A Novel Formulation of the Fractional Derivative with the Order $\alpha \ge 0$ and without the Singular Kernel

Round 1

Reviewer 1 Report

Dear authors,

Plecase added the suggation.

Sincerely.

 

Comments for author File: Comments.pdf

Author Response

Dear 

Please see the attached file

Regards

Author Response File: Author Response.docx

Reviewer 2 Report

This paper is disorganized and the results are poorly presented. The authors advise to reformulate it and present and explain the results in a better way.

Author Response

Dear

Please see the attached file

Regards

Author Response File: Author Response.docx

Reviewer 3 Report

In the present study, the authors have introduced the new definition of a fractional derivative with non-negative fractional order and without  a non-singular kernel. Also, with the help of some theorems and results, the new fractional derivative is compared with the other existing fractional derivatives. The obtained results look interesting and may helpful for future work. However, there are some modifications are needed to publish this work in reputed journals.

1.      I advise the authors to read the entire manuscript carefully and correct all typographical and grammatical errors.

2.      The history of fractional calculus and its recent developments in the field of science and technology should be mentioned in the introduction part of the manuscript.

3.      Abstract is not contructed in a scientific manner.

4.      The Caputo fractional derivative derived Eq. (1) is wrong. Correct it.

5.      More description about the obtained results should be incorporated in the separate section.

6.      Check both tables for the proper usage of column headings. I strogly suggest authors to read the complete manuscript once again and correct all such mistakes.

7.      What is the impact of considering the fractional derivatives with the non-singular kernel instead of those with the singular kernel while obtaining the solutions of fractional differential equations?

8.      Mention the value of the spatial variable in table 2.

9.      The frame labeling is not clear in the plots. Make it visible by increasing font size.

10.  English should be checked throughout the paper.

11.   The introduction section needs to improve professionally. Particularly, fractional calculus, fractional differential equations and their applications need to be discussed. In this regard, authors can refer to the following papers: An efficient analytical approach with novel integral transform to study the two-dimensional solute transport problem. Ain Shams Engineering Journal, (2022) 101878, Analysis of fractional Swift‐Hohenberg equation using a novel computational technique. Mathematical Methods in the Applied Sciences, 43(4) (2020), 1970-1987, An efficient technique for fractional coupled system arisen in magnetothermoelasticity with rotation using Mittag–Leffler kernel. Journal of Computational and Nonlinear Dynamics, 16(1) (2021), An efficient technique to analyze the fractional model of vector-borne diseases, Physica Scripta, 97(5) (2022), 054004.

 

Briefly, the author has derived some interesting results but requires significant modifications. However, it needs major revision.

Author Response

Dear 

Please see the attached file

Regards

Author Response File: Author Response.docx

Round 2

Reviewer 2 Report

The paper has been significantly improved, I don't see any need for the first two new paragraphs in the introduction.

Author Response

Dear Professor

Thank you so much

We added this introduction at the request of one of the reviewers 

Warm regards

Reviewer 3 Report

The authors have satisfactorily addressed most of my concerns. So paper can be accepted in present form. 

Author Response

Dear Professor 

Thank you so much