Spectral Analysis of One Class of Positive-Definite Operators and Their Application in Fractional Calculus
Durdimurod Kalandarovich Durdiev
Round 1
Reviewer 1 Report
In the paper under review, the spectral analysis of one class of integral operators, associated to the boundary value problems for differential equations of fractional order has studied. In particular, the authors have showed the positive definiteness of studying operators, which makes it possible to select areas in the complex plane where there are no eigenvalues of these operators.
In my opinion the paper deserve to be publish but only after some major revision. First of all, the authors should modify the abstract section, in particular they should give what type of problem they are discussing here. Also they should give how they used the Mittag-leffler function in their paper.
Secondly, the author should address the following correction:
1. What is mean by the sentences given before in section 2? The authors should clarify it.
2. Check the convergence condition of the first equation in section 2.
3. Write conditions on the parameters atleast once.
4. The introduction section should be modify.
5. The Aim and structure of the paper should be given.
6. The English of the paper should be recheck, it need a major check to avoid the miss spellings.
7. The commas and full-stops should be given in their correct places.
My third recommendation is, in the revised paper, the authors should include a new section “Concluding Remarks and Observations”. Particularly, the novelty and relevance of the newly introduced classes should be emphasized, pointing out their potential applications and perspective for future research. Such information can be found throughout the manuscript, but it must be particularly highlighted in the conclusions.
Finally, double check the references, the references must be adjust in one patron and also need to be arrange in alphabetically or numerically (check instruction for author).
Moderate editing of English language required
Author Response
All comments made to these reviewers were taken into account.The authors express their gratitude to the reviewer for comments that contributed to the improvement of the manuscript.
Reviewer 2 Report
See the attached file.
Comments for author File: 
Comments.pdf
Extensive editing of English language required.
Author Response
The reviewer writes that the operator studied in the work is not of interest.I specifically made a link to the monograph of a very famous mathematician, where this operator is studied and talks about my work on studying this operator and the role of this operator in fractional calculus is very clearly indicated.
Round 2
Reviewer 1 Report
The authors did not addressed all of my suggestion.
The conclusion section is still missing. I cant recommend the paper in its present form.
English of the paper is fine
Author Response
The reviewer's comments have been taken into account.Added an item with conclusions.
Reviewer 2 Report
I recommend publishing the article.
Extensive editing of English language required
Author Response
The reviewer's comments have been taken into account.Added an item with conclusions.
