Asymptotic Solution of a Singularly Perturbed Integro-Differential Equation with Exponential Inhomogeneity
Round 1
Reviewer 1 Report
Please see attached file
Comments for author File: 
Comments.pdf
Author Response
The following are comments and suggestions:
Abstract: Rewrite abstract and try to focus on what has been done in the paper rather than writing that this work has not been done before.
Answer. In the case of a stable spectrum, we can construct a regularized asymptotics without using a fundamental solution. In the case of an unstable spectrum it is necessary to take into account its point features. In this case, inhomogeneity plays an essential role. It significantly affects the type of singularities in the solution of the initial problem. The fundamental solution allows us to construct asymptotics regardless of the nature of the inhomogeneity (it can be both slowly changing and rapidly changing, for example, rapidly oscillating). The approach developed in the paper is universal with respect to arbitrary inhomogeneity.
Introduction: It is based on old literature. It should enhance by providing relevant recent references by comparing this work which must novelty of the present work.
Answer. There is literature on the construction of asymptotics of problems with an unstable spectrum. However, in them the role of instability is played by zeros of the coefficient a(t). In our case, the unstable spectral value is of inhomogeneity. The problem has not been considered in this formulation before.
Page 2, line 21: Either delete this text “More general cases will be studied in our next papers.” Or write something here, what kinds of cases will consider in the next papers?
Answer. We have removed the words about further research into this problem.
Page 2, Equation (3): Check that V(s,s,ϵ)=1 or there is one t?
Answer. This is the initial condition for the fundamental solution. We have written it down in more detail.
Page 2, Equation (4): Delete first line expression including from second line of Equation (4).
Answer. We removed the first equation (4) and left only the second.
Page 3, lines 1 and 2: Rewrite sentence clearly.
Answer. Сorrected
Page 3, line 20: Meaning not clear properly “this algorithm has an independent meaning”.
Answer. The proposal has been removed from the text.
Page 3, line 25: In Condition 2), what is Re?
Answer. The coefficient a(t) can be complex. Re a(t) means its real part.
Page 7, line 1: Correct spacing in this line.
Answer. Corrected.
Page 9, last line: Use one type of matrix notation either ( ) or [ ].
Answer. We have corrected [ ] to ( ).
Page 11, line 4: Rewrite sentence clearly “Hence we get that”.
Answer. Consequently, the following relation follows from the resulting inequality.
Mathematical Expressions write in proper way:
In Equation (14), delete “-“ from the end of second line and better write third line with 2 nd line.
Answer. Corrected.
Equation (18) should write in one line only.
Answer. Corrected.
In Equation (24), delete from the end of first line.
Answer. Corrected.
In Equation (26), delete from the end of first line.
Answer. Corrected.
Page 8, Lines 19-20, write expression in one line by deleting one “+”.
Answer. Corrected.
Provide proper space between expression and condition of equation, for example, see Equation (31). Make similar spaces in other equations.
Answer. Corrected.
Page 10, from bottom lines 5-6, write expression in one line by deleting one “-”.
Answer. Corrected.
Page 11, lines 2-3, write expression in one line by deleting one “-”.
Answer. Corrected.
Conclusion: It is not given. It should provide in the paper.
Answer. Аdded.
References: Provide at least 5 relevant references from last 5 years.
Answer. Аdded.
Reviewer 2 Report
see the report.
Comments for author File: Comments.pdf
Author Response
Responses to reviewer's comments - 2
Remarks:
1) The authors must check the paper in terms of grammatically and typos.
Answer. Grammar errors and typos corrected.
2) The authors must mention about the practical importance of the paper.
Answer. The results of the work can be applied to those areas in which asymptotics are used, especially if it is related to the instability of the spectrum.
3) The presentation of the paper can be improved by arranging the statements of theorems.
Answer. The presentation of the theorems given is ordered.
4) The authors can improve the problem by using fractional operators in the future studies.
Answer. Information for thought.
5) The authors must add a conclusion section to emphazise the importance of the findings.
Answer. Аdded conclusion section.
6) The authors must add the following recent papers to extend the introduction and references.
Answer. The list of references used is supplemented by 13 references.
Reviewer 3 Report
Comments and Suggestions for Authors
The Manuscript ID axioms-2210405, titled “Asymptotic Solution of a Singularly Perturbed Integro-Differential Equation with Exponential Inhomogeneity,” belongs to the field of Mathematical Analysis, subfield Asymptotic solutions of The Integrodifferential Equations.
Concerning the scientific content of the paper
1. In my opinion, the Abstract must be developed, emphasizing the novelty of the research work.
2. The Introduction must be consistent, and reading the manuscript, in my opinion, this goal is not achieved. The Introduction is not sufficiently developed for the mentioned research field, there are mentioned only 4 references[1-4].
3. The manuscript has only 10 references, being very poor. Even the bibliography is very poor references 5-8 are not cited in the text. Why?? The last mention of reference 4 is done on page 13 ( the first mention of reference 4 is on page 2). From page 2 to page 13 is not mentioned any references in the range 5-8. The authors jump to reference 10 on page 13 and return to reference 9 on page 14 of the manuscript. Why???
4. The paragraph “3. Solvability of iterative problems” is not mentioning any reference and the paragraph “4. Asymptotic convergence of formal solutions to the exact solution” is mentioning only reference 4. Why???
5. In the paragraph “5. Construction of an asymptotic solution to the original problem (1)” is mentioned reference 4 on page 13 and on the same page is mentioned reference 10 ( I don’t understand the “jump”). While on page 14 is mentioned reference 9 ( “jumps” and “returns” ???).
6. The paragraphs “Applications” and “Conclusions” are missing, causing the reader not to understand the goal of the research.
7. Because of this omission mentioned before the manuscript is missing the specific scientific fields where the work results can be used.
8. Can the authors explain what brings new the work presented in the manuscript compared with the previously published https://doi.org/10.1515/math-2021-0021.
9. The manuscript is not emphasizing the novelty of the work.
About the presentation of the paper concerning the Manuscript Type MDPI journal template
1. The References don’t respect 100% the form demanded by the Manuscript Type MDPI journal template.
For the aspects mentioned above, I strongly advise a major revision of the paper.
Author Response
Responses to reviewer's comments - 3
- In my opinion, the Abstract must be developed, emphasizing the novelty of the research work.
Answer. The novelty of the research paper is described in the words «The problem belongs to the class of singularly perturbed equations with an unstable spectrum and has not been considered before in the presence of an integral operator. A particular difficulty is its investigation in the neighborhood of the zero spectral value of inhomogeneity».
- The Introduction must be consistent, and reading the manuscript, in my opinion, this goal is not achieved. The Introduction is not sufficiently developed for the mentioned research field, there are mentioned only 4 references[1-4].
Answer. The introduction quite clearly describes the formulation of the problem, with an indication of the difficulties encountered in it. The lemma by which these difficulties will be overcome is also described. The bibliography on this subject is not numerous. We indicate only those devoted to this topic. There are no other sources.
- The manuscript has only 10 references, being very poor. Even the bibliography is very poor references 5-8 are not cited in the text. Why?? The last mention of reference 4 is done on page 13 ( the first mention of reference 4 is on page 2). From page 2 to page 13 is not mentioned any references in the range 5-8. The authors jump to reference 10 on page 13 and return to reference 9 on page 14 of the manuscript. Why???
Answer. We have added some sources on this topic. The remark about the order of references is reasonable. We have put the links in order.
- The paragraph “3. Solvability of iterative problems” is not mentioning any reference and the paragraph “4. Asymptotic convergence of formal solutions to the exact solution” is mentioning only reference 4. Why???
Answer. The observation is correct. We have added literature describing theorems about the solvability of iterative problems.
- In the paragraph “5. Construction of an asymptotic solution to the original problem (1)” is mentioned reference 4 on page 13 and on the same page is mentioned reference 10 ( I don’t understand the “jump”). While on page 14 is mentioned reference 9 ( “jumps” and “returns” ???).
Answer. The jumpiness of the links is caused by our laxity in keeping the order of the links. We have corrected this carelessness.
- The paragraphs “Applications” and “Conclusions” are missing, causing the reader not to understand the goal of the research.
Answer. The results of the work can be applied to those areas in which asymptotics are used, especially if it is related to the instability of the spectrum. We have completed the conclusion.
- Because of this omission mentioned before the manuscript is missing the specific scientific fields where the work results can be used.
Answer. In the case of a stable spectrum, we can construct a regularized asymptotics without using a fundamental solution. In the case of an unstable spectrum it is necessary to take into account its point features. In this case, inhomogeneity plays an essential role. It significantly affects the type of singularities in the solution of the initial problem. The fundamental solution allows us to construct asymptotics regardless of the nature of the inhomogeneity (it can be both slowly changing and rapidly changing, for example, rapidly oscillating). The approach developed in the paper is universal with respect to arbitrary inhomogeneity.
- Can the authors explain what brings new the work presented in the manuscript compared with the previously published https://doi.org/10.1515/math-2021-0021.
- The manuscript is not emphasizing the novelty of the work.
Answer. Paragraphs 6 and 7 are the answer to this observation.
Round 2
Reviewer 1 Report
The authors have tried incorporate comments and suggestions. But they should check all equations. In many same equations, they have "+" or "-" sign at the end as well as beginning of next mathematical expression.
Moreover, conclusion should rewrite since it is not showing, what has been done in the paper.
Author Response
- The authors have tried incorporate comments and suggestions. But they should check all equations. In many same equations, they have "+" or "-" sign at the end as well as beginning of next mathematical expression.
Answer. In the text of the article, in all mathematical expressions, the signs “+”, “-”, “=”, “” are removed from the end and left at the beginning of the next line.
- Moreover, conclusion should rewrite since it is not showing, what has been done in the paper.
Answer. The conclusion of the article is supplemented with suggestions «However, in this case, different exponential inhomogeneities may have different points of instability belonging to the considered time interval. Therefore, it will not be easy to single out essentially special singularities in the solution of the original problem and to carry out further regularization of the problem. Using our method of constructing asymptotics with the help of a fundamental solution, this difficulty can also be overcome. The specific implementation of this case is the subject of a separate article».
- Are the conclusions supported by the results?
Answer. The result obtained shows that the main term of the asymptotics (35) of the original problem (1) for the fundamental solution is constructed by the algorithm of the regularization method.
This phrase was written by us on Note 2.
Reviewer 3 Report
The revised form of the Manuscript ID axioms-2210405, titled “Asymptotic Solution of a Singularly Perturbed Integro-Differential Equation with Exponential Inhomogeneity,” belongs to the field of Mathematical Analysis, subfield Asymptotic solutions of The Integrodifferential Equations.
Concerning the scientific content of the paper
All the comments and suggestions were corrected in the revised form of the manuscript by the authors.
About the presentation of the paper concerning the Manuscript Type MDPI journal template
1. The References don’t respect 100% the form demanded by the Manuscript Type MDPI journal template.
For the aspects mentioned above, I advise a minor revision of the paper.
Author Response
- Does the introduction provide sufficient background and include all relevant references
Answer. In the introduction of the article, the background of the research is quite adequately described, and they are reflected in the corresponding links.
- Are the conclusions supported by the results?
Answer. The result obtained shows that the main term of the asymptotics (35) of the original problem (1) for the fundamental solution is constructed by the algorithm of the regularization method.
This phrase was written by us on Note 2.
In addition, the conclusion of the article is supplemented by sentences «However, in this case, different exponential inhomogeneities may have different points of instability belonging to the considered time interval. Therefore, it will not be easy to single out essentially special singularities in the solution of the original problem and to carry out further regularization of the problem. Using our method of constructing asymptotics with the help of a fundamental solution, this difficulty can also be overcome. The specific implementation of this case is the subject of a separate article».
