Thank you for this nice discussion/explanation. I learnt a lot especially in relation with transfer funtions. I am not too old on fractional calculus but I was soon interessted about time-fractional Navier-Stokes equations. There is a plenty of papers studing the subject from all perspectives(theoritical and numerical..). I was looking for a motivation for the problem. Why should I use the fractional derivative with respect to time instead of the simple one derivative? . The motivation given by Zhou and Peng is that such model is suitable for fractal medias without more explanations. Others just motivate generally fractional models. What do you think about that?

Professor Jocelyn Sabatier , considering your comments. When talking about numerical methods, I was using methods proposed by Article Numerical Solution of Fractional Differential Equations: A S...

Which are mostly based on the Grünwald–Letnikov derivative for which we can set the initial condition to any value. I know how the initial value can give a different response depending on the definition which is a problem you tackle previously. Thus, should these numerical methods be avoided? what can we use instead?

Dear Adrian-Josue Guel-Cortez

I have not read the paper you cite, but I think that the problem is not linked to the numerical methods proposed, but on the way the initial conditions are taken into account. It is wrong that you can only consider initial condition (value at one moment of the past). If you have a look on the Grünwald–Letnikov derivative , (see for instance the 5th relation of https://en.wikipedia.org/wiki/Grünwald-Letnikov_derivative) you see that variable m goes from 0 to infinity, thus all the function past is considered. Thus you can decide to ignore it and consider only samples starting from t=0, and thus you condider an initial condition at t=0. But the result will be wrong if the function have a past prior t=0.

On the contrary, the Grünwald–Letnikov derivative is very interesting because if well impemented, it shows that all the past of the function from -infinity must be taken into account.

It is wrong to believe that an initial condition is sufficient. Fractional systems have elephant memory, all the time, even at time t = 0 and whatever the method used to simulate them.

Dear Khalid Akhlil

I think that the motivation by Zhou and Peng is good. You can find a explanation in the Sapoval's book "Universality et fractales". This is a book in french and I don't know if there exist an english version. First you have to consider the link between diffusion and random walk (an equation diffusion results in a randow walk of a particule: solution of the diffusion equation can be viewed as the probabilty to find the particule at place x and time t) and to know that diffusion produces behaviours in t^(1/2).

In this book, you can read page 255 something like this: " Place a random walker on a wire. Along this wire, the random walker travels a distance varying in t ^ (1/2). But if this wire is placed in a (fractal) space of dimension 5/3 (fractal dimension), the distance traveled will be proportional to (t ^ (1/2)) ^ (3/5) = t ^ (3/10) ".

But notice that fractional models is not the only solution to capture fractional (power law) behaviours (see the following papers):

[1] - Sabatier J., Farges C., Tartaglione V. (2020), Some alternative solutions to fractional models for modelling long memory behaviors, Mathematics, 8, 196; 2020

[2] - Sabatier, J. Fractional state space description: A particular case of the volterra equations. Fractal Fract. 2020, 4, 23. [23. Sabatier,

[3] - J. Sabatier Non-singular kernels for modelling power law type long memory behaviours and beyond. Cybern. Syst. 2020, 51, 383–401

[4] - J. SABATIER, Beyond the particular case of circuits with geometrically distributed components for approximation of fractional order models: Application to a new class of model for power law type long memory behaviour modelling, Journal of Advanced Research, doi.org/10.1016/j.jare.2020.04.004

Dear @ Jocelyne Sabatier,

thank you for the response. I am used to work with french but I can not find the book. I don't know if it is possible to get a pdf. I will see your papers refering to fractional behaviours.

Merci beaucoup.

Fractional calculus is not suitable for fractals since they have different measures.

Dear Alireza Khalili Golmankhaneh

Could you please describe in more detail what you want to say?

Sure, In fractional calculus we use length as a measure but we know that we can not consider it for fractals , for example, the Cantor set has zero length or the Koch curve has infinity length then one must use proper measure for fractal (e.g Hausdorff measure).

Dear Jocelyn Sabatier,

A strange way to build a discussion. Let me remind you that in the discussion about another question you agreed with me and already seen problems with fractional derivatives in physics (https://www.researchgate.net/post/What-are-the-advantages-of-using-fractional-systems-instead-of-integer-order-systems#view=5fc3dc09ce836c7d3a3abf4d).

In my opinion, there is only one conclusion from the old discussion - the application of fractional derivatives in physics is a deception.

But now in this discussion you no longer see any problems.

If you think that fractal derivatives can be used in physics, then, please as example get the stationary heat transfer equation with fractional derivatives based on the energy balance for an elementary volume, as it was done in textbooks.

Dear A. S. Kravchuk

No, it's not strange. I always leave room for doubt. So I opened up this discussion to see if anyone could contradict the view I have on these fractional models (incosistence). But nothing of the sort has emerged so far.

Since you seem to me to be more expert than me in physics, do you know the following works?

1. Metzler, R.; Joseph Klafter, J. The random walk's guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 2000, 339, 1–77.

2. Barkai, E.; Metzler, R.; Klafter, J. From continuous time random walks to the fractional Fokker-Planck equation. Phys. Rev.E 2000, 61, 132–138.

They show that for a continuous time random walk process with the power-law waiting time, probability density function and jump lengths with finite variance, the corresponding probability density function to find the particle at position x at time t in the Fourier–Laplace domain is the same as the solution of the time fractional diffusion equation with Riemann–Liouville fractional derivative .

What are you thinking of these works that seems to show that a fractional differential equation results in a physical phenomenon?

Dear Jocelyn Sabatier,

you can familiarize with the substantiation of the falsehood of using fractional derivatives in physical processes by the example of differential equation of the motion of a mathematical pendulum (https://www.researchgate.net/publication/346492959_IMPOSSIBILITY_OF_REPLACING_ORDINARY_DERIVATIVES_BY_FRACTIONAL_ONES_IN_THE_EQUATION_OF_SMALL_OSCILLATIONS_OF_A_PENDULUM).

The preprint on the impossibility of using fractional derivatives for the diffusion equation will be finished in a few days. The method of substantiating the incorrect use of fractional derivatives in the diffusion equation is practically analogous to a mathematical pendulum. Unfortunately, I am not familiar with the papers that you mention, but you can easily make conclusions on it by yourself after reading my mentioned future preprint.

Dear Sirs,

I wrote a short lecture on why fractional derivatives cannot be used even in the case of differential equations with dimensionless functions (https://www.researchgate.net/publication/346629474_TO_THE_ISSUE_OF_IMPOSSIBILITY_OF_USING_FRACTIONAL_DERIVATIVES_IN_DIFFERENTIAL_EQUATIONS_WITH_DIMENSIONLESS_FUNCTIONS).

Dear Sirs,

As promised I finished the lecture about fractional derivatives in the diffusion equation (https://www.researchgate.net/publication/346927202_DEMONSTRATION_OF_GROUNDLESS_APPLICATION_OF_FRACTIONAL_DERIVATIVES_IN_THE_DIFFUSION_EQUATION). Also analyzed the viscoelasticity (https://www.researchgate.net/publication/346905393_SIMPLEST_PROOF_OF_UNREASONABLE_REPLACING_THE_ORDINARY_DERIVATIVE_BY_A_FRACTIONAL_ONE_IN_THE_SIMPLEST_MODELS_OF_LINEAR_VISCOELASTICITY).

Firts of all, I wish to all of you an happy new year full of health and happiness! 

A. S. Kravchuk Thank you for your contribution, I also think that fractional model are not physically consistent and are now good for me only for dynamic behaviour fitting, and that we should not try to make them say more. But the proof you propose are questionable for two reasons. - To show that there is a dimension problem, you supose that the constants in the analysed equation keep the dimensions of their integer versions. But it not what it is claimed in the litterature. - Is it claimed that the dimension of the constants is adapted (for instance t^beta). In such a situation, you say that such a constant is no more a constant but a function, and really I dont understand why (Constant)^beta is a function. (even if I understand that I see no physical meaning to t^beta). We can very well build a model like this, use it to capture a dynamic behavior, but I agree to avoid giving it a physical meaning.

Here is my last paper:

Article Fractional Order Models Are Doubly Infinite Dimensional Mode...

Is it physically consistent to use this kind of model ?

There is one solution to restore a physical consistence : avoid the confusion or rather the implicit link that exists between fractional (differentiation based) models and fractional behaviors (of physical nature). Fractional differentiation based models is only one solution to model fractional (or power law) behaviours. Consideration of fractional (or power‐law) behaviors without being limited to fractional models provides countless opportunities for research in the field of model analysis, identification and control. Have a look in the following paper:

Article Modelling Fractional Behaviours Without Fractional Models