On the Regularized Asymptotics of a Solution to the Cauchy Problem in the Presence of a Weak Turning Point of the Limit Operator
Round 1
Reviewer 1 Report
Noticed technical inaccuracies should be corrected:
p.1 Introduction, line 7 \epsilon=0
p.2 line 16 \farc{1}{\epsilon}
p.2 line 11 function h{t} must be defined earlier
p.2 (2.1) u(0,\epsilon)
p.2 line 3 below specify index i values
p.3 line 7 below specify index s values
p.12 line 1 delete in
p.13 line \div
p.14 (6.2) skipped T before ds_1
p.15 line 2 below unpaired bracket
p.15 when describing an example, it is better not to repeat the conditions specified after (2.1), but to add explicitly the parameter values for the example
Throughout the text:
supplement and correct punctuation marks in formulas
Eliminate Russian words: pp.7,11, 13(3 times),14,17
Improve English
Author Response
Replies to the reviewer's comments are in tex and pdf files. Highlighted in red
Author Response File: 
Author Response.pdf
Reviewer 2 Report
On the regularized asymptotics of a solution to the Cauchy problem in the presence of a weak turning point of the limit operator by Yeliseev Alexander.
An asymptotic solution of a singularly perturbed Cauchy problem is constructed in the case of a weak turning point for the limit operator.
The problem of constructing an asymptotic in the solution of a singularly perturbed Cauchy problem in the case of a weak turning point of the limit operator A (t) causes particular difficulties. A feature of this problem is that two eigenvalues ​​of the operator A (t) intersect for some values ​​of the variable t. And for other values ​​of t, the eigenvalues ​​are different from each other. In this case, it is said that the spectrum of the operator is not stable. Moreover, it is assumed that the operator A (t) remains diagonal for all t. This leads to considerable difficulties in constructing an asymptotic series in powers of ε of solving the problem. The author successfully coped with this problem in our opinion.
This work is undoubtedly of scientific interest. All theorems are stated and proved.
Nevertheless, we give some remarks which, from our point of view, must be eliminated.
Remarks:
§1. Introduction
1.The sentence "The point ε for a singularly perturbed Cauchy problem is singular in the sense that the classical theorems the existence of a solution to the Cauchy problem does not take place at this point" assumes the point ε = 0? (Typo?)
2. In the sentence “In [4], singularities are presented in the case of a simple turning point, when a single point of the spectrum of the operator A (t) has the form” does a separate point of the spectrum mean an eigenvalue of the operator?
3. It is advisable in the introduction to indicate the difference between the results of this paper and the work [6].
§2. Statement of the problem. Description of the main singularities of the problem
4. To clarify in (2.2): f (t) is an arbitrary function? Clarify the purpose of the Lagrange - Sylvester interpolation polynomial.
5. To clarify in (2.3): J1 (t, ε), J2 (t, ε) are singularity functions?
6. In (2.9) give a decoding of the symbols ⊕, ⊗ (direct sum and tensor product)?
§6. Application
7. In formula (6.1) indicate that J = end is a vector, where J1 (t, ε), J2 (t, ε) are functions satisfying (2.3) .
8. What does the phrase “The system (6.1) in the general case is not explicitly solved” mean?
Nevertheless, these remarks do not detract from the significance of this work, and after their elimination, the article can be recommended for publication in your journal.
Author Response
Reply to reviewer comments
Changes to the PDF file are highlighted in red in the revised file
- The sentence "The point ε for a singularly perturbed Cauchy problem is singular in the sense that the classical theorems the existence of a solution to the Cauchy problem does not take place at this point" assumes the point ε = 0? (Typo?)
Fixed
- In the sentence “In [4], singularities are presented in the case of a simple turning point, when a single point of the spectrum of the operator A (t) has the form” does a separate point of the spectrum mean an eigenvalue of the operator?
Printed « point of the spectrum of the operator A (t) has the form”
Fixed “the only eigenvalue of the operator A (t)…”
Moreover, it is assumed that the operator A(t) has a diagonal form for any t….
- It is advisable in the introduction to indicate the difference between the results of this paper and the work [6].
The introduction adds a distinction to the work [6] of this article
- To clarify in (2.2): f (t) is an arbitrary function? Clarify the purpose of the Lagrange - Sylvester interpolation polynomial.
Explanations entered in the text.
- To clarify in (2.3): J1 (t, ε), J2 (t, ε) are singularity functions?
Clarification introduced
- In (2.9) give a decoding of the symbols ⊕, ⊗ (direct sum and tensor product
Character definitions entered in the text
- In formula (6.1) indicate that J = end is a vector, where J1 (t, ε), J2 (t, ε) are functions satisfying (2.3) .
Clarification entered in the text.
- What does the phrase “The system (6.1) in the general case is not explicitly solved” mean?
Means that the solution cannot be written as an explicitly defined function.
I consider it inappropriate to clarify the text
Author Response File: 
Author Response.docx
Reviewer 3 Report
The article describes the asymptotic solution of a linear Cauchy problem based on regularization approach.
I believe that description of the results could be improved, by adding related works, the organization of the article, the annex with proofs, and correcting some terms which appears to be non english language.
Author Response
Corrected the terms, removed the Russian words. The proofs of the theorems are carried out carefully, some proofs are presented in the appendix. I agree that the article is somewhat heavy in the presentation. But asymptotic solutions to complex problems always sin in this way. According to the literature, the subject of the article is a consistent study of the turning point from a common perspective. thank
Author Response File: Author Response.pdf
