New Soliton Solutions of Time-Fractional Korteweg–de Vries Systems

Round 1

Reviewer 1 Report

Review Manuscript ID: universe-1846714
Type of manuscript: Article
Title: New Soliton Solutions of Time-Fractional Korteweg-de Vries Systems
Authors: Mubashir Qayyum, Efaza Ahmad, Muhammad Bilal Riaz*, Jan Awrejcewicz, Tauseef Saeed
Submitted to the section: Mathematical Physics, Research on Optical Soliton Perturbation
Date: 28 July 2022

The manuscript describes some examples of Laplace homotopy perturbation method applied to fractional derivatives of KdV-related nonlinear evolution equations. There seems to be no novel results in the paper. English is not too bad, but it definitely needs to be improved significantly even before sending out for peer review.

Author Response

Review Manuscript ID: universe-1846714

Manuscript Title: New Soliton Solutions of Time-Fractional Korteweg-de Vries Systems

Dear Editor/Reviewers,

Thank you for taking out time to review our manuscript in detail. We have followed up on the advice received and have revised our manuscript. The new version takes into account individual comments by each reviewer/editor.

 

Reviewer 1 Comments:

The manuscript describes some examples of Laplace homotopy perturbation method applied to fractional derivatives of KdV-related nonlinear evolution equations. There seems to be no novel results in the paper. English is not too bad, but it definitely needs to be improved significantly even before sending out for peer review.

 

Our response: In this work, we proposed a new approach for the solution of nonlinear time-fractional coupled KdV systems. In this approach, Laplace transform is hybrid with homotopy perturbation in fractional scenario for nonlinear coupled system of PDEs. Proposed method is tested against different nonlinear time-fractional systems, including generalized Hirota-Satsuma, and dispersive long wave KdV systems. Validation of obtained results is performed through comparison with existing solutions. Moreover, for convergence check, proposed algorithm is applied on the entire fractional domain and different solutions along with errors are computed for multiple values of fractional parameters. Numerical analysis through tables and figures is also performed to reassert the capability of the proposed technique in terms of accuracy in fewer computations. To the best of authors knowledge, fractional KdV systems have not been attempted before through this hybrid of Laplace transform along with homotopy perturbation.

 

Also, manuscript is rechecked for grammar and typos as per reviewer advise. 

 

 

 

 

Reviewer 2 Report

The authors present numerical evidences about the efficiency of the homotopy perturbation method in solving a system of time-fractional KdV equations. The results and the conclusion seem to be correct. However the following point should be improved before publication:

1. Omega is used in eq. (3) but neither defined or introduced.

2. I1, I2 and I3 are used in eq. (4) but not introduced before.

3. The relation between the functions R, S and T is not clear, they appear to be independent in (3). Why do the author consider several functions?

4. The figures should be commented. For instance, the errors display a clear structure in Fig. 2, 5 and 8. What is the origin of this structure? What determines the regions with small error? What is the lessons of Fig. 3, 6 and 9?

5. The tables should be avoided, their replacement by figures or the explanation of their lesson improves the readability of the manuscript.

6. There few more works devoted to the fractional KdV equations which should be commented, e.g. El-Ajou et al. J. Comp. Phys. 293 (2015) 81; Kurulay, Bayram Comm. Nonl. Sci Numer. Sim. 15 (2010) 1777; Wang Chaos Solitons & Fractals 35 (2008) 843.

Author Response

Review Manuscript ID: universe-1846714

Manuscript Title: New Soliton Solutions of Time-Fractional Korteweg-de Vries Systems

Dear Editor/Reviewers,

 

Thank you for taking out time to review our manuscript in detail. We have followed up on the advice received and have revised our manuscript. The new version takes into account individual comments by each reviewer/editor.

 

Reviewer 2 Comments:

The authors present numerical evidences about the efficiency of the homotopy perturbation method in solving a system of time-fractional KdV equations. The results and the conclusion seem to be correct. However, the following point should be improved before publication:

 

1. Omega is used in eq. (3) but neither defined or introduced.

 

            Our response: Omega is representing spatial domain. Omega is now      defined in section III as per reviewer suggestion.

 

1. I1, I2 and I3 are used in eq. (4) but not introduced before.

 

Our response: I1, I2 and I3 are initial conditions and now defined in section III as per reviewer advise.

 

1. The relation between the functions R, S and T is not clear, they appear to be independent in (3). Why does the author consider several functions?

 

Our response: R, S and T are three unknown functions in the system of three coupled PDEs which need to be computed (see Example 3 for clarification). Furthermore, R, S and T depends on Ӷ and t only but they are interlinked with each other because systems are mostly coupled. 

 

1. The figures should be commented. For instance, the errors display a clear structure in Fig. 2, 5 and 8. What is the origin of this structure? What determines the regions with small error? What is the lessons of Fig. 3, 6 and 9?

 

Our response: Fig. 2, 5 and 8 represents the errors in all three systems. These figures clearly indicate that error is comparatively larger where the turbulence/disturbance in the long dispersive wave. Also, near the origin (where disturbance/turbulence occurs), system is more unpredictable but with time when waves disperse from the origin it becomes uniform and solution is more predictable.

Fig 3, 6, and 9 are showing the effect of fractional parameter on the solutions in the current systems (example 1-3). Effect of fractional parameter on the solution components is commented in detail (See results and discussion section).

 

1. The tables should be avoided, their replacement by figures or the explanation of their lesson improves the readability of the manuscript.

 

Our response: Tables are given for the numerical analysis of the proposed method. These tables provide information about the different aspects of the proposed method and may increase the interest of the audience for better use in their fields. Table shows the following aspects of the proposed methodology.

• comparison of current results with the ones given in the literature shows the efficiency and supremacy of the proposed scheme (see table 1,3 and 5).
• Application of the proposed method in the entire fractional domain (see table 2, 4 and 6).

We also improve the discussion as per reviewer suggestion to improve the readability of the current manuscript.

 

1. There are few more works devoted to the fractional KdV equations which should be commented, e.g. El-Ajou et al. J. Comp. Phys. 293 (2015) 81; Kurulay, Bayram Comm. Nonl. Sci Numer. Sim. 15 (2010) 1777; Wang Chaos Solitons & Fractals 35 (2008) 843.

 

Our response: Literature review has been improved by discussing other related work as suggested by respected reviewer.

Round 2

Reviewer 1 Report

Second Review Manuscript ID: universe-1846714
Type of manuscript: Article
Title: New Soliton Solutions of Time-Fractional Korteweg-de Vries Systems
Authors: Mubashir Qayyum, Efaza Ahmad, Muhammad Bilal Riaz*, Jan Awrejcewicz, Tauseef Saeed
Submitted to the section: Mathematical Physics, Research on Optical Soliton Perturbation
Date: 17 August 2022

The authors seem to attempt to improve the manuscript. However, some issues need attention.

* The functions sech, tanh, cosh, etc. are not written in italics mode.

* The plots in Figure 1 need more mesh grid points.

* Space is needed between parameters/variables and functions, e.g., eq. (38).

* The plots in Figure 4 need better resolutions, perhaps by adding more mesh grid points.

* What do the arrows mean in Figures 3, 6, and 9?

* What can you say more about and/or from the solutions plot presented in Figures 1, 4, and 7?

* Why reference no. [15] is written in capital letters?

* The same remark for the journal title in [33].

* Please check whether you could cite some works of E. van Groesen from Universiteit Twente and M. Kovalyov from Alberta Canada since they also worked on the KdV equation and its related models.

Author Response

Second Review Manuscript ID: universe-1846714

Manuscript Title: New Soliton Solutions of Time-Fractional Korteweg-de Vries Systems

 

Dear Editor/Reviewers,

 

Thank you for taking out time to review our manuscript in detail. We have followed up on the advice received and have revised our manuscript. The new version takes into account individual comments by each reviewer/editor.

 

Regards,

 

Response to Reviewer

 

Reviewer Comments:

The authors seem to attempt to improve the manuscript. However, some issues need attention.

 

1. The functions sech, tanh, cosh, etc. are not written in italics mode.

 

            Our response: Correction has been made and as per reviewer       suggestion.

 

1. The plots in Figure 1 need more mesh grid points.

 

Our response: Figure 1 is replaced with new plot which contains all the mesh grid points.  We have changed all the 3D plots and included more mesh grade points to improve the resolution as per reviewer suggestion. (See 3D plots in the revised version)

 

1. Space is needed between parameters/variables and functions, e.g., eq. (38).

 

Our response: Spacing in equations are corrected. 

 

1. The plots in Figure 4 need better resolutions, perhaps by adding more mesh grid points.

 

Our response: Figure 4 is replaced with new one containing more mesh grids as per reviewer advice. We have changed all the 3D plots and included more mesh grade points to improve the resolution as per reviewer suggestion.

 

1. What do the arrows mean in Figures 3, 6, and 9?

 

Our response: Arrow show the direction of solution component with an increase in fractional parameter.

 

1. What can you say more about and/or from the solutions plot presented in Figures 1, 4, and 7?

 

Our response: In all three examples it is observed that error is comparatively larger near the turbulence/disturbance in the waves. Also, near the origin of turbulence, system is more unpredictable but with time when waves disperse from origin it becomes uniform and solution is more predictable.

 

1. Why reference no. [15] is written in capital letters?

 

Our response: References are corrected in revised version.

 

1. The same remark for the journal title in [33].

 

Our response: References are corrected in revised version.

 

1. Please check whether you could cite some works of E. van Groesen from Universiteit Twente and M. Kovalyov from Alberta Canada since they also worked on the KdV equation and its related models

 

Our response: Thanks for suggestion the suggested work help to improve the work and new citation is included in the revised version.