Analysis of the Multi-Dimensional Navier–Stokes Equation by Caputo Fractional Operator

Round 1

Reviewer 1 Report

Please find the attached file.

Comments for author File: Comments.pdf

Author Response

1. The abstract does not contain the main idea and the new findings. The authors mentioned “study” in three sentences and are several grammatical mistakes in the abstract: Suggestion implemented. Abstract is rewritten and it contains the findings of the paper as well. Highlighted for your kind reference.

2. I suggest keeping only the related studies and adding the following: https://doi.org/10.3390/sym14061102

Implemented. Added the suggested article and removed some references too. Highlighted for your kind reference.

3. In the pre-requisites section, the subsections need all to be the same design (2.1 and 2.2 are bold, but 2.3 and so on are not). Also, check the English language and grammar of this section carefully:

      Implemented. All corrections have been done and highlighted for your kind reference.

4. In section 3, correct the equations after setting Lagrange multiplier:

Implemented and equations is being corrected. (Brackets were at wrong place now is being corrected.)

5. The proof in section 4 is correct, but the language and the design of the section need to be worked: Implemented and tried to polish the language. Highlighted for your kind reference.

6. In section 5, the constraints of Lipschitz condition need to be explained after equation:

Implemented and highlighted for your kind reference.

7. Could you please explain the meaning of “iteration” after the equation (43):

      Corrected 

8. The paper is not designed carefully. The graphical solutions are placed after the conclusion and between references, Also, it is better to discuss those graphical solutions one by one and give an explanation to show the meaning of the solutions in the graphs:

Implemented. The design was corrected, it was due to some latex command error. Now figures are at their proper place and an explanation of graphs are included in conclusion section. Highlighted for your kind reference.

Reviewer 2 Report

The results of the present paper are new and interesting to a certain extent. However, it should be particularly revised before its final decision is made.

1. The main innovation of this paper should be highlighted in the introduction part.

2. In the Introduction, the research status of the Caputo Fractional Operator can be introduced in detail. The language and the proof process can be simply expressed because the paper is long.

3. In the revised manuscript, the proper Notations should be introduced. For example, What does mean that $C^k$ in equation (1)?

4. Please check equation (5). If you chose Y as a fractional order, equation (1)  $gamma$ is a fraction derivative. The derivations are confusing for readers. Please justify it. 

5. In the revised manuscript, proper citations should be included. For example, some recent references are highlighted, and add one remark about your main contributions. 

Author Response

# For Reviewer 1 1. The main innovation of this paper should be highlighted in the introduction part: Implemented and included in abstract part of the paper. Highlighted for your kind reference.

2. In the Introduction, the research status of the Caputo Fractional Operator can be introduced in detail: Implemented in pre-requisite section. Highlighted for your kind reference.
3. In the revised manuscript, the proper Notations should be introduced. For example, What does mean that $C^k$ in equation (1): Implemented and corrected the notation.
4. Please check equation (5). If you chose Y as a fractional order, equation (1) $gamma$ is a fraction derivative. The derivations are confusing for readers. Please justify it:

Thanks for such a close observation. The mistake has been removed now. Suggestion implemented.

5. In the revised manuscript, proper citations should be included. For example, some recent references are highlighted: Implemented. Latest work has been highlighted for your kind reference.

Reviewer 3 Report

The authors studied the multi-dimensional Navier-Stokes equation based on the Caputo fractional operator. They used the Sumudu transform method to analyze the behavior of a Navier-Stokes equations system, but their proposal's novelty is not apparent. 

1. What is the difference between their work and others like https://doi.org/10.1186/s13662-021-03250-x, https://doi.org/10.1016/j.asej.2016.04.009 or https://doi.org/10.1038/s41598-022-20911-3? What is the novelty?

2. Why do the authors choose the Caputo fractional derivative? This operator has a singularity when the beta is equal to the tau. It can be interpreted as an incomplete memory effect. Could this affect the analysis of multi-dimensional Navier-Stokes?

3. Figs. 1-3 must improve; the quality is poor.

4. Figs. 1-3 are included in the manuscript, but they are not cited in the text. Why?

5. The authors carried out different approaches to study the rate of change in flow, but the interpretation is not included. Please tell readers more details about your results.

Author Response

Now, if we talk about the novelty of our article then we have applied Caputo derivative and Sumudu transform with an iterative technique to solve the NS system. The methodology is unique and not common. We have also checked the existence of the applied method and validated the methodology. This work is strive to find an approximate solution of the NS system by another new methodology and approach.

1. Why do the authors choose the Caputo fractional derivative? This operator has a singularity when the beta is equal to the tau: We used Caputo derivative (knowing that it has singularity) because it includes traditional initial and boundary conditions in the formulation of the physical problems. (Not included in the paper, this is for your kind reference.)
2. 1-3 must improve; the quality is poor: Implemented and are replaced.
3. 1-3 are included in the manuscript, but they are not cited in the text. Why?: Implemented, explanation included in conclusion section.
4. The authors carried out different approaches to study the rate of change in flow, but the interpretation is not included: Implemented, conclusion is revised.

Along with above explanations and implementations, I would like to receive further suggestions for improvements as well.

Round 2

Reviewer 1 Report

Authors didn't revise the paper appropriately. Moreover, they didn't pay attention to the points in my previous report. All they did, is adding the answers "corrected" and "Implemented" ...etc.

For example, I asked "Could you please explain the meaning of “iteration” after the equation (43)?", and their answer was "Corrected" and there were no changes in the manuscript!! Similarly for the rest of the questions.

In addition, authors need to work intensively on the manuscript language or ask a native speaker to revise it. Moreover, the design of the paper still bad, and the graphs are still placed in between the references, and not in their correct places.

Finally, I can't recommend the publication of the manuscript without revising the paper appropriately.

Author Response

Please see attached file.

Author Response File: Author Response.docx

Reviewer 2 Report

No need for further comments. The revised manuscript is accepted. 

Author Response

Thanks

Round 3

Reviewer 1 Report

It seems that the authors made a good effort this time and revised the paper accordingly. However, there are still several typos in the paper as well as some misprinted parentheses after equation (14). 

Now, I can recommend the paper for publication, but I STRONGLY suggest a deep proofreading to correct all typos and grammatical errors.

Best of luck!

Author Response

After equation 14, changes are done.