Electrical Circuits RC, LC, and RLC under Generalized Type Non-Local Singular Fractional Operator

Round 1

Reviewer 1 Report

In the paper, the authors propose analytical solutions using the M-fractional derivative. The work is interesting, but many questions can be asked to the authors. I do not completely agree with the authors.

1) What are the advantages of using an M-fractional derivative instead of the Caputo derivative or the Riemann-Liouville derivative. 

2) Give the references for Eq. (2.1) and Eq. (2.2).

3) In Eq. (2.1) and (2.2), the authors just copy and past classical transformation but haven't cite Aguilar's paper in this section authors of these transformations; it is not acceptable. I am not sure the same transform done with Caputo derivative should be valid with M-derivative. If it is, please prove it clearly.

4) In Eq. (2.5) and other parts of the text, the authors have used the M-derivative, but they haven't previously given the detail concerning the Laplace transform of this derivative. Thus Eq. (2.8) , (2.13), (2.18), (2.24), (2.29), (2.34), (2.41) are questionable.

5) From Figure 1 to figure 20, the authors should analyze the fractional-order derivative impacts. It is not made in the paper.

6) The authors should give the details of the Laplace transform of the derivative D^{2\alpha,\rho,\gamma}. Please give us the form. I disagree with the form of the Laplace transform. In my opinion it should be s^{2\alpha}L(q)-s^{2\alpha-1}q(0). If this form is confirmed, the authors should check all the Laplace transform in the paper. I see the original paper, but Eq. (77) is seriously questionable, too (ref.2). Classically for the RLC circuit, the authors should obtain as Laplace transform the following term \frac{A}{Bs^{2\alpha}+Cs^{\alpha}+D}. The inverse of this form is an open problem in the RLC circuits; for example, it can be inverted using series representations.

7) What are the novelties of the paper? The authors just replace the work made with Caputo derivative by using M-derivative. What is the advantage of your derivative? In my opinion Caputo derivative is complete. Explain clearly the motivations of the replacements into the Introduction.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

1/ In my opinion, quality of presentation must be improved. The authors present many graphs with varying parameters but do not discuss their contents. The number of graphs can be lower but each graph can be supported by discussion and conclusion.

2/ Frankly speaking, I do not feel the physical meaning of 3 parameters of fractional-order derivatives. Please, tell a reader what is the physical meaning of fractional-order parameters. I understand that the authors can get closer representation of experimental data on the graphs, but someone using 4,5,6.. parameters of fractional derivative will be able to do a better approaximation in the future. So, why do you need the derivative with 3 parameters to describe the simple RC circuit, not 1, 2, 4, 5,.. parameters?

3/ Have you considered the derivative definition from parameter ranges point of view? That is, some values might be not applicable because circuit is not causal, time-invariant, etc...

4/ Please describe better the experiment in your paper. Otherwise, I don't know what was done and I am not able to refer to your results in my research.

5/ The origin of the circuit modeling is in the quasi-static approximation of Maxwell's equations. Circuit modeling gives the interpretation of the Maxwell's equations. For instance, you can imagine that each small volume of space consists of small capacitor and inductance. When I see the proposed capacitor model with 3-parameter fractional-order derivative, I do not see the space modelled by electrodynamics.

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

I like to tell the authors the works is not so novel as they mentioned. The present work exists with Caputo derivative in the literature, but authors have merits to check the results with M- derivative. Good luck.

Author Response

Thank you very much for your good comments and all helpful
suggestions you recommended in the first revision. We will also consider everything you suggest for our future works.

Reviewer 2 Report

The authors responded to my suggestions.

Author Response

Thank you very much for your good and kind comments.