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Article
Publication date: 3 June 2019

Abdul-Majid Wazwaz

The purpose of this paper is concerned with developing two integrable Korteweg de-Vries (KdV) equations of third- and fifth-orders; each possesses time-dependent…

Abstract

Purpose

The purpose of this paper is concerned with developing two integrable Korteweg de-Vries (KdV) equations of third- and fifth-orders; each possesses time-dependent coefficients. The study shows that multiple soliton solutions exist and multiple complex soliton solutions exist for these two equations.

Design/methodology/approach

The integrability of each of the developed models has been confirmed by using the Painlev´e analysis. The author uses the complex forms of the simplified Hirota’s method to obtain two fundamentally different sets of solutions, multiple real and multiple complex soliton solutions for each model.

Findings

The time-dependent KdV equations feature interesting results in propagation of waves and fluid flow.

Research limitations/implications

The paper presents a new efficient algorithm for constructing time-dependent integrable equations.

Practical implications

The author develops two time-dependent integrable KdV equations of third- and fifth-order. These models represent more specific data than the constant equations. The author showed that integrable equation gives real and complex soliton solutions.

Social implications

The work presents useful findings in the propagation of waves.

Originality/value

The paper presents a new efficient algorithm for constructing time-dependent integrable equations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 29 no. 6
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 25 June 2019

Abdul-Majid Wazwaz and Gui-Qiong Xu

The purpose of this paper is to develop a new time-dependent KdV6 equation. The authors derive multiple soliton solutions and multiple complex soliton solutions for a…

Abstract

Purpose

The purpose of this paper is to develop a new time-dependent KdV6 equation. The authors derive multiple soliton solutions and multiple complex soliton solutions for a time-dependent equation.

Design/methodology/approach

The newly developed time-dependent model has been handled by using the Hirota’s direct method. The authors also use the complex Hirota’s criteria for deriving multiple complex soliton solutions.

Findings

The examined extension of the KdV6 model exhibits complete integrability for any analytic time-dependent coefficient.

Research limitations/implications

The paper presents a new efficient algorithm for constructing extended models which give a variety of multiple real and complex soliton solutions.

Practical implications

The paper introduced a new time-dependent KdV6 equation, where integrability is emphasized for any analytic time-dependent function.

Social implications

The findings are new and promising. Multiple real and multiple complex soliton solutions were formally derived.

Originality/value

This is an entirely new work where a new time-dependent KdV6 equation is established. This is the first time that the KdV6 equation is examined as a time-dependent equation. Moreover, the complete integrability of this newly developed equation is emphasized via using Painlevé test.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 29 no. 11
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 26 January 2022

Abdul-Majid Wazwaz

This study aims to introduce a variety of integrable Boussinesq equations with distinct dimensions.

Abstract

Purpose

This study aims to introduce a variety of integrable Boussinesq equations with distinct dimensions.

Design/methodology/approach

The author formally uses the simplified Hirota’s method and lump schemes for exploring lump solutions, which are rationally localized in all directions in space.

Findings

The author confirms the lump solutions for every model illustrated by some graphical representations.

Research limitations/implications

The author examines the features of the obtained lumps solutions.

Practical implications

The author presents a variety of lump solutions via using a variety of numerical values of the included parameters.

Social implications

This study formally furnishes useful algorithms for using symbolic computation with Maple for the determination of lump solutions.

Originality/value

This paper introduces an original work with newly useful findings of lump solutions.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 32 no. 9
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 9 August 2021

Abdul-Majid Wazwaz

This paper aims to introduce a new (3 + 1)-dimensional fourth-order integrable equation characterized by second-order derivative in time t. The new equation models both…

Abstract

Purpose

This paper aims to introduce a new (3 + 1)-dimensional fourth-order integrable equation characterized by second-order derivative in time t. The new equation models both right- and left-going waves in a like manner to the Boussinesq equation.

Design/methodology/approach

This formally uses the simplified Hirota’s method and lump schemes for determining multiple soliton solutions and lump solutions, which are rationally localized in all directions in space.

Findings

This paper confirms the complete integrability of the newly developed (3 + 1)-dimensional model in the Painevé sense.

Research limitations/implications

This paper addresses the integrability features of this model via using the Painlevé analysis.

Practical implications

This paper presents a variety of lump solutions via using a variety of numerical values of the included parameters.

Social implications

This work formally furnishes useful algorithms for extending integrable equations and for the determination of lump solutions.

Originality/value

To the best of the author’s knowledge, this paper introduces an original work with newly developed integrable equation and shows useful findings of solitons and lump solutions.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 32 no. 5
Type: Research Article
ISSN: 0961-5539

Keywords

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

Article
Publication date: 6 February 2017

Mohammad Heydari, Ghasem Barid Loghmani and Abdul-Majid Wazwaz

The main purpose of this paper is to implement the piecewise spectral-variational iteration method (PSVIM) to solve the nonlinear Lane-Emden equations arising in…

Abstract

Purpose

The main purpose of this paper is to implement the piecewise spectral-variational iteration method (PSVIM) to solve the nonlinear Lane-Emden equations arising in mathematical physics and astrophysics.

Design/methodology/approach

This method is based on a combination of Chebyshev interpolation and standard variational iteration method (VIM) and matching it to a sequence of subintervals. Unlike the spectral method and the VIM, the proposed PSVIM does not require the solution of any linear or nonlinear system of equations and analytical integration.

Findings

Some well-known classes of Lane-Emden type equations are solved as examples to demonstrate the accuracy and easy implementation of this technique.

Originality/value

In this paper, a new and efficient technique is proposed to solve the nonlinear Lane-Emden equations. The proposed method overcomes the difficulties arising in calculating complicated and time-consuming integrals and terms that are not needed in the standard VIM.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 27 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 11 January 2021

Abdul-Majid Wazwaz

This paper aims to develop a new (3 + 1)-dimensional Painlevée-integrable extended Sakovich equation. This paper formally derives multiple soliton solutions for this…

Abstract

Purpose

This paper aims to develop a new (3 + 1)-dimensional Painlevée-integrable extended Sakovich equation. This paper formally derives multiple soliton solutions for this developed model.

Design/methodology/approach

This paper uses the simplified Hirota’s method for deriving multiple soliton solutions.

Findings

This paper finds that the developed (3 + 1)-dimensional Sakovich model exhibits complete integrability in analogy with the standard Sakovich equation.

Research limitations/implications

This paper addresses the integrability features of this model via using the Painlevée analysis. This paper reports multiple soliton solutions for this equation by using the simplified Hirota’s method.

Practical implications

The study reports three non-linear terms added to the standard Sakovich equation.

Social implications

The study presents useful algorithms for constructing new integrable equations and for handling these equations.

Originality/value

The paper reports a new Painlevée-integrable extended Sakovich equation, which belongs to second-order partial differential equations. The constructed model does not contain any dispersion term such as uxxx.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 31 no. 9
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 17 June 2020

Gangwei Wang and Abdul-Majid Wazwaz

The purpose of this paper is to concern with introducing symmetry analysis to the extended Sakovich equation.

Abstract

Purpose

The purpose of this paper is to concern with introducing symmetry analysis to the extended Sakovich equation.

Design/methodology/approach

The newly developed Sakovich equation has been handled by using the Lie symmetries via using the Lie group method.

Findings

The developed extended Sakovich model exhibit symmetries and invariant solutions.

Research limitations/implications

The present study is to address the two main motivations: the study of symmetry analysis and the study of soliton solutions of the extended Sakovich equation.

Practical implications

The work introduces symmetry analysis to the Painlevé-integrable extended Sakovich equation.

Social implications

The work presents useful symmetry algorithms for handling new integrable equations.

Originality/value

The paper presents an original work with symmetry analysis and shows useful findings.

Details

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

Keywords

Article
Publication date: 25 August 2022

Abdul-Majid Wazwaz, Wedad Albalawi and Samir A. El-Tantawy

The purpose of this paper is to study an extended hierarchy of nonlinear evolution equations including the sixth-order dispersion Korteweg–de Vries (KdV6), eighth-order…

Abstract

Purpose

The purpose of this paper is to study an extended hierarchy of nonlinear evolution equations including the sixth-order dispersion Korteweg–de Vries (KdV6), eighth-order dispersion KdV (KdV8) and many other related equations.

Design/methodology/approach

The newly developed models have been handled using the simplified Hirota’s method, whereas multiple soliton solutions are furnished using Hirota’s criteria.

Findings

The authors show that every member of this hierarchy is characterized by distinct dispersion relation and distinct resonance branches, whereas the phase shift retains the KdV type of shifts for any member.

Research limitations/implications

This paper presents an efficient algorithm for handling a hierarchy of integrable equations of diverse orders.

Practical implications

Multisoliton solutions are derived for each member of the hierarchy, and then generalized for any higher-order model.

Social implications

This work presents useful algorithms for finding and studying integrable equations of a hierarchy of nonlinear equations. The developed models exhibit complete integrability, by investigating the compatibility conditions for each model.

Originality/value

This paper presents an original work with a variety of useful findings.

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

Article
Publication date: 4 September 2019

Abdul-Majid Wazwaz

The purpose of this paper is concerned with investigating three integrable shallow water waves equations with time-dependent coefficients. The author obtains multiple…

Abstract

Purpose

The purpose of this paper is concerned with investigating three integrable shallow water waves equations with time-dependent coefficients. The author obtains multiple soliton solutions and multiple complex soliton solutions for these three models.

Design/methodology/approach

The newly developed equations with time-dependent coefficients have been handled by using Hirota’s direct method. The author also uses the complex Hirota’s criteria for deriving multiple complex soliton solutions.

Findings

The developed integrable models exhibit complete integrability for any analytic time-dependent coefficients defined though compatibility conditions.

Research limitations/implications

The paper presents an efficient algorithm for handling time-dependent integrable equations with analytic time-dependent coefficients.

Practical implications

This study introduces three new integrable shallow water waves equations with time-dependent coefficients. These models represent more specific data than the related equations with constant coefficients. The author shows that integrable equations with time-dependent coefficients give real and complex soliton solutions.

Social implications

The paper presents useful algorithms for finding integrable equations with time-dependent coefficients.

Originality/value

The paper presents an original work with a variety of useful findings.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 30 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

1 – 10 of 31