Homotopy Perturbation ρ-Laplace Transform Method and Its Application to the Fractional Diffusion Equation and the Fractional Diffusion-Reaction Equation
Round 1
Reviewer 1 Report
In my opinion, the paper presents some details. A MINOR revision is required to make the manuscript worth publishing.
It is proper to add some convergence theories to support the numerical results, if it is possible. In addition, an overall review may be needed for fixing the grammatical errors in the manuscript.
What is the effectiveness the proposed method over the existing method? Comparison should be done with the latest reported method in (2016/2017/2018) papers.
I recommend a revision of the introduction. In my opinion, you should explain your results in more detail in the introduction
Check the format of the journal and made all the references according to the journal style.
Future research direction will be shown in Conclusion.
I want to read speedly the last version of paper before publishing if it possible for you.
Author Response
Dear Pr.
See, the responses attached. Please, see the final version of the manuscript with the Editor.
Best personal regards.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The work is very interesting and has a subject of great relevance in the present day. We have the fractional calculus, on the other hand we have the diffusion process and reaction-diffusion process.
I liked the work. The work is interesting, and information that holds the reader's attention. But, I have some suggestion and considerations to improvement of this work. Some modifications are necessary, I listed below:
(1) Please, delete the "0"-Section called "Lead Paragraph", since the same information is written in the last paragraph of the introduction.
(2) After all the equation, there is a comma or a period mark. Please, verify all equation!
Example: In eqs. 1, 2, 3, 6, 7, 8... 10, 13, 14 (...) there are commas "," after the mathematical expressions.
Correct example eq. 5.
(3) Please, change the notation Γ(.) to Γ(...).
(4) After the eq. 5 is missing a reference, see in the text "[?]".
(5) What is $\gamma^n$ in equation 3? Don't was defined!
(6) In eq. 4 the Laplace transform was applied in g(t) (see the left side), in the right side appears the f(t)-function! Please, correct the notation!
(7) In my mind, the ρ-Laplace transform not is new! If you take a variable change t'=t^{\rho}/\rho you obtain the usual Laplace transform (in f(t') function), to me is only a different notation.
ps: To create a new transform function is necessary define a inverse transformation in complex plane.
(8) The title of work is very long, if you think in something better, please change. But not is a problem.
(9) The figures 3, 4 and 6 needs improvement. The authors need to move the caption inside of figure to other position, because the curves are erasing the values of \rho (see figures 3, 4 and 6).
(10) The introduction and eq. 13 needs some references. I will list a sequence of references on fractional calculus that will certainly help better the article, as well as attracting the attention of a different audience.
----- I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (Academic Press, New York, 1998).
---- Approximate solutions to fractional subdiffusion equations. The European Physical Journal Special Topics, v. 193, n. 1, p. 229-243, 2011. APA
-----Derivatives with non-singular kernels from the Caputo-Fabrizio definition and beyond: Appraising analysis with emphasis on diffusion models. Front. Fract. Calc, v. 1, p. 270-342, 2017.
----- Non-Gaussian Distributions to Random Walk in the Context of Memory Kernels. Fractal and Fractional, v. 2, n. 3, p. 20, 2018.
-----Response functions in linear viscoelastic constitutive equations and related fractional operators.Mathematical Modelling of Natural Phenomena, 14(3), 305. (2019)
-----Fractional Prabhakar Derivative in Diffusion Equation with Non-Static Stochastic Resetting.Physics, 1(1), 40-58. (2019)
-----A fractional Fokker–Planck equation for non-singular kernel operators.Journal of Statistical Mechanics: Theory and Experiment,2018(12), 123205.
-----Mittag-Leffler input stability of fractional differential equations and its applications. Discrete & Continuous Dynamical Systems-S, 636-643.
I liked very much,
Thank you for your attention.
Author Response
Dear Pr.
See, the responses attached. Please, see the final version of the manuscript with the Editor.
Best personal regards.
Author Response File: Author Response.pdf
Round 2
Reviewer 1 Report
Now, the manuscript is suitable for publication since all recommendations and suggestions of the referee are satisfied and implemented in the revised text.
