Generalization of the Regularization Method to Singularly Perturbed Integro-Differential Systems of Equations with Rapidly Oscillating Inhomogeneity

Round 1

Reviewer 1 Report

See attachment

Comments for author File: Comments.pdf

Author Response

Replies to the reviewer 1.

The author rightly notes that in the work there are no references to the possibility of applying other methods to the problem under consideration (for example, the averaging method or multiple scale asymptotic procedure method). Regarding the application of the averaging method, we note that in our example the inhomogeneities do not have a finite time average. As for the method of many scales, it is possible that it can be applied, but in the available mathematical literature we have not come across any works on the application of this method to singularly perturbed systems containing an integral operator.

2. The method of boundary functions essentially uses the exponential decrease of solutions outside the boundary layer, which can be ensured only in the case when the points of the spectrum of the matrix A(t) lie strictly in the left half-plane of the complex variable. In our case/ the points of the spectrum can be purely imaginary. The application of this method is also hindered by a rapidly oscillating imhomogeneity, which actually corresponds to the presence of purely many points in the spectrum.

3. The work is indeed theoretical. The possible of applying it to problems depends on the availability of an appropriate real model, which we, unfortunately, do not have at the moment.

Author Response File: Author Response.docx

Reviewer 2 Report

The investigations in this paper are not clear and applicable . The authors tried to suggest some the asymptotic of the solution of the studied problem. They provide an example, and they tried to illustrate on this example the practical application of the theoretical result. But this example illustrates the opposite. It is shown by this example that the asymptotic of the solution could not be obtained in an explicit form, it contains a solution of a system of  integral equations, which could not be solved explicitly (including in this particular case, not tolking about the genera case). So, this asymptotic is not applicable and the provided results are not applicable and not interesting to other readers of the journal.

Additionally, the paper is not written carefully. For example, in the abstract, there is a sentence without any verb. The key words are written twice.

Author Response

Replies to the reviewer 2.

1. The integro-differential systems considered in our work have not been previously studied from the point of view of obtaining asymptotics. The results are theoretical. Perhaps in the future they will find practical application.
2. The purpose of this work is to develop an algorithm for constructing the asymptotics of finite order. In the example given at the end of the article, the leading term of the asymptotics is obtained in final form. Upon careful examination, one can make sure that all functions written out in it are calculated exactly. If desired, we can obtain higher-order asymptotic solutions using the algorithm we have developed. But this is not necessary, since in practice the leading term of the asymptotics is usually used.
3. An indication of the negligence of the presentation of the article is valid only in relation to the repeated spelling of key words. As for the article as a whole, in our opinion, it is written neatly.

Best regards, authors.

Author Response File: Author Response.docx

Reviewer 3 Report

The paper under review is a very interesting study on systems of singularly perturbed integro-differential equations with a rapidly oscillating right-hand side, including an integral operator with a slowly varying kernel.

Mathematics are correct but I would like that the authors compare their approach with other recent one appear in the literature for this problem. I mean the techniques stated in

https://0-doi-org.brum.beds.ac.uk/10.2478/amns.2020.2.00055

Do you think that a fractional approach can help you to go further?

I would like to read a discussion on it.

Author Response

Reviewer's note.

Mathematics are correct but I would like that the authors compare their approach with other recent one appear in the literature for this problem. I mean the techniques stated in

Do you think that a fractional approach can help you to go further?

I would like to read a discussion on it.

Answer.

Your approach is based on the operator  being the compression operator and in our case, an integral equation of the type (14) looks like this:

therefore, the function at is not a compression operator. Unfortunately, your approach cannot be applied to singularly perturbed integral equations.

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

I am satified with the answers by the authors

Author Response

Thank you.

Reviewer 2 Report

The authors neither answered to my review adequately nor changed the paper.

They insist they construct an asymptotic without any considerations about the closeness of this so called “leading term” with the exact solution. It makes the practical application useless. Any construction of an asymptotic has to done with a theoretical proof of the error which is not done in this paper. Or at least an example showing thia error has to be provided. Nothing is done in this paper. Definitely, the provided algorithm is not interesting to the most readers of the journal, it is not practically applicable. Thus, the paper definitely has to be rejected.

Author Response

Reviewer's comments are of a general nature. In evaluating the remainder

there is no need, as it is described in detail in our previous works (see, for example, [16]).

We believe that the review is not objective. If the editorial board has a different opinion, we are ready to remove our article and send it to another journal.

Best regards, authors.