A New Mixed Fractional Derivative with Applications in Computational Biology

Round 1

Reviewer 1 Report

The paper is the generalization of fractional derivative recently proposed by the same author termed as GHF fractional operators. The proposed paper give a new definition of fractional derivative or integrals that with an appropriate selection of the various parameters cover all the various definitions of the existing of derivatives and integrals including Caputo's and the Riemann-Liouville fractional integrals, Caputo-Fabrizio (CF) fractional operators, Atanaga-Baleneu (AB) and weighted operators. The CF and AB fractional operators have the appealing that the kernels are nonsingular. Laplace transform of such new operators is also presented. The application of GHF fractional operator is used for solving classical problem describing the evolution of population in human body. Because the large variety of the parameters the different behaviour of such a problem may be investigated.

The paper is quite good and it may be published in the MDPI journal.

Minor comments:

1) Pag. 2, last row "with non-singular given by" has to be rewritten as (with non-singular kernel given by"

2) When you introduce the various parameters, please, define clearely if they are reals, complex or natural numbers

3) A statement on the versatilityof the new GHF fractional derivative in eq.(32) is necessary

4) A concluding remark section is welcome.

Moderate editing of English language required

Author Response

See the attached file.

Author Response File: Author Response.pdf

Reviewer 2 Report

New mixed fractional derivative is discussed. Interesting results are obtained, but there are significant drawbacks. The proof of some statements is incorrect, and some are not given at all. Assumptions of weight $w$ are made and then violated. The error of the proposed approximation of FDE solutions is not obtained. The biological meaning of the fractional derivative parameters in the equation of cell population (32) is not given. I think that the manuscript in its present form cannot be published and requires significant revision.

1. Page 2. The word is missing in "with non-singular given by" 

2. Line 74. Typo, instead of $f$ there should be $u$.

3. Line 74. The proof is not correct. The Taylor series expansion of the function $wu$ is given in a certain neighborhood of $t$, but under the integral $\tau$ can leave this neighborhood since $\tau\in[a,t]$.

4. Page 5. Formula (6) goes beyond the text margins.

5. Page 5. It is a very strange situation when the main statement - Theorem 3.2, is a consequence of the auxiliary statement - Lemma 3.1. The proof of Theorem 3.2 is not given.

6. Page 6. It is necessary to clarify what the symbol * means.

7. Page 6. It is necessary to write in more detail how equality (14) is obtained.

8. Page 6. The word "Fondamental" is probably misspelled.

9. Page 8, line 113. In accordance with the assumptions made, $w'>0$, and therefore it cannot be $w=1$.

10. Page 8. It is not specified for what values of $t$ and for what functions $f$ equation (20) is considered.

11. Page 10. Since $w'>0$, then $w=e^{-t}$ is impossible.

12. For the proposed approximation of solutions to equation (20), it is necessary to indicate the error of this approximation.

12. Page 12. When applying the results obtained to computational biology, it is necessary to somehow explain what biological meaning the parameters $p,r,w,\delta$ have, otherwise there is no point in using FDE.

The word is missing in "with non-singular given by"

The word "Fondamental" is probably misspelled.

Author Response

See the attached file.

Author Response File: Author Response.pdf

Reviewer 3 Report

Paper deals with important task. It has a scientific novelty. It has a logical structure. The paper is technically sound. The experimental section must be improved. The proposed approach is logical, results must be described in more detail.

Positive sides:

1.       The new mixed fractional derivative in the Caputo and Riemann-Liouville sense was proposed.

2.       Also the associated fractional integral operator was defined.

3.       Fundamental properties of the proposed operators were investigated.

4.       The method for finding an approximate solution of differential equation with mixed fractional derivative was proposed.

5.       An example was considered

Suggestions:

1.       It is necessary to more clearly describe the motivation of the article, the practical significance of the proposed operators, and to argue why exactly this type of fractional differentiation and integration operators should be used when building mathematical models of dynamic processes.

2.       Table 1 shows the error values of the method at constant parameter (p, r, ᵟ) values. It would be appropriate to expand the table and show the accuracy of the method for different sets of these parameters.

3.       It is also advisable to provide a table to compare the execution time of the method with a change in the discretization step of the time variable.

4.       Section 7 gives an approximate solution to the problem (32) with two cases of initial conditions. However, there is no comparison with real data or with solutions to this problem obtained by other researchers.

5.       It is advisable to expand the description of the obtained results and emphasize the advantages of using the proposed operators.

6.       The author uses many self-citations, their number exceeds 25%.

7.       It would be appropriate to expand the overview of the application of the fractal approach in the modeling of physical processes. For example, pay attention to publications DOI: 10.1016/j.rinp.2021.105103, DOI: 10.1080/01495739.2019.1623734 and DOI: 10.1007/978-3-031-04812-8_9

8.       The is no Conclusion section.

Author Response

See the attached file.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

All my previous comments have been corrected and I have no others.

Reviewer 3 Report

The authors took into account most of the comments