Solving a Higher-Dimensional Time-Fractional Diffusion Equation via the Fractional Reduced Differential Transform Method

Round 1

Reviewer 1 Report

Manuscript presents interesting results regarding the use of numerical methods to solve fractional differential equations. My only suggestion is to add, for one of the example cases, a comparison with previously reported method in order to show an equivalent/better accuracy and precision of the proposed method and an equal/lower computational time. As a suggestion, would be a comparison to the method reported by the manuscript bellow. Igor Podlubny, Aleksei Chechkin, Tomas Skovranek, YangQuan Chen, Blas M. Vinagre Jara. Matrix approach to discrete fractional calculus II: Partial fractionaldifferential equationsIgor Journal of Computational Physics 228 (2009) 3137–3153.

Author Response

1. In the first example I indicate: " this is exactly the same exact solution obtained using fractional variational homotopy perturbation iteration method (FVHPIM) with modified Riemann-Liouvile derivative".
2.  I also add this sentence in example 1: "In comparison with the approximate solution obtained by homotopy decomposition method (HDM) via modified beta derivative
equation, FRDTM gives direct exact solution with simple computations"

Abuasad, S.; Hashim, I. Homotopy decomposition method for solving one-dimensional time-fractional diffusion equation. AIP
Conference Proceedings 2018, 1940, 020126

Author Response File: Author Response.pdf

Reviewer 2 Report

1. The abstract of the paper should be extended to be more informative.
2. The introductory section should be amended to cover: the main motivation of the work, extensive review literature, and the cons/pros of the method.
3. No need for Sec 2.2, all researchers in the field are aware of the elementary properties of the Gamma function.
4. The authors should clearly state the sufficient conditions on the function (say u) to have FRDT.
5. The methodology is too short and not clear, the methodology part should be extended with clear steps.
6. No analysis of the method was presented.
7. The authors should append an algorithm for the method.
8. The specification of the machine used to run the codes should be included, also which platform is used to write the codes.
9. All figures should be saved in eps format to reduce the size of the file to be more convenient to the reader.
10. Proofread the whole manuscript for typos and grammatical errors.

Author Response

Please, see the attachment.

Author Response File: Author Response.pdf

Reviewer 3 Report

1/ Line 19-20 – Authors claim that ‘’The definition used usually for solving different types of fractional differential equations is the Caputo fractional derivative.” I don’t think so. Prove this statement, please. Analyze references to this topic in terms of the number of derivative-definition usage.

2/ Authors should answer, why the Caputo derivative is used in their research. The problem is that their paper is not related to any physical process (i.e., electromagnetism, fluid mechanics, etc.); hence, I am not able to say if any other definition is better…

3/ Line 24-26 – What the authors want to say by this sentence? - ‘’Countless methods for solving linear and nonlinear fractional partial differential equations based on different fractional derivatives.”

4/ Line 26-41 – The authors write what was done by other researchers, but the reader don’t know how it is related to the topic of the paper. I think that this introduction is not well written.

5/ Finally, the introduction presents next sections of the paper but a reader don’t know what the purpose of this research is and why it is important. There are millions of equations which mathematicians can solve but somehow time-fractional diffusion equation is solved by the authors. Why? Why the proposed solution is important for the community? To sum up, the manuscript needs better introduction and abstract.

6/ Section 2 includes many vague sentences.

7/ Section 4 describes some relations to the physics. But there is nothing about memory processes…

8/ Section 4 includes many solution examples, but actually I don’t know how to use them. Furthermore, I do not understand what the purpose is to present so many graphs with insufficient explanation of these results? Is there any physics behind? For many equations and parameters, one can generate various results but it does not mean that it is worth publication…

Author Response

Please, see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The manuscript can be now accepted.

Author Response

Thank you very much 

Reviewer 2 Report

The authors successfully addressed most of the comments raised in my early report, nevertheless, some minor corrections are needed:

1. The figures on Page 13 and Page 17 need to be amended, please use "PlotRange->All" to get the maximum range of the height.
2. No Refs in 2021, the review literature should be updated, the authors may cite the following recent  relevant work on FDEs:

https://doi.org/10.3390/fractalfract5030100

https://doi.org/10.1007/s40819-021-00958-y

https://doi.org/10.1515/ijnsns-2020-0124

https://doi.org/10.1002/num.22756

Author Response

Thank you very much for your important remarks:

1. The figures on Page 13 and Page 17 need to be amended, please use "PlotRange->All" to get the maximum range of the height.

I solve this problem, please, see figures in page 13 and 17.

2. No Refs in 2021, the review literature should be updated, the authors may cite the following recent  relevant work on FDEs:

I add all suggested references, please, see line : 41-48.

Reviewer 3 Report

The authors answered most of my questions in the revised version of the manuscript. However, I think that the manuscript still requires corrections in terms of language style. Furthermore, the authors still do not try to interpret the results presented in figures. Perhaps, this opinion stems from my background in applied physics, so can be skipped if the journal is focused on mathematical considerations... However, some readers would also like to know what is the relation between curves in figures and transport processes analyzed in the manuscript. That is, how can I control transport processes analyzed in the manuscript? Can I get any interesting properties for some parameters? How to tune various coefficients? This is still a weak point of the manuscript that can be improved.

Author Response

1. However, some readers would also like to know what is the relation between curves in figures and transport processes analyzed in the manuscript. That is, how can I control transport processes analyzed in the manuscript? Can I get any interesting properties for some parameters? How to tune various coefficients? This is still a weak point of the manuscript that can be improved.

I add some analyzed for figures 1-4:

please, see line 172-184