The correlation functions of many quantities of physical interest show ``fractional" behavior, in that the scaling dimensions aren't integers or even fractions at all, in general. So fractional behavior in this sense can be understood as emerging from fluctuations. Experiments in fluid mdchanics or in magnetic materials may have reached the precision required.

So it's not necessary to impose fractional models, in order to obtain fractional behavior.

Do you have real data or a paper in which this kind of data are discussed ? in Understand that I have been working in the fractional field for 30 years, and I have too often heard "that's fractional" without anyone having proven to me that this was indeed the case.

I do not have data but I know some researcher in engineering trying to use fractional to model the system along with experimental data.

One example may be found here: https://www.researchgate.net/publication/264312618_Avalanche_shape_and_exponents_beyond_mean-field_theory

Once more, fluctuations naturally lead to fractional, or real, scaling exponents..

Maybe first we need to talk about the criterion of fractional data? I suspect that today no one knows an "indicator" that would make it possible to clearly identify the data as fractional or not.

it can be assumed that data obtained from a theoretical fractional model should exhibit fractional behavior?...

Regarding "practical" data,the data on the coronavirus [https://github.com/pcm-dpc/COVID-19] (or other medical data) may be fractional in nature. But let's go back to the fractionality criterion...

Dear professor, there are several definitions from discrete fractional calculus could be used in your question. Especially, if we search respond from your question, we can use Atangana-Baleanu-Caputo fractional model for your two different concepts. Especially, Atangana Baleanu Caputo fractional operator could be used and it solved with Laplace transform technique or Sumudu transform. I suggested 5 different references for your question:

[1] Baleanu, D., Machado, J. A. T., & Luo, A. C. (Eds.). (2011). Fractional dynamics and control. Springer Science & Business Media.

[2] Baleanu, D., Diethelm, K., Scalas, E., & Trujillo, J. J. (2012). Fractional calculus: models and numerical methods (Vol. 3). World Scientific.

[3] Sun, H., Zhang, Y., Baleanu, D., Chen, W., & Chen, Y. (2018). A new collection of real world applications of fractional calculus in science and engineering. Communications in Nonlinear Science and Numerical Simulation, 64, 213-231.

[4] Kumar, S., Chauhan, R. P., Momani, S., & Hadid, S. (2024). Numerical investigations on COVID‐19 model through singular and non‐singular fractional operators. Numerical Methods for Partial Differential Equations, 40(1), e22707.

[5] Khan, H., Jarad, F., Abdeljawad, T., & Khan, A. (2019). A singular ABC-fractional differential equation with p-Laplacian operator. Chaos, Solitons & Fractals, 129, 56-61.

" you can look this references about your question. " Good work,

lets hope collabaration in next projects Jocelyn Sabatier

Dear Stam Nicolis

Sorry for my late reply. Thank you very much for your replies. I agree with you taht it's not necessary to impose fractional models to get fractional behaviours.

This is precisely what I have been demonstrating for several years.

Article Some Alternative Solutions to Fractional Models for Modellin...

Fractional models can even lead to dangerous interpretations.

Thank you very much also for the paper you cite on avalanche. This is really a class of process that provide a power law behaviour. I modelled it with a nonlinear model in the folowing chapter.

Book Non-linear Models

But I look for input driven systems, in order to apply an input-output model. Not easy with avalanche.

Also, could please tell me what you mean by fluctuations?

Dear Mykhailo Mysyk

Thank you for your reply. You are right. This is a question that I ask to myself and I try to give some replies here:

Book Power-Law Type Dynamic Behaviours

The data of Corona are not fractional. Please have a look at:

Data Paper analysis

This is a bit what I criticize about this field: seeing fractionalism everywhere without proving it.

Fractional ordered PDEs and fractional physics informed neural networks can be applied with numerical analysis to optimise the equations governing a system. One approach would be to minimising the loss function between the integer-ordered data driven system and fractional ordered physics driven model of the system, where the physics driven model could be considered as data with noise.

50% of the fractional calculus is nothing but a scam in mathematics.

I confirm this, it's true.

Dear Pushpendra Kumar , Dear Ramzi Bouzaiene

Unfortunately, it’s not just in the field of mathematics. There is a lack of self-criticism in this area in which we "believe" in fractional calculation. To a point where many things are fractionalized without real justification. It's a bit in this sense that I asked this question. I sometimes compare this area to a sectarian movement and when you no longer "believe", as in any sectarian movement, they seek to “kill” you. This is what they tried to do to me since I said that there were limitations and drawbacks to fractional models and that I showed that many other models could also have fractional behavior.

Dear Prof. Jocelyn Sabatier

If you remember this question that I asked before,

https://www.researchgate.net/post/Fractional_calculus_Hub_of_uncertainty_and_confusion

There are nearly 1100 reads and some answers, but not a single exact answer for what I was looking for.

Let us take the case of mathematical modelling. Many of the researchers write that we should prefer fractional-order models over integer-order models because fractional-order model outputs are better than integer-order outputs. Then, they solve the model and plot several meaningless graphs (50% times), choosing many random fractional orders without any real data. It is not true all the time. Just recently I published the following paper where it is clearly visible that random choose of fractional-orders is inaccurate: https://www.researchgate.net/publication/376236178_Forecasting_of_HIVAIDS_in_South_Africa_using_1990_to_2021_data_novel_integer-_and_fractional-order_fittings

There should be a separate parameter estimation for each fractional order case (if you see Table 5).

Also, those who blindly claim that fractional-order models are more accurate than integer-order models, should not forget a famous Quote of Prof. George E.P. Box (1919-2013) "All models are approximations. Essentially, all models are wrong but some are useful."

To be very honest, I do not have an answer to your question. If I find something, I will come back here.

Dear Pushpendra Kumar

Sorry I miss your question. I try to propose you a reply (perheap it will not be satisfactory for you) soon.

Best regards.

@sabatier

I think It's not about "believe", it's about publishing more and having more influence in the literatüre. I think there is a mafia-like organization (secretian movement can be seen more lovely!) based on self-interest. A community of people who misunderstand networking!