Novel Results on Legendre Polynomials in the Sense of a Generalized Fractional Derivative
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe paper, after minor revision according to the attached report, deserve to be published, Then, the corresponding author have 4 working days to upload the revised version for the final revision.
Best regards.
Comments for author File: 
Comments.pdf
The paper, after minor english misprints, can be upload for the fional decision.
Best regards
Author Response
Dear Respected Reviewer and Handling Editor,
We would like to thank you for all your comments and suggestions that improved our research article entitled “Novel Results on Legendre Polynomials in the Sense of Generalized Fractional Derivative” at your respected journal. We have implemented all your suggestions and comments in the revised version of our article.
Comment 1: pg.1 line 27, for a better diffusion of the paper I suggest to add ”fractional derivative”
Response 1: We would like to thank the reviewer for this comment and information. We greatly appreciate that. We have implemented the reviewer’s suggestion.
Comment 2: pg.1 line 35, if the authors agree, I suggest to write ”differential equations [3, X,
Y, Z].”, where:
[X] Al-Askar F.M., Impact of fractional derivative and brownian motion on the solutions of the Radhakrishnan-Kundu-Lakshmanan equation, Journal of Function Spaces,
vol.2023, art.n.8721106, (2023);
[Y] Balachandran K., Matar M., Annapoorani N., Prabu D., Hadamard functional
fractional integrals and derivatives and fractional differential equations, Filomat, 38 (3),
779–792, (2024);
[Z] Chems Eddine N., Ragusa M.A., Repovs D.D., Correction to: On the concentrationcompactness principle for anisotropic variable exponent Sobolev spaces and its applications (Fractional Calculus and Applied Analysis, (2024), 27, 2, (725-756), 10.1007/s13540-
024-00246-8), Fractional Calculus and Applied Analysis, doi:10.1007/s13540-024-00282-4
, (2024);
Response 2: We would like to thank the reviewer for this comment and suggestion. We greatly appreciate that. We have implemented the reviewer’s suggestion.
Comment 3: pg.2 line 74, in general a paper should be self-contained, in this sense I suggest to spend a line to write the definition of the Gamma function;
Response 3: We would like to thank the reviewer for this comment and suggestion. We greatly appreciate that. We have implemented the reviewer’s suggestion and we have added the definition of Gamma function.
Comment 4: pg.3 line 91, in my opinion is not very elegant and precise ”are mentioned in Section
2. ”, so I suggest to wrote ”as: the GDFD with its essential properties are exposed in
Section 2.”;
Response 4: We would like to thank the reviewer for this comment and suggestion. We greatly appreciate that. We have implemented the reviewer’s suggestion and we have corrected that as suggested.
Comment 5: pg.11 line 304, I suggest delete this line because from the previous to the next lines
calculations are clear;
Response 5: We would like to thank the reviewer for this comment and suggestion. We greatly appreciate that. We have implemented the reviewer’s suggestion and we have deleted that line as suggested.
Comment 6: pg.20 line 498, please, write clearly what means ” no, 01”.
Response 6: We would like to thank the reviewer for this comment and suggestion. We greatly appreciate that. We have corrected that typo as suggested.
Reviewer 2 Report
Comments and Suggestions for AuthorsThe paper is partially well-written and subject of the paper is interesting. However, before any acceptance, the authors need to address the following points:
1- Moreover, language expressions need to be polished. Please check over the whole manuscript and polish the language. Less errors on these will make the paper more readable. Some of them are as follow:
-In Thm. 4: “Let W be any…”
-You have three different forms for the name “Rodriques”.
-Delete all “the” before “Equation (number)”.
2- Page 2, line 75: Please check “0<=α<1”. I think α cannot be zero, but can be 1. Also, why “t>0”? While the Legendre functions defined on [-1,1]. The same is repeated in Eq. (9).
3- Please be careful in using “.” and “,” especially in Math formulae. For instance, on page 2, line 74, after the equation you need a “,” Not “.”. Please check all manuscript.
4- In Ref. [12] it is discussed that the “conformable derivative” is not a fractional operator. So, how do you ensure that your GDFD is meaningful?
5- The author should correct all occurrences of "Caputo" in the paper to read
"Liouville-Caputo" (because Liouville considered such fractional derivatives
many decades earlier in 1832). See cf. page 136 of the following paper:
DOI: http://dx.doi.org/10.25073/jaec.202153.340
6- Please use “\mathbb{R}” for showing real numbers in (1) and elsewhere. The same for natural numbers is needed: “\mathbb{N}”.
7- In Eq. (1) you wrote: “β>-1, β\in R^+”. So why do you need to write “β>-1” here? Only the second condition us sufficient.
8- How do you can generalize ‘Definition 3” to have α-singular point of a differential equation? Also, the concept “generalize α-analytic” is not give before this definition.
9- In Eq. (9), I think “p” should be a positive integer?
10- It is better do not start your sentences with Math formulae such as line 151, on Page 5.
11- Please make an alignment between the coefficient “c_i” on page 6, above and so on.
12- In Remark 7, line 372: What does mean the integral notation “I_α^{-1}”? Please clarify. The “.” should be after “.” Not before “dt”.
13- What is “I_α^{c}” in line 444 on page 19?
14- More descriptions about the results of Tables and Figures should be given.
15- The list of Ref. should be checked again for typos. The journal’ names should be abbreviated. All Refs. should have the same format. Ref. [13] has no vol/page number. Ref. [14] is not published after 2 years?
16- Enhance your introduction with some latest works related to applications of Legendre functions:
https://doi.org/10.1016/j.chaos.2021.110779
https://doi.org/10.1016/j.apnum.2023.12.004
Based on the above reasons, I suggest major revision. After revision, I will available to recommend it for publication in MCA.
Comments on the Quality of English Language
A minor check is needed.
Author Response
Dear Respected Reviewer and Handling Editor,
We would like to thank you for all your comments and suggestions that improved our research article entitled “Novel Results on Legendre Polynomials in the Sense of Generalized Fractional Derivative” at your respected journal. We have implemented all your suggestions and comments in the revised version of our article.
Comment 1:
Moreover, language expressions need to be polished. Please check over the whole manuscript and polish the language. Less errors on these will make the paper more readable. Some of them are as follow:
-In Thm. 4: “Let W be any…”
-You have three different forms for the name “Rodriques”.
-Delete all “the” before “Equation (number)”.
Response 1: We would like to thank the reviewer for these comments and information. We greatly appreciate that. We have implemented all reviewer’s suggestions.
Comment 2: Page 2, line 75: Please check “0<=α<1”. I think α cannot be zero, but can be 1. Also, why “t>0”? While the Legendre functions defined on [-1,1]. The same is repeated in Eq. (9).
Response 2: We would like to thank the reviewer for this comment and suggestion. We greatly appreciate that. It was a typo, and we have corrected it as suggested.
Comment 3: Please be careful in using “.” and “,” especially in Math formulae. For instance, on page 2, line 74, after the equation you need a “,” Not “.”. Please check all manuscript.
Response 3: We would like to thank the reviewer for this comment and suggestion. We greatly appreciate that. We have implemented the reviewer’s suggestion.
Comment 4: In Ref. [12] it is discussed that the “conformable derivative” is not a fractional operator. So, how do you ensure that your GDFD is meaningful?
Response 4: We would like to thank the reviewer for this comment. The conformable derivative does not satisfy the semigroup property while our derivative GDFD satisfies that. In addition, the results of GDFD matches the results from Cuputo and Riemann-Liouville derivatives' results. The GDFD has been already employed successfully in various recent research studies in Mathematical Physics and Engineering Sciences.
Comment 5:
The author should correct all occurrences of "Caputo" in the paper to read
"Liouville-Caputo" (because Liouville considered such fractional derivatives
many decades earlier in 1832). See cf. page 136 of the following paper:
DOI: http://dx.doi.org/10.25073/jaec.202153.340
Response 5: We would like to thank the reviewer for this comment and suggestion. We greatly appreciate that, and we totally agree with you. We have implemented the reviewer’s suggestion.
Comment 6: Please use “\mathbb{R}” for showing real numbers in (1) and elsewhere. The same for natural numbers is needed: “\mathbb{N}”.
Response 6: We would like to thank the reviewer for this comment and suggestion. We greatly appreciate that. We have implemented this suggestion.
Comment 7: In Eq. (1) you wrote: “β>-1, β\in R^+”. So why do you need to write “β>-1” here? Only the second condition us sufficient.
Response 7: We would like to thank the reviewer for this comment. The constraint β>-1 is provided in the above definition to make sure that the Gamma function is well-defined positive and finite because it is well-known that this special function is defined for all complex numbers except for the non-positive integers. Otherwise, if β is less than or equal to , the Gamma function will not hold and it can be infinite which is not suitable for the definition of the above fractional derivative.
Comment 8: How do you can generalize ‘Definition 3” to have α-singular point of a differential equation? Also, the concept “generalize α-analytic” is not give before this definition.
Response 8: We would like to thank the reviewer for this comment and suggestion. We greatly appreciate that. We have implemented the reviewer’s suggestion, and we have defined it as suggested.
Comment 9: In Eq. (9), I think “p” should be a positive integer?
Response 9: We would like to thank the reviewer for this comment and suggestion. We greatly appreciate that. It was a typo, and we have corrected that as suggested.
Comment 10: It is better do not start your sentences with Math formulae such as line 151, on Page 5.
Response 10: We would like to thank the reviewer for this comment and suggestion. We greatly appreciate that. We have implemented this suggestion.
Comment 11: Please make an alignment between the coefficient “c_i” on page 6, above and so on.
Response 11: We would like to thank the reviewer for this comment and suggestion. We greatly appreciate that. We have implemented this suggestion.
Comment 12: In Remark 7, line 372: What does mean the integral notation “I_α^{-1}”? Please clarify. The “.” should be after “.” Not before “dt”.
Response 12: We would like to thank the reviewer for this comment and suggestion. We greatly appreciate that. We have implemented this suggestion. It is a generalized \alpha-integral function of ℎ where the lower bound of integration is -1.
Comment 13: What is “I_α^{c}” in line 444 on page 19?
Response 13: We would like to thank the reviewer for this comment. It is a generalized \alpha-integration according to the given function.
Comment 14: More descriptions about the results of Tables and Figures should be given.
Response 14: We would like to thank the reviewer for this comment. This study is theoretical-based study and in the future studies, we will have many extensions with numerical studies with figures and tables.
Comment 15: The list of Ref. should be checked again for typos. The journal’ names should be abbreviated. All Refs. should have the same format. Ref. [13] has no vol/page number. Ref. [14] is not published after 2 years?
Response 15: We would like to thank the reviewer for this comment. It was a typo, we have corrected it. We have also checked all references.
Comment 16: Enhance your introduction with some latest works related to applications of Legendre functions.
Response 16: We would like to thank the reviewer for this comment. We have implemented his suggestions.
Round 2
Reviewer 2 Report
Comments and Suggestions for AuthorsThe paper now suitable for publication in MCA.
All the best
