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1 – 10 of 31Syed Tauseef Mohyud‐Din, Elham Negahdary and Muhammad Usman
The purpose of this paper is to present a numerical solution of a family of generalized fifth‐order Korteweg‐de Vries equations using a meshless method of lines. This method uses…
Abstract
Purpose
The purpose of this paper is to present a numerical solution of a family of generalized fifth‐order Korteweg‐de Vries equations using a meshless method of lines. This method uses radial basis functions for spatial derivatives and Runge‐Kutta method as a time integrator and exhibits high accuracy as seen from the comparison with the exact solutions.
Design/methodology/approach
The study uses a meshless method of lines. This method uses radial basis functions for spatial derivatives and Runge‐Kutta method as a time integrator.
Findings
The paper reveals that this method exhibits high accuracy as seen from the comparison with the exact solutions.
Originality/value
This method is efficient method as it is easy to implement for the numerical solutions of PDEs.
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Keywords
Perumandla Karunakar and Snehashish Chakraverty
This study aims to find the solution of time-fractional Korteweg–de-Vries (tfKdV) equations which may be used for modeling various wave phenomena using homotopy perturbation…
Abstract
Purpose
This study aims to find the solution of time-fractional Korteweg–de-Vries (tfKdV) equations which may be used for modeling various wave phenomena using homotopy perturbation transform method (HPTM).
Design/methodology/approach
HPTM, which consists of mainly two parts, the first part is the application of Laplace transform to the differential equation and the second part is finding the convergent series-type solution using homotopy perturbation method (HPM), based on He’s polynomials.
Findings
The study obtained the solution of tfKdV equations. An existing result “as the fractional order of KdV equation given in the first example decreases the wave bifurcates into two peaks” is confirmed with present results by HPTM. A worth mentioning point may be noted from the results is that the number of terms required for acquiring the convergent solution may not be the same for different time-fractional orders.
Originality/value
Although third-order tfKdV and mKdV equations have already been solved by ADM and HPM, respectively, the fifth-order tfKdV equation has not been solved yet. Accordingly, here HPTM is applied to two tfKdV equations of order three and five which are used for modeling various wave phenomena. The results of third-order KdV and KdV equations are compared with existing results.
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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…
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.
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The purpose of this paper is concerned with developing two-mode higher-order modified Korteweg-de Vries (KdV) equations. The study shows that multiple soliton solutions exist for…
Abstract
Purpose
The purpose of this paper is concerned with developing two-mode higher-order modified Korteweg-de Vries (KdV) equations. The study shows that multiple soliton solutions exist for essential conditions related to the nonlinearity and dispersion parameters.
Design/methodology/approach
The proposed technique for constructing a two-wave model, as presented in this work, has been shown to be very efficient. The employed approach formally derives the essential conditions for soliton solutions to exist.
Findings
The examined two-wave model features interesting results in propagation of waves and fluid flow.
Research limitations/implications
The paper presents a new and efficient algorithm for constructing and studying two-wave-mode higher-order modified KdV equations.
Practical implications
A two-wave model was constructed for higher-order modified KdV equations. The essential conditions for multiple soliton solutions to exist were derived.
Social implications
The work shows the distinct features of the standard equation and the newly developed equation.
Originality/value
The work is original and this is the first time for two-wave-mode higher-order modified KdV equations to be constructed and studied.
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Abdul-Majid Wazwaz, Weaam Alhejaili and Samir El-Tantawy
The purpose of this study is to form a linear structure of components of the modified Korteweg–De Vries (mKdV) hierarchy. The new model includes 3rd order standard mKdV equation…
Abstract
Purpose
The purpose of this study is to form a linear structure of components of the modified Korteweg–De Vries (mKdV) hierarchy. The new model includes 3rd order standard mKdV equation, 5th order and 7th order mKdV equations.
Design/methodology/approach
The authors investigate Painlevé integrability of the constructed linear structure.
Findings
The Painlevé analysis demonstrates that established sum of integrable models retains the integrability of each component.
Research limitations/implications
The research also presents a set of rational schemes of trigonometric and hyperbolic functions to derive breather solutions.
Practical implications
The authors also furnish a variety of solitonic solutions and complex solutions as well.
Social implications
The work formally furnishes algorithms for extending integrable equations that consist of components of a hierarchy.
Originality/value
The paper presents an original work for developing Painlevé integrable model via using components of a hierarchy.
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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 dispersion…
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.
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Initial value problems for the one‐dimensional third‐order dispersion equations are investigated using the reliable Adomian decomposition method (ADM).
Abstract
Purpose
Initial value problems for the one‐dimensional third‐order dispersion equations are investigated using the reliable Adomian decomposition method (ADM).
Design/methodology/approach
The solutions are obtained in the form of rapidly convergent power series with elegantly computable terms.
Findings
It was found that the technique is reliable, powerful and promising. It is easier to implement than the separation of variables method. Modifications of the ADM and the noise terms phenomenon are successfully applied for speeding up the convergence of non‐homogeneous equations.
Research limitations/implications
The method is restricted to initial value problems in which the space variable fills the whole real axis. Modifications are required to deal with initial boundary value problems. Further, the input initial condition is required to be an infinitely differentiable function and obviously, the convergence radius of the decomposition series depends on the input data.
Practical implications
The method was mainly illustrated for linear partial differential equations occuring in water resources research, but the natural extension of the ADM to solving nonlinear problems is extremely useful in nonlinear studies and soliton theory.
Originality/value
The study undertaken in this paper provides a reliable approach for solving both linear and nonlinear dispersion equations and new explicit or recursively‐based exact solutions are found.
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Jin-Jin Mao, Shou-Fu Tian and Tian-Tian Zhang
The purpose of this paper is to find the exact solutions of a (3 + 1)-dimensional non-integrable Korteweg-de Vries type (KdV-type) equation, which can be used to describe the…
Abstract
Purpose
The purpose of this paper is to find the exact solutions of a (3 + 1)-dimensional non-integrable Korteweg-de Vries type (KdV-type) equation, which can be used to describe the stability of soliton in a nonlinear media with weak dispersion.
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 non-integrable KdV-type equation. The used approach formally derives the essential conditions for these solutions to exist.
Findings
The results show that the equation exists rogue waves, homoclinic breather waves and soliton waves. To better understand the dynamic behavior of these solutions, the authors analyze the propagation and interaction properties of the these solutions.
Originality/value
These results may help to investigate the local structure and the interaction of waves in KdV-type equations. It is hoped that the results can help enrich the dynamic behavior of such equations.
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This study aims to investigate two newly developed (3 + 1)-dimensional Kairat-II and Kairat-X equations that illustrate relations with the differential geometry of curves and…
Abstract
Purpose
This study aims to investigate two newly developed (3 + 1)-dimensional Kairat-II and Kairat-X equations that illustrate relations with the differential geometry of curves and equivalence aspects.
Design/methodology/approach
The Painlevé analysis confirms the complete integrability of both Kairat-II and Kairat-X equations.
Findings
This study explores multiple soliton solutions for the two examined models. Moreover, the author showed that only Kairat-X give lump solutions and breather wave solutions.
Research limitations/implications
The Hirota’s bilinear algorithm is used to furnish a variety of solitonic solutions with useful physical structures.
Practical implications
This study also furnishes a variety of numerous periodic solutions, kink solutions and singular solutions for Kairat-II equation. In addition, lump solutions and breather wave solutions were achieved from Kairat-X model.
Social implications
The work formally furnishes algorithms for studying newly constructed systems that examine plasma physics, optical communications, oceans and seas and the differential geometry of curves, among others.
Originality/value
This paper presents an original work that presents two newly developed Painlev\'{e} integrable models with insightful findings.
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