The author uses the simplified Hirota’s method and lump technique for determining multiple soliton solutions and lump solutions as well. The author shows that the developed (2+1)- and (3+1)-dimensional models are completely integrable in in the Painlené sense.

Findings

The paper reports new Painlevé-integrable extended equations which belong to the shallow water wave medium.

Research limitations/implications

The author addresses the integrability features of this model via using the Painlevé analysis. The author reports multiple soliton solutions for this equation by using the simplified Hirota’s method.

Practical implications

The obtained lump solutions include free parameters; some parameters are related to the translation invariance and the other parameters satisfy a non-zero determinant condition.

Social implications

The work presents useful algorithms for constructing new integrable equations and for the determination of lump solutions.

Originality/value

The paper presents an original work with newly developed integrable equations and shows useful findings of solitary waves and lump solutions.

Details

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

Keywords

1 – 10 of 98