I prefer, instead of discussing the advantages of one operator over another, to consider that both are tools that can be useful and effective in problems of different nature.

Sir Jocelyn Sabatier

Dear Zeeshan Ali

I cannot that that is wrong, but I work in the field since 1993 and I have never been bothered by "singular kernel" in the sense you cite. But colleagues more mathematician and more theorist than me should have another point of view.

But one more time we have to have a look under. If you work with singular kernels, you work with kernels whose behaviour in high freqency (laplace trandform is neede) is of the form 1/s^nu. This kind of kernel have infinitly fast dysnamics. It was pointed out by Prof Jordan Hristov and me during online CMFCA conference by this week-end. You can also have an illustration of that, in a recent paper from my side:

https://www.researchgate.net/publication/339068107_Some_Alternative_Solutions_to_Fractional_Models_for_Modelling_Power_Law_Type_Long_Memory_Behaviours?_sg=DIPV_fTFCKs4zlOB8uEPcyIzS4vy1euhDP-yMTt6rksZg_DNbrl6_NGPbXoGaF7CL7TO5WMF4EgbuX8L2znzsQQqabDOoYpcKHW3lf0o.JAdgXSkQHTXijVlaUfspTuxkY2xZkc5O1R8vo1fy6ntlz-E6epyul0t7gnMMKLmweXiGKSyFLNqGfAAhL9qMTw

Also see the list of paper that I published recently, more or less connected to this problem. And that is avalaible at the end of my CFMCA talk

Presentation Some proposals for a renewal in the field of fractional beha...

Thus, this is one of the interest I see in non singular kernels. They can be still be improved, but they pemits to eliminate one physical problem: infinitly fast dynamics.

The so-called fractional derivatives (operators) appear in mathematical modelling of dynamic processes as memory integrals, that is the models are integro-differential equations. The perplexities in the approach want to be applied , singular or not singular, come from the fact that many people try to solve unrealistic models obtained by replacements . If somebody, starts building-up a model from fist principles, constitutive equation, balance equations, boundary conditions, etc, than we can see that the results is quite different.

Many problems start from the fact that many people, without physical meaning, use the fractional operators "independently", as a "flying butterflies" , that can be applied everywhere they like. But the principle question is : why ?

Think, and then apply !

There is a "genetic" origin of all these perplexities : Many good people, educated as pure mathematicians, know how to solve equations, but cannot explain how they were created. I have this point of view from many contacts: trying to explain the origin, the common reaction is " Please, stop, I cannot understand this physical explanation".

Thus, if somebody tries to devote his efforts on mathematical modelling , but his physical knowledge is not enough, then the solution is : self-education.

Try to understand how the physics of the phenomenon at issue was interpreted with mathematical tools. All models are approximations, so it is very important to understand what was assumed as important and what was neglected.

As to the question what type of operator should be used, I will give a simple example:

Everybody knows, how a discharge of a simple (linear) capacitor is modelled by integer order derivative, electrical engineers know that.

Now, if the material used is not linear (focussing of super capacitors, for example), then the first step is see what type of function can approximate the measured data (discharging current), with the modern techniques this is possible. The, decide what type of functions can approximate these points, that is what would the approximation of this relaxation behaviour. This approximating function should form the kernel of your fractional operators. However, be careful . Not all functions could serve as memory kernels.

In the attached article, there is a table with some conditions.

Have a nice reading and successful self-education in mathematical modelling.

Yes, Sir Jocelyn Sabatier, you are right. I have seen many applications and advantages of the singular kernel in real-world problems. And researchers can argue about singular kernel and show its benefits by giving different examples. But for non-singular, no one has an answer. Even they have used it in many papers. Thank you so much for sharing your presentations.

Yes, Sir Jordan Hristov, you are right about us. But actually, I have not use non-singular kernel before. Now I want to use it. I have studied many papers and tried to understand it. But the problem is that what type of complexities there are, which can not be replicated by power-law kernel. There are notable drawbacks of Singular kernel in some real-world models. So what are that drawbacks, no one mentioned it.

Thank you so much for suggestion and for the paper :-)

Thanks for the reply. I repeat again, try to understand what of relaxation behaviour you model: power law or not. Then, start to build the model, or find a book where this model is developed systematically. Then, decide what type of memory should be use. There is no necessary the fractional operator (time derivative) to replace the time derivative in the existing integer-order model. It may appear as additional damping term.

In the dominating modelling, especially with the non-singular operator, in fcat, there is modelling. Just replacements of the derivatives and solutions. This not modelling, since there are guarantees that these models are physically correct.

See the attachments.

A small correction to my previous message

It should be :

In the dominating modelling, especially with the non-singular operator, in fact, there is NO modelling.

Thank you Prof Jordan.

I prefer, instead of discussing the advantages of one operator over another, to consider that both are tools that can be useful and effective in problems of different nature.

Thank you so much, Professor Dr. Jordan Hristov , for the suggestions, recommended papers and presentations.

Yes Professor Dr. Juan Napoles . Thank you :-)

see

Article New fractional derivatives with nonlocal and non-singular ke...

Thank you Sir Dr. Maged Gumaan Bin-Saad

Dear Zeeshan, You are right. Some researchers are against to the fractional derivative of the non-singular kernel. These non-singular kernel derivatives suffer from many disadvantages that should be prevented from being used.

They fail to fulfil the basic fractional calculus theorem since they do not admit the presence of an integral corresponding convolution of which the derivative is the left-inverse; and the value of the derivative is always zero at the initial time t = 0, placing an unnatural limitation on the differential equations and models in which such derivatives can be used.

For more detail, please see the following article.

Firs of all, it is highly required to understand where these non-singular operators appear in mathematical models, then to judge are they correct or not. Applying voluntary to different functions is calculation without meaning. The same is valid for the operators with singular kernels. Please, avoid serious standpoints and judgments when understanding what operators mean is not your strong side.

Thank you Shabir Ahmad for sharing the paper.

Ok Sir Jordan Hristov . Thank you so much for your guidance.

Dear Juan Napoles

Dear All,

Exactly, but for that, it is necessary to have studied and well understood the limits and interests of the various operators and models.

And I think it is not yet. Fractional classical operators and the resulting models seems to be (for some) surrounded by a sort of mystical aura that would make them untouchable. On my side, I tried to adopt a critical attitude. Sometimes there is a price to pay with that (I was fire from my research group for that) but we gain freedom. You can find that in my recent paper:

https://www.researchgate.net/publication/339068107_Some_Alternative_Solutions_to_Fractional_Models_for_Modelling_Power_Law_Type_Long_Memory_Behaviours?_sg=rEvDjXWPf2uiSbiCymRuj5t33H6kE4dsADAUGBW4K34FBnRsGd87CTCpP1qt4p1ShuaxVDGyFWecmpHW3NZEg1uOQPBNh5AO-sF2WwIb.ADDQrjJWRbmHqd1Lxwy-DF9iTbgYqP0iUO-U5hWwrET0n1jnvTonIhxrk1CEjadeivBBz0-6oWjq410N-e8v-w

On the other side, many other kernels should be analyzed. Non singularity is only one interesting property with these kernels, but many others interesting properties should be obtained to overcome the drawbacks associated to classical kernels and models and to achieve physical consistency.

Also, it seems to me more relevant and broader to consider fractional behaviours and not only fractional models and operators. Fractional behaviours are connected to what it is experienced, what nature is made of. Fractional operators and models are only one solution to model these fractional behaviours. Other exist.

That is what I tried to explain in my talk last week at the OCMFCA (see also references ate the end of the slides).

https://www.researchgate.net/publication/346647380_Some_proposals_for_a_renewal_in_the_field_of_fractional_behaviour_studies_and_modeling_FIRST_ONLINE_CONFERENCE_ON_MODERN_FRACTIONAL_CALCULUS_AND_ITS_APPLICATIONS_OCMFCA-2020_-_Plenary_talk_-_December_

Thus we still have not explore enough the domain to know what is better or not and why.

Yes Sir Dr. Jocelyn Sabatier

Function and cofficients in partial differential modelling in physics could be discountinuous or their derivative are discontinuous. For example bondary layer in turbulence modelling or capularity modelling and fractional derivatives are suitable rather than discountinious scheme

In fact, a fractional integral is a product of convolution with a singular kernel, namely (t-s) ^ (- a), a> 1. Otherwise they are just nonsense.

Too restrictive standpoint with respect to the memory kernel. Not all non-local problems with convolution operators involve, from the physical modelling point view, singular kernels. Predetermination of the type of memory kernel is actually ignorance of the physical background. The type of the memory kernel comes from the physics of the no-local phenomenon modelled by it .

Yes, Prof. Hristov's guidance is important, let us go back to physics. I would suggest everyone read this paper Article Sir Isaac Newton Stranger in a Strange Land

Book Fractional Calculus View of Complexity: Tomorrow's Science

Thank you YangQuan Chen,

Thinking and physical reasoning are the primary issues that should be done prior any general standpoints

Greeting for the year of Iron Ox