Numerical Solution for Fuzzy Fractional Differential Equations by Fuzzy Multi-Step Methods

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The authors proposed a numerical solution for FFDEs, using the fuzzy multi-step method. However, there are many unclear contents in the paper that need to be revised.

1. Why study this content, what is the purpose of the research, and what is the significance of studying this content? It should be clearly explained in the text, such as abstract, introduction, main text, etc.
2. So far, what research has been conducted on FFDEs? These research results should be sorted out in detail, such as what content has been studied, what methods have been used, and what results have been obtained. Especially the achievements related to the paper need to be explained in detail. For FFDEs, what methods have other scholars used to solve the FFDEs proposed in this paper? Revise the introduction and main text.
3. Also, was the fuzzy multi-step method proposed by the author? If not, relevant research results should be cited in the introduction section.
4. Why is this method proposed throughout the article to solve FFDEs, and what problems can this method solve? What are the advantages compared to other methods? That is to say, what are the innovative points of your entire article? Please list them in the introduction section.
5. There are some details that need to be carefully revised, such as 3-5 keywords, the first few characters left blank in each paragraph, and the format of citing references in the introduction, which should be modified according to the format specifications of this journal. Suggest checking the entire text.
6. On page 7 of the main text, an=1, an-1=-1, an=0,k=0,1,...,n-2, unclear expression, suggested modification.
7. In 3.1.1, Xk=a+kh, k=0,1,...,n, does n have a specific meaning, and what does it refer to?
8. How to prepare the results obtained by applying the method described in the paper in Examples 6.1, 6.2, and 6.3? What indicators should be used to evaluate the method you proposed? There should be corresponding textual explanations.
9. What specific modeling problems can the method proposed by the author be applied to solve FFDEs?

Author Response

Reviewer 1

Thank you for your review of our paper.  Our responses to your points are as follows.

 

• Why study this content, what is the purpose of the research, and what is the significance of studying this content? It should be clearly explained in the text, such as the abstract, introduction, main text, etc.

Our response:

The abstract and introduction sections have been rewritten to explain why this content is studied, its significance, and our aim regarding this content, especially in the introduction.

 

• So far, what research has been conducted on FFDEs? These research results should be sorted out in detail, such as what content has been studied, what methods have been used, and what results have been obtained. Especially the achievements related to the paper need to be explained in detail. For FFDEs, what methods have other scholars used to solve the FFDEs proposed in this paper? Revise the introduction and main text.

Our response:

We have answered these questions in the introduction section in the 1-paragraph. (see page 1)

 

• Also, was the fuzzy multi-step method proposed by the author? If not, relevant research results should be cited in the introduction section.

Our response:

Yes. We proposed.

 

• Why is this method proposed throughout the article to solve FFDEs, and what problems can this method solve? What are the advantages compared to other methods? That is to say, what are the innovative points of your entire article? Please list them in the introduction section.

Our response:

 

• It is my personal opinion that the multi-step methods are more interesting methods in solving ordinary differential equations, and it will be very good when we can solve the fuzzy and especially fractional differential equations with these methods because they have the high order of accuracy in the obtained results.

There are some details that need to be carefully revised, such as 3-5 keywords, the first few characters left blank in each paragraph, and the format of citing references in the introduction, which should be modified according to the format specifications of this journal. Suggest checking the entire text.

Our response:

Keywords, citation references in the introduction have been updated.

 

• On page 7 of the main text, an=1, an-1=-1, an=0, k=0,1,...,n-2, unclear expression, suggested modification.

Our response:

Index n is revised to k. (see page 7.)

 

• In 3.1.1, X_k=a+kh, k=0, 1,...,n, does n have a specific meaning, and what does it refer to?

Our response:

Yes. Index n is the number of equal subintervals that we have divided in [a, b]. (see page 7)

 

• How to prepare the results obtained by applying the method described in the paper in Examples 6.1, 6.2, and 6.3? What indicators should be used to evaluate the method you proposed? There should be corresponding textual explanations.

Our response:

We have solved these equations with MATLAB in all cases based on the type of differentiability of them and the best obtained results are given in the tables. If you would like we can test the codes on any other models that you want.

 

• What specific modeling problems can the method proposed by the author be applied to solve FFDEs?

Our response:

This method is able to solve linear and nonlinear fuzzy differential equations, such as evaluating weapon systems, and population models.

 

 

Author Response File: Author Response.docx

Reviewer 2 Report

Comments and Suggestions for Authors

The paper titled "Numerical Solution for Fuzzy Fractional Differential Equations By Fuzzy
Multi-Step Methods" is reviewed. The paper is very nicely written and the authors had done the appreciable work.

1. The sentence "In thie present work, we develop the fuzzy Adams-Bashforth" can be revised as ..... This work developed the fuzzy Adams-Bashforth....
2. Avoid the usage of words I/WE, instead of these you can prefer the words "The authors/researchers in this work"
3. The literature section can be enhanced by discussing a few other fuzzy aided and fuzzy fractional order methods available in the literature.

4. The statistical results regarding the proposed theory can enhance the quality of the paper.

Author Response

Reviewer 2

Thank you for your review of our paper.  Our responses are given in a point-by-point manner below.

 

• The sentence “In this present work, we develop the fuzzy Adams-Bashforth” can be revised as….

This work developed the fuzzy Adams-Bashforth…

Our response:

This sentence is corrected in lines 4-6 of the abstract.

 

• Avoid the usage of words I/WE, Instead of these, you can prefer the words “ the  authors/researchers in this work”

Our response:

All sentences in this work are rewritten in passive form, then the words I/We are deleted, and in some places, it is needed to use the author is used. For example, in line 8 of the conclusion section.

 

• The literature section can be enhanced by discussing a few other fuzzy aided and fuzzy fractional order methods available in the literature.

Our response:

 

• The abstract and introduction section is rewritten more carefully, and other fuzzy fractional methods for solving fuzzy fractional differential equations.

 

• The statistical results regarding the proposed theory can enhance the quality of the paper.

Our response:

You are right. Thank you very much.

 

Author Response File: Author Response.docx

Reviewer 3 Report

Comments and Suggestions for Authors

Fuzzy Fractional Differential Equations (FFDEs) are one of the interesting areas of research in fuzzy mathematics, and they are applied in various branches of engineering and applied science. Since the invention of FFDEs, many methods have been developed for numerical solutions of FFDEs by reformulating them to crisp equations. In this paper, the authors proposed two fuzzy methods, namely the fuzzy generalized Adams-Bashforth method and the fuzzy generalized Adams-Moulton method to find the numerical solution of the FFDEs directly in the fuzzy form without reformulating them to a crisp form. They further established the symmetry properties and stability of the solutions, to show the consistency effectiveness, and reliability of developed methods. The concepts, methods, and results presented in the paper seem original, interesting, and supported by numerical simulations.  The quality and organization of the paper are comparable to the standard of any reputed international Journal. Hence, I recommend this paper for publication in Symmetry.

Author Response

Reviewer 3

 

Comments and Suggestions for Authors

Fuzzy Fractional Differential Equations (FFDEs) are one of the interesting areas of research in fuzzy mathematics, and they are applied in various branches of engineering and applied science. Since the invention of FFDEs, many methods have been developed for numerical solutions of FFDEs by reformulating them to crisp equations. In this paper, the authors proposed two fuzzy methods, namely the fuzzy generalized Adams-Bashforth method and the fuzzy generalized Adams-Moulton method to find the numerical solution of the FFDEs directly in the fuzzy form without reformulating them to a crisp form. They further established the symmetry properties and stability of the solutions, to show the consistency, effectiveness, and reliability of the developed methods. The concepts, methods, and results presented in the paper seem original, interesting, and supported by numerical simulations.  The quality and organization of the paper are comparable to the standard of any reputed international Journal. Hence, I recommend this paper for publication in Symmetry.

Thank You.

 

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The paper has been greatly improved through revision, and I think it is acceptable.