To the Problem of Discontinuous Solutions in Applied Mathematics
Round 1
Reviewer 1 Report
The paper deals with discontinuous solutions of common problems. Non-linear functions and non-linear derivatives have been introduced and two problems have been solved using the new approach. It seems, this approach plays the same role as the well-known local solutions describing the edge effect.
The paper would benefit from the description of the experimental studies outlined in Figs 3 and 6. Specifically, it would be useful to describe the method of loading of structures.
No problems with language.
Author Response
Response to the Reviewer’s Comments
Vasiliev Valery and Lurie Sergey
July 27, 2023
We are very grateful to the reviewers, who carefully read and evaluated the manuscript.
Please find enclosed the revised version of the paper. We have made a number of changes in the manuscript following given comments. Changes in the manuscript are highlighted with yellow color.
Reviewer #1
The paper deals with discontinuous solutions of common problems. Non-linear functions and non-linear derivatives have been introduced and two problems have been solved using the new approach. It seems, this approach plays the same role as the well-known local solutions describing the edge effect.
The paper would benefit from the description of the experimental studies outlined in Figs 3 and 6. Specifically, it would be useful to describe the method of loading of structures.
Thank You very much for evaluation and attention to this manuscript.
The following given comment we added the figure 4, which shows the experimental facility for investigating of a transversal displacements of stretched string under the action of a transversal local force. The using equipment provides a constant controlled thread tension. During the test, the transverse displacements of the thread are measured using an optical system with an accuracy of 0.1 mm.
We included the additional text in the manuscript.
As for the experimental study of the membrane, we have added a link to the work devoted to the non-singular solution for the membrane and the experimental study of its deflections under the action of a local force. In this work, an experiment on loading a membrane is presented and discussed, in which a controlled tension is provided.
We added the following work in the list of Reference.
- Vasiliev V.V. and Lurie S.A. Generalized Solution of the Problem on a Circular Membrane Loaded by a Lumped Force, Mech. Solids. 2016, Vol 51, No. 3, p. 334-338.
Thank you very much.
Authors:
Valery Vasiliev and Sergey Lurie
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The paper is dedicated to singular solutions in differential equations, commonly encountered in mathematics and applied physics. It is meticulously studied in the works of mathematicians. However, when it comes to applied mechanics such solutions often contradict common physical sense. To address these issues authors propose using alternative, named after Landen, definitions of derivatives instead of traditional Newton-Leibniz definitions. That permits regularization of singularities.
Details of these procedures are illustrated by comparing solutions for deformations of a string and a beam. Another example is for axisymmetric deformations of a thin round plate.
Most derivations are clear, convincing, and easy to follow. The only possible exception is the example at the end of part 2. There is some discrepancy, which can be detected by direct substitution. Please, check it or disregard this comment if I’m wrong.
A couple comments about references. Ref. 4 is most likely about a book by Vol’mir. I know that but younger guys may have difficulty finding it just by the title. Also, if Ref. 5 is to the book by Abramowitz&Stegun, referencing its 1964 English edition will be more appropriate.
Generally, the paper is written in clear conventional English. One minor exception is the first introductory paragraph.
I would change “The paper is concerned with” by “The paper addresses” or something like that. Concerned has some negative flavor.
The sentence, starting with “Nonlocal functions and nonlocal derivatives which are not specified…” is too long. Try to break it up in shorter sentences for better understanding.
In many places punctuation is questionable, particularly usage of commas before right adjectives. These rules are not rock-solid, but a simple rule is that in most cases there is a comma before an adjective clause starting with “which”. Having said that, even native English speakers, which I am not, often have disagreements about placing commas.
Author Response
Response to the Reviewer’s Comments
Vasiliev Valery and Lurie Sergey
July 27, 2023
We are very grateful to the reviewers, who carefully read and evaluated the manuscript.
Please find enclosed the revised version of the paper. We have made a number of changes in the manuscript following given comments. Changes in the manuscript are highlighted with yellow color.
Reviewer #2
The paper is dedicated to singular solutions in differential equations, commonly encountered in mathematics and applied physics. It is meticulously studied in the works of mathematicians. However, when it comes to applied mechanics such solutions often contradict common physical sense. To address these issues authors propose using alternative, named after Landen, definitions of derivatives instead of traditional Newton-Leibniz definitions. That permits regularization of singularities.
Details of these procedures are illustrated by comparing solutions for deformations of a string and a beam. Another example is for axisymmetric deformations of a thin round plate.
- Most derivations are clear, convincing, and easy to follow. The only possible exception is the example at the end of part 2. There is some discrepancy, which can be detected by direct substitution. Please, check it or disregard this comment if I’m wrong.
The authors are grateful to the referee for a careful reading of the paper. Indeed, in the second part of the work, the authors made an unfortunate mistake in the text of the article.
This part has been corrected and highlighted in the new version of the article. All the given examples and results refer, of course, to the correct version of the manuscript.
- A couple comments about references. Ref. 4 is most likely about a book by Vol’mir. I know that but younger guys may have difficulty finding it just by the title. Also, if Ref. 5 is to the book by Abramowitz&Stegun, referencing its 1964 English edition will be more appropriate.
We have replaced Ref. 5 in References with the English edition as recommended by the reviewer.
Unfortunately, we did not find the opportunity to replace link 4. since only in the indicated work we found the Equation (22) used in this work.
Comments on the Quality of English Language
Generally, the paper is written in clear conventional English. One minor exception is the first introductory paragraph.
I would change “The paper is concerned with” by “The paper addresses” or something like that. Concerned has some negative flavor.
The sentence, starting with “Nonlocal functions and nonlocal derivatives which are not specified…” is too long. Try to break it up in shorter sentences for better understanding.
In many places punctuation is questionable, particularly usage of commas before right adjectives. These rules are not rock-solid, but a simple rule is that in most cases there is a comma before an adjective clause starting with “which”. Having said that, even native English speakers, which I am not, often have disagreements about placing commas.
Thank You very much for evaluation and attention to this manuscript.
The reviewer's recommendations have been fully implemented. Corrections and additions have been made to the text of the introductory paragraph, which highlighted with yellow color.
Thank you very much.
Authors:
Valery Vasiliev and Sergey Lurie
Author Response File: Author Response.pdf
Reviewer 3 Report
The work is written in clear language, understandable to the reader. it touches upon the problems of singular solutions that are important for applications.
It is emphasized that singular solutions can be quite mathematically substantiated but are often unacceptable from a physical point of view in applied problems of mechanics and physics.
The simplicity and clarity of presentation, despite the complexity of the problem under consideration, will undoubtedly be attractive to readers.
The authors make the assumption that the singularity of solutions to problems of mathematical physics is associated with the use of the mathematical apparatus of differential calculus based on the calculus of infinitesimals
At the same time, the authors mention that the definition of derivatives is a non-trivial problem and point out the ambiguity of the definition of derivatives and the existence of other definitions. An example is the definition proposed by J. Landen in 1764., which predates the differential calculus L_N. This question in itself is interesting.
The authors propose a new conceptual approach related to the introduction of non-local functions that depend not only on the value of the original function at a point, but on the value of the second derivative as well (and, in general, on the values of higher-order derivatives).
In the future, the authors propose to use the Landen’s definition of the derivative to construct nonlocal derivartives of nonlocal functions.
It is shown that when determining solutions within the framework of the proposed approach, it is possible to introduce and determine local fields that are found up to the scale parameter as solutions of the Helmholtz equations
Clear and fairly easy-to-understand examples are given in which singular solutions are known solutions to classical test problems of equations of mathematical physics.
For a minor revision, I have the following remarks, which will eliminate the technical error. Eliminating this error can undoubtedly be useful for the potential readers
I recommend that the authors check the end of section 2. I believe that instead of the formula (17)
(17) v=(P/2tk)(exp(-kx)-1) - x(P/2t)
should be the formula v=(P/2tk)(exp(-kx)-1) -(l/2 -x) (P/2t)),
valid for the region 0<x<l/4
The authors should check all the ratios in this part of the work.
I believe that this inaccuracy is associated with a technical error, which is easily eliminated.
The paper contains new scientific results which concern a wide range of applied problems in physics and mechanics, and allow us to formulate a procedure for regularizing singular solutions in them. Due to the simplicity and clarity of presentation, the article will attract readers, because it does not contain complex calculations. Which sometimes make it difficult to understand. I suggest acceptance of this paper for publication.
It is suggested to check the text for possible errors
Author Response
Response to the Reviewer’s Comments
Vasiliev Valery and Lurie Sergey
July 27, 2023
We are very grateful to the reviewers, who carefully read and evaluated the manuscript.
Please find enclosed the revised version of the paper. We have made a number of changes in the manuscript following given comments. Changes in the manuscript are highlighted with yellow color.
Reviewer #3
Comments and Suggestions for Authors
The work is written in clear language, understandable to the reader. it touches upon the problems of singular solutions that are important for applications.
It is emphasized that singular solutions can be quite mathematically substantiated but are often unacceptable from a physical point of view in applied problems of mechanics and physics.
The simplicity and clarity of presentation, despite the complexity of the problem under consideration, will undoubtedly be attractive to readers.
The authors make the assumption that the singularity of solutions to problems of mathematical physics is associated with the use of the mathematical apparatus of differential calculus based on the calculus of infinitesimals
At the same time, the authors mention that the definition of derivatives is a non-trivial problem and point out the ambiguity of the definition of derivatives and the existence of other definitions. An example is the definition proposed by J. Landen in 1764., which predates the differential calculus L_N. This question in itself is interesting.
The authors propose a new conceptual approach related to the introduction of non-local functions that depend not only on the value of the original function at a point, but on the value of the second derivative as well (and, in general, on the values of higher-order derivatives).
In the future, the authors propose to use the Landen’s definition of the derivative to construct nonlocal derivartives of nonlocal functions.
It is shown that when determining solutions within the framework of the proposed approach, it is possible to introduce and determine local fields that are found up to the scale parameter as solutions of the Helmholtz equations
Clear and fairly easy-to-understand examples are given in which singular solutions are known solutions to classical test problems of equations of mathematical physics.
For a minor revision, I have the following remarks, which will eliminate the technical error. Eliminating this error can undoubtedly be useful for the potential readers
I recommend that the authors check the end of section 2. I believe that instead of the formula (17)
(17) v=(P/2tk)(exp(-kx)-1) - x(P/2t)
should be the formula v=(P/2tk)(exp(-kx)-1) -(l/2 -x) (P/2t)),
valid for the region 0<x<l/4
The authors should check all the ratios in this part of the work.
I believe that this inaccuracy is associated with a technical error, which is easily eliminated.
The paper contains new scientific results which concern a wide range of applied problems in physics and mechanics, and allow us to formulate a procedure for regularizing singular solutions in them. Due to the simplicity and clarity of presentation, the article will attract readers, because it does not contain complex calculations. Which sometimes make it difficult to understand. I suggest acceptance of this paper for publication.
Thank You very much for evaluation and attention to this manuscript.
The following given comment we revised part two starting with formula (16) and continuing through the end of the section. We carefully checked the entire text of the article as recommended by the reviewer.
Corrected text is highlighted with yellow color.
Thank you very much.
Authors:
Valery Vasiliev and Sergey Lurie
Author Response File: Author Response.pdf
