Hereditary Mathematical Model of the Dynamics of Radon Accumulation in the Accumulation Chamber

Round 1

Reviewer 1 Report

Comments for author File: Comments.pdf

Author Response

The manuscript builds a new mathematical model for studying the hereditary mechanism of accumulation of radioactive radon gas in a chamber with gas-discharge counters at several observation points in Kamchatka. The proposed hereditary model of radon accumulation in the chamber, based on fractional derivatives, is a generalization of the previously known model with an integer derivative (classical model). The results show that due to the order of the fractional number of the derivative, as well as the quadratic nonlinearity in the model equation, the results of numerical simulation give a better approximation of the experimental data of radon monitoring than classical models.

The manuscript is informative, all the statements are proven correctly and I certainly propose to publish it.

Here are just a few comments that we recommend correcting before publishing the manuscript:           

Point 1: page 5, line 170: instead of "that Γ(α) from (4) is" write "that Γ(α) is";

Response 1: Agree. Corrected. Thanks for the note. 

Point 2: page 5, line 176: instead of "here the Gerasimov-Caputo fractional derivative [36,37] is of order 0 < α < 1." write "here @0αt is the Gerasimov-Caputo fractional derivative of order 0 < α < 1 [36,37].";

Response 2: Agree. Corrected.

Point 3: page 6, formula (11): in our opinion, the designation  is not successful, since if α(t) is a constant, then the parameter θ does not participate in the definition of  (see formula (6));

Response 3: Disagree. Θ - the parameter is needed only to correct the physical dimension of the model term, where a fractional derivative is used instead of the usual derivative. The importance of its use was noted by us recently, from [41]. Its use is advisory in nature, and makes sense only for applied tasks. However, Θ is not in the integrand, i.e. does not refer directly to the definition of a fractional operator of the Gerasimov-Caputo type used by us. This means that it is not necessary to mark it in a short form of recording .

 

Point 4: in many places, instead of a comma, you need to put a dot: in formulas (7), (8) and so on;

Response 4: Agree. Corrected.

Point 5: page 7, line 219: instead of ";" write ".".

Response 5: Agree. Corrected. Thanks for the note.

We would like to thank you for your valuable comments, which helped improve the quality of the article.

Author Response File: Author Response.docx

Reviewer 2 Report

The authors demonstrate the effectiveness of fractional calculus in the study of hereditary phenomena and in particular in the use of the Gerasimov-Caputo fractional derivative that allows to highlight this character. The importance of a 222radon concentration analysis and monitoring tool is well known. It is appropriate for better information to specify that different radon isotopes exist in nature and to understand whether and how the proposed model may be used to discriminate against other types of radioactive elements. The manuscript is very interesting and lends itself to a wide audience of readers. I think it is of high scientific value and well structured. However, authors are required to specify the unit of time given in figures 3, 10 and 13. In addition, the authors must represent the trend of the fractional order alpha at 200 units of time in Figure 10 c) to show that the value is always less than 1. Finally, it would be appropriate to point out some bibliographical references related to the study of other sites for example:

Neri M, Ferrera E, Giammanco S, et al. (2011) Space ground radon distribution as a tool to recognize active failure on an active volcano: the example of Mt. Etna (Italy). J Environ Radioact 102: 863-870. https://doi.org/10.1016/j. jenvrad.2011.05.002

Barberio, M.D.; Gori, F.; Barbieri, M.; Billi, A.; Devoti, R.; Doglioni, C.; Petitta, M.; Riguzzi, F.; Rusi, S. Diurnal and Semidiurnal Cyclicity of Radon (222Rn) in Groundwater, Giardino Spring, Central Apennines, Italy. Water 2018, 10, 1276. https://doi.org/10.3390/w10091276

Author Response

The authors demonstrate the effectiveness of fractional calculus in the study of hereditary phenomena and in particular in the use of the Gerasimov-Caputo fractional derivative that allows to highlight this character. The importance of a 222radon concentration analysis and monitoring tool is well known. It is appropriate for better information to specify that different radon isotopes exist in nature and to understand whether and how the proposed model may be used to discriminate against other types of radioactive elements. The manuscript is very interesting and lends itself to a wide audience of readers. I think it is of high scientific value and well structured.

 

Point 1: However, authors are required to specify the unit of time given in figures 3, 10 and 13.

            Response 1: For Figure 3, specifying the unit of time (as it is done, for example, for Figure 4) will not work. Unfortunately, in the files with the data of interest to us from the PRTR point for that period, information about the measurement time was not recorded or it was lost.

In Figures 10 and 13, this is the current simulation time and is a dimensionless quantity. Figures 10 and 13 are rather technical in nature. Figure 10 shows how the values of the simulation parameters change during the numerical experiment. And figure 13 clearly shows the memory effect that manifests itself during the simulation. Both of them are ultimately related to Figure 12 where they are compared: the model curve and the processed observed data. And for figure 12 it was necessary to specify the unit of time. However, in fig. 10 and 13 is not necessary.

Point 2: In addition, the authors must represent the trend of the fractional order alpha at 200 units of time in Figure 10 c) to show that the value is always less than 1.

Response 2: We are interested in precisely the specified interval , since it is within this framework that the burst of radon is determined, which we want to describe. So, unfortunately, it won't work. To do this, in example 4, you will have to redefine all parameter functions (not only alpha) so that, on the considered time interval , you get exactly the same behavior. And this means exactly the same solution of the model equation on this interval. But this will greatly complicate the form of the functions in Example 4. And to present in the article one more graph, where we build the alpha function up to 200, will be redundant.You can show that alpha will always be less than 1 by simply substituting the limit values:

Point 3: Finally, it would be appropriate to point out some bibliographical references related to the study of other sites for example:

Neri M, Ferrera E, Giammanco S, et al. (2011) Space ground radon distribution as a tool to recognize active failure on an active volcano: the example of Mt. Etna (Italy). J Environ Radioact 102: 863-870. https://doi.org/10.1016/j. jenvrad.2011.05.002

Barberio, M.D.; Gori, F.; Barbieri, M.; Billi, A.; Devoti, R.; Doglioni, C.; Petitta, M.; Riguzzi, F.; Rusi, S. Diurnal and Semidiurnal Cyclicity of Radon (222Rn) in Groundwater, Giardino Spring, Central Apennines, Italy. Water 2018, 10, 1276. https://doi.org/10.3390/w10091276

Response 3: Agree. Links to manuscripts added.

We would like to thank you for your valuable comments and references to interesting research, which served to better understand the results of the article and improve it.

Author Response File: Author Response.docx

Reviewer 3 Report

The work is written very carefully. Works that are so refined in terms of editing are rare (already at the beginning of the review process). The presented methods and the obtained results show that the work makes a significant contribution to the field discussed in it. An additional confirmation of the above is the patent mentioned by the authors at the end of the work. As I have already mentioned, it is difficult to find any editing errors - a few that I managed to find have been marked in the pdf file.

I have two questions for the authors:

1. Why the "mod" function is discriminated :-) (other functions, e.g. sin, cos, ... are written in simple font, and mod in italics)

2. what was the reason for using the Matlab program (most papers published in Mathematics are based, if they use this type of software at all, on the Mathematica program).

3. Have the authors considered the possibility of using other methods to solve the equation that describes the problem under study (both in its classical case and in the fractional case)? I wrote about several approaches in the previously mentioned pdf file.

Comments for author File: Comments.pdf

Author Response

The work is written very carefully. Works that are so refined in terms of editing are rare (already at the beginning of the review process). The presented methods and the obtained results show that the work makes a significant contribution to the field discussed in it. An additional confirmation of the above is the patent mentioned by the authors at the end of the work. As I have already mentioned, it is difficult to find any editing errors - a few that I managed to find have been marked in the pdf file.

I have two questions for the authors:

           

Point 1: Why the "mod" function is discriminated :-) (other functions, e.g. sin, cos, ... are written in simple font, and mod in italics)

Response 1: Mistake in Latex code. Fixed. Thanks for the note))

 

Point 2: what was the reason for using the Matlab program (most papers published in Mathematics are based, if they use this type of software at all, on the Mathematica program).

Response 2: Co-author and programmer in this project Tverdyi D.A. knows the Matlab program much better than the Mathematica program, and has more development experience in Matlab. This is the main reason. Earlier papers [38,42,45] used the Maple program, and it would be possible to continue using Maple, but to solve the problem in this study, it was necessary to develop a more functional and flexible program.

 

Point 3: Have the authors considered the possibility of using other methods to solve the equation that describes the problem under study (both in its classical case and in the fractional case)? I wrote about several approaches in the previously mentioned pdf file.

Response 3: At this stage, no, it has not been considered. We used the well-studied by the authors in [45] mathematical apparatus for the numerical solution, and the numerical solution schemes developed there.

We would like to thank you for your valuable and interesting comments, which helped to improve the quality of the article.

Author Response File: Author Response.docx