On the Stability and Numerical Scheme of Fractional Differential Equations with Application to Biology

Round 1

Reviewer 1 Report

This paper first purpose is to investigate the qualitative properties including stability, asymptotic stability as well as Mittag-Leffler stability of solutions of FDEs with the new generalized Hattaf fractional derivative which encompasses the popular forms of fractional derivatives with non-singular kernels. The second aim is to develop a new numerical method in order to approximate the solutions of such FDEs. The developed method recovers the classical Euler numerical scheme for ordinary differential equations (ODEs). In addition, the obtained analytical and numerical results are applied to a biological nonlinear system arising from epidemiology.
The paper presents some interesting theoretical analysis and applications for fractional differential equations and the results are correct and valuable. It can be accepted by Computation provided the author can address the following issues in the minor revision.
Comments:
1. In Lemma 2.5, the author should introduce the ranges of x (t) and y (t).
2. It is suggested to have a separate section to conclude the research results at
the end of paper.
3. The author is recommended to connect his/her study to the latest developments in the literature on theoretical and numerical studies of FDEs: [1,2,3] for inverse problem associated with FDEs and [] for numerical methods for FDEs. This will help increase the readership and enhance the impact of this paper. References

[1] X. Cao, Y.-H. Lin and H. Liu, Simultaneously recovering potentials and
embedded obstacles for anisotropic fractional Schr ̈odinger operators. Inverse Probl. Imaging 13 (2019), no. 1, 197–210.
[2] X. Cao and H. Liu, Determining a fractional Helmholtz equation with un-known source and scattering potential. Commun. Math. Sci. 17 (2019), no. 7, 1861–1876.
[3] R.-Y. Lai and Y.-H. Lin, Inverse problems for fractional semilinear elliptic equations. Nonlinear Anal. 216 (2022), Paper No. 112699, 21 pp..
[4] S. Yang, Y. Liu, H. Liu and C. Wang, Numerical methods for semilinear
fractional diffusion equations with time delay. Adv. Appl. Math. Mech. 14 (2022), no. 1, 56–78

Author Response

Dear reviewer,
Thank you for your review. I answer your questions and implemented the suggestions you made.

1.  I fixed it.
2.  I added a conclusion section, see the revised version.
3. I added it. Thank you very much for your interesting comments.

Reviewer 2 Report

The properties (reductions etc) of the new generalized Hattaf fractional (GHF) derivative should be given in detail. 

The following papers should be read to improve the introduction/literature part

1. “A note on the fractional hyperbolic differential and difference equations”, Applied Mathematics and Computation, 217(9), 4654-4664 (2011).

2. “On the explicit solutions of fractional Bagley-Torvik equation arises in Engineering”, An International Journal of Optimization and Control: Theories Applications (IJOCTA), 9(3), 52, (2019).

3. "Analytical solutions of some nonlinear fractional-order differential equations by different methods". MATHEMATICAL METHODS IN THE APPLIED SCIENCES, 44(9), (2021) 7526-7537., Doi: 10.1002/mma.6313

4. “On the Numerical Solution of Fractional Hyperbolic Partial Differential Equations”, Mathematical Problems in Engineering, 2009, 1-11(2009).

The considered numerical method is not clear.

However, the manuscript has the merit to be accepted for a wider audience.

Author Response

Dear reviewer,
Thank you for your review. I answer your questions and implemented the suggestions you made.

1. I fixed it.
2.  I added it.
3.  The numerical method considered is clarified. Thank you very much for your interesting comments.

Reviewer 3 Report

x/

In this paper, the author considers the application of his own definition of fractional derivatives to the analysis of fractional differential equations stability and their numerical simulation.

Thus, all the paper is focused on the definition of this new derivative, given by equation (1). There are many new definitions of fractional derivatives. Generally, the motivation is related to the use of a non singular kernel.

Definition (1) is derived from the Caputo derivative. This classical derivative is very popular because of its so-called physical initial conditions. In many papers of the last ten years, it has been proved that x(0) cannot be used to predict the future behavior of any FDE. The main reason is that x(t) is not the state of the FDE, only its pseudo-state. In fact, the true internal state is infinite dimensional: x(t) is only the weighted integral of the infinite dimensional internal state. Consequently, many mathematical proofs based on x(t) and the Caputo derivative are questionable.

The author has not defined the initial conditions of his fractional derivative. Nevertheless, I think that the problem is the same with this derivative: in equations (6) and (7), x(t) is not the true state of the FDEs and in my opinion, the mathematical proofs of the paper are questionable.

A major issue of fractional calculus is the definition of FDE initial conditions which are necessary to solve correctly the Cauchy problem. The use of a non singular kernel does not change this fundamental problem. A fractional system is characterized by a long memory phenomenon, which means that fractional derivatives are non local derivatives, since they depend on all the past behavior: consequently, they cannot be summarized by x(0), which is the property of integer order local derivatives.

I think that the proposed numerical scheme of equation (26) is correct, because it is based on the fractional integral equation (25), derived from the Riemann-Liouville integral definition (3). Nevertheless, the initial condition x(0) of equation (25) is not correct, for the previous reasons.

Author Response

Dear reviewer,
Thank you for your review. I answer your questions and implemented the suggestions you made.

Thank you very much for your interesting comments. I changed "state" to " pseudo-state". The other remarks will be the main goal of my future work. 

Reviewer 4 Report

The author presents a new numerical method in this manuscript. First he study is to investigate the qualitative properties including stability, asymptotic stability as well as Mittag-Leffler stability of solutions of FDEs with the new generalized Hattaf fractional derivative which encompasses the popular forms of fractional derivatives with non-singular kernels. These qualitative properties are obtained by constructing a suitable Lyapunov function. Secondly, he develops a new numerical method in order to approximate the solutions of such FDEs. The developed method recovers the classical Euler numerical scheme for ordinary differential equations (ODEs). Finally, the obtained analytical and numerical results are applied to a biological nonlinear system arising from epidemiology.

The paper is very interesting and can be accepted after mirror revision. To do so, the authors should address the following issues:

1.     Similarity of the paper is too high and should be less then 20%.

2.     The author is advised to remove the abbreviation from the abstract.

3.     Introduction can also be improved by dealing with a comparative approach with the previous studies.

4.     The author should add the following references:

a)     Investigation of fractional order bacteria dependent disease with the effects of different contact rates

b)     Hermite–Hadamard Type Inequalities for Interval-Valued Preinvex Functions via Fractional Integral Operators

c)     Dynamics of fractional order delay model of coronavirus disease

d)     Numerical analysis of Atangana-Baleanu fractional model to understand the propagation of a novel corona virus pandemic

e)     Analysis of time-fractional Kawahara equation under Mittag-Leffler Power Law

5.     The author is advised to conclude their work at the end of the paper.

Author Response

Dear reviewer,
Thank you for your review. I answer your questions and implemented the suggestions you made.

1. I reduced it.
2.  I removed it.
3. I fixed it.
4. I added it.
5.  I added a conclusion section. Thank you very much for your interesting comments.

Round 2

Reviewer 3 Report

The author has not modified his paper according to my comments.

He says he has replaced state variable by pseudo state variable !

He will use my comments for further papers !

Thus I have not changed my opinion.

However, I can add some other remarks.

What is the interest to introduce new derivatives without singular kernel ? What is the problem with the usual kernel ?

In my opinion, the main issue of fractional derivatives and more generally with fractional calculus is the initial condition problem. The kernel is not the origin of this problem. My advice would be to consider this problem and try to solve it.