An Approach for Numerical Solutions of Caputo–Hadamard Uncertain Fractional Differential Equations

Round 1

Reviewer 1 Report

Uncertain fractional differential equations are studied. In particular uncertain differential equations in the sense of Liu are considered. In these equations the role of the Wiener process of the Ito stochastic differential equations is taken by a Liu process. Some elementary theorems about these equations are deduced and some explicit formulae and numerical approximations useful to solve some sample equations are given. 

My concerns with this paper are :

1 The results presented have no motivation,

2  The use of English is not always satisfactory. Many sentences must be rewritten.

After being revised carefully the paper can be accepted.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

In the article under review, the authors dealt with the numerical algorithm for numerical solving the Caputo-Hadamard uncertain fractional differential equations, they introduced the definition of $\alpha$-path to establish the relationship between the Caputo-Hadamard uncertain fractional differential equation and Caputo-Hadamard fractional differential equation, found that the $\alpha$-path is the inverse uncertainty distribution of the Caputo-Hadamard uncertain fractional differential equation, provided a formula for calculating the expected value of a monotonic function for the solution of the Caputo-Hadamard uncertain fractional differential equations and designed the numerical algorithms for calculating the inverse uncertainty distribution and the expected value of solution of the Caputo-Hadamard uncertain fractional differential equations, and provided several numerical examples to illustrate the effectiveness and accuracy of t he proposed algorithms.

After reading and checking the full article, I didn't find any mathematical and logic mistakes. The proposed methods and techniques are interesting for readers who are working on the related fields. Therefore, I recommend the article to be accepted for publication in the Fractal and Fractional after providing minor revision.

Comments and suggestions:

(1) The authors should read and check the full article very carefully to correct possible grammar and spelling mistakes;

(2) Rewrite the Abstract section to make it more concise and precise;

(3) Further strengthen the motivation for writing this article in the Introduction section;

(4) Polish the English writing of the full article;

(5) Some recently published articles focus on the topic of the article are missing in the references. I suggest the authors to cite the following items in the text and in the references such that the readers can better understand the latest progress in this research field:

https://doi.org/10.1155/2020/6598682

https://doi.org/10.1016/j.rinp.2021.103953

https://doi.org/10.1088/1402-4896/ac0bce

Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Authors have considered the uncertain fractional differential equation (UFDE) is an effective tool to describe dynamic changes in an uncertain environment. In most cases, it is hard to obtain analytical solutions for nonlinear Caputo-Hadamard UFDE. The presented work is good. I recommend for publication.

Some revision is needed like:

1. There are many grammatical mistakes. English should be improved.

2. In the introduction why uncertain problems are so important?

3. Relate your work with recent work like: 

8) Existence and stability of impulsive coupled system of fractional integrodifferential equations DOI10.1515/dema-2019-0035
9) Almost periodic dynamics of a discrete Nicholson's blowflies model involving a linear harvesting term Difference equations: New trends and applications in biology, medicine and biotechnologyDOI
10.1186/1687-1847-2012-158

10) Investigation of the p-Laplacian nonperiodic nonlinear boundary value problem via generalized Caputo fractional derivativesDOI
https://doi.org/10.1186/s13662-021-03228-9

11) Generalized fractional derivatives generated by a class of local proportional derivativesDOI
https://doi.org/10.1140/epjst/e2018-00021-73. Explain graphical presentation.4. Enlist applications of randomness problem in real-world procedure.

Author Response

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Author Response File: Author Response.pdf

Reviewer 4 Report

REVIEW REPORT

TITLE: An approach for numerical solutions of Caputo-Hadamard uncertain fractional differential equations

My comments are as follows:

The obtained results are correct, to my knowledge. However, the present work requires significant revision in order to publish in reputed journals.

1.      The author advises reading the entire manuscript and correcting all typographical and grammatical errors.

2.      The abstract is not organized according to its scientific relevance.

3.      The keywords need to be changed.

4.     For all the equations, numbers should be added to all the mathematical definitions and formulae used in the entire manuscript.

5.       The novelty of the present work needs to be highlighted.

6.   Check all the mathematical expressions throughout the paper. The representation of each term in the mathematical equations should be done properly.

7.    The introduction section needs to improve professionally. Particularly, fractional calculus, fractional differential equations and their applications need to be discussed. 

8.      The keyword financial system should be removed.

9.      More description of the obtained results should be highlighted.

10.  The results and discussion section need to be improved with more description.

11.  The references are not sufficient for mathematical modeling and numerical-based research. The author needs to work on this.

12.  Some information is missing in the introduction section; the authors need to make sure all the references are uniform and include all the missing information.

13.  The future direction of the present work should be highlighted to help the readers.

Comments for author File: Comments.pdf

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 4 Report

Accept