Dear Miled

I'm going to try go give an answer to your question. Firstly, I do not understand the need for a meaning. Any derivative is an operator or even physical device that gives some change on a given input. Only for order 1 or 2, you can give a geometrical or mechanical interpretation. For any order you cannt give it. For the fractional case, there are several. I remember two. Prof. Podluby's geometric interpretation and the probabilistic interpretation of Prof. Tenreiro Machado. Recently, I proposed a more clear iterpretation based on the Bode diagrams. An integer order derivative is represented by a straight line that increases 20dB/dec (amplitude) and by a constant phase (90ª). A fractional derivative is represented by a staright line that increases or decreases alpha. 20dB/dec in amplitude and has a constant phase. See the papers at

https://www.mdpi.com/2227-7390/7/2/150

Article Which Derivative?

When you apply Mathematics to any natural ohenomenon you are trying to get a "picture" of reallity, but, much more important, you are looking for a tool that allows you to go ahead. Let me give an example. The speech production tube can be modelled by a set of partial differential equations. However, they are not very useful. They were substitued by a discrete-time model that allows us to analysis, synthesis and recognize speech and that is the base for modern communications.

Send me a mail to [email protected] and I'll send you some papers with sucessful applications of fractional models. The problem we frequently find is which derivative use and there is much confusion in the sbject.

In short timescales, most phenomena scale linearly in time. That is, the small increment dF of some quantity F is proportional to the tome increment dt. The proportionality factor dF/dt is the 'ordinary' derivative.

However, in some cases the increment dF is proportional to some power of dt, namely (dt)^a, where the exponent a can be any number (for simplicity let's assume a is between 0 and 1). This is where fractional derivatives dF/(dt)^a become useful.

The key example of application is the diffusion. In this case the average displacement of a molecule is proportional to (dt)^1/2 - and so the proportionality factor (diffusion coefficient) is a fractional derivative. Diffusion-like models are widely spread in biology, occuring wherever random processes are involved - from biochemical reactions through migration of animals to stochastic models of brain activity, and are certainly the most important application of fractional derivatives. I am not aware of any non-mathematical applications of fractional derivatives with power other than 1/2.

Dear Pawel

There are many applications. One, very famous is the paper "capacitor theory" by Svante Westerlund. I attach two of my papers. I found factional behaviour, for the first time in ECG, and later in other biomedical signals. Prof. Tenreiro Machado has many papers with applications.

I suggest you transform to frequency space whereby a fractional derivative of power a, introduces a factor (i w )^a in frequency w- space. Then you can define Greens functions etc , provided all integrals are bounded. From the Greens functions you can interpret signal propagation effects associated with the fractional derivative. The physics is then best expressed as the system's response to a driving source function in frequency space. It is very interesting that such effects appear in biological systems.

Often one finds that the calculus at hand is limiting. That is it does not supply adequate tools to advance our analysis. Take a classic example - the integration of standard quantum mechanics with special relatability theory. Standard quantum mechanics is not invariant under the Lorentz transform. In order to address this issue a new calculus had to be developed. Dirac in developing quantum field theory - the integration of relativity theory and standard quantum theory - introduce what has become known as the "Dirac Operator" D, which has the property that DD=the standard Laplacian. That is D is the square root of the Laplacian. In fact today a Dirac operator is defined as any operator that is the square root of an elliptical partial differential operator. They are elliptic pseudo differential operators.

Not only did new operators need to be defined but new structures that the operator acted upon - Spinors and later arbitrary Clifford algebra bundles. The Dirac operator is not a differential operator but what is called a pseudo differential operator. The introduction of the calculus of distributions in the 1940's, Sobolev spaces of distribution which led to the introduction of pseudo differential operators in the 1950's introduced a framework that allowed great progress and unification in mathematical physics, differential geometry and the analysis of partial differential equations.

Even to define the common "Dirac delta function" used commonly in engineering and physics requires distributions since the Dirac delta is not a function and cannot be realized as one so in reality has no physical meaning - it is at the heart of a lot of physical theories. However, the calculus of distributions (which are functionals not functions) and pseudo differential operators gives the tools in which to model, analyze and understand modern physical theory. That is independent of the fact the underlying constructs may or may not have a direct "physical meaning."

The power of many forms of calculus in the sciences is not the "physical, biological, etc., meaning" of the base concept (like a Dirac delta or Dirac operator) but the ability to develop a model to produce a prediction of a theory based on that model so it can be tested.

Dear Miled,

First of all I recommend you to take a look at the paper:

Article Matrix approach to discrete fractional calculus

In this paper you clearly find the main mathematical difference between integer order and fractional order derivatives. In summary, integer order derivative needs specified number(s) of previous time steps, according to the order, while fractional order derivative needs all of the previous time steps.

According to my studies, one of the most important advantages of fractional calculus in biological subjects is related to the efficiency of this mathematical tool in the characterization of material's behavior, specially viscoelastic materials. Fractional based models need less parameters to describe the material behavior and they are valid in a wider range of frequencies. You can find more details in these papers:

Article Fractional Derivatives Embody Essential Features of Cell Rhe...

Article Viscoelastic behavior through fractional calculus: An easier...

Article Fractional order models of viscoelasticity as an alternative...

Dear Dr.Miled El Hajji

Please find attached two closely related articles which give a possible physical interpretation of fractional time derivatives in a framework of viscoelastic responses of composite materials.

Perhaps these works could be of some help.

Best regards,

Roberto Suarez-Antola

There is another (maybe many but one that I know of) thread on this same topic that might be interesting to look at. You can find this thread of discussions by navigating through ResearchGate to the member Rajnee Tripathi and look for the question asked by this member called "Why we need to do Fractional Calculus". If you click the button "show previous answers" enough times you will find many contributions to this discussion in addition to those in your own thread.

Fractional derivatives by time mean deviated arguments. If consider the simplest development equation (dx/dt=kx) with deviated arguments, it becomes complex and has different solutions depending on deviation including critical regimes ( Zhirmunsky A. V., Kuzmin V. I. (1990). Critical Levels of Natural Systems Development. USSR, Leningrad: Nauka). As concern to spatial fractional derivatives, they describe processes and systems with fractal geometry. The simplest 1-D dissipation equation reveals oscillating solutions as far as fractional derivatives are taken into account, especially with time.

Fractional derivative.

Where does the physical distinction between fractional and continuous disappear?

Process modeling levels:

1. Space-time, forms of existence of matter.

2. Parametric - mass, force, temperature, speed, acceleration, charge and much more.

3. Pulse - force and time, mass and speed, etc.

4. Energy E - action in time-space, energy.

The distinction between fractional and continuous disappears at the energy level: dЕ / dh = n * h, h is the Planck constant, n is the whole.

Derivatives of energy: dЕ / dх, dЕ / dt, dЕ / dV, dЕ / dТ and others, these are whole physical, chemical, biological, sociological, economic, intellectual parameters.

This is the physical meaning of fractional derivatives.

Fractional (non-integer order) calculus can provide a concise model for the description of the dynamic events that occur in biological tissues. Such a description is important for gaining an understanding of the underlying multiscale processes that occur when, for example, tissues are electrically stimulated or mechanically stressed. The mathematics of fractional calculus has been applied successfully in physics, chemistry, and materials science to describe dielectrics, electrodes and viscoelastic materials over extended ranges of time and frequency. In heat and mass transfer, for example, the half-order fractional integral is the natural mathematical connection between thermal or material gradients and the diffusion of heat or ions. Since the material properties of tissue arise from the nanoscale and microscale architecture of subcellular, cellular, and extracellular networks, the challenge for the bioengineer is to develop new dynamic models that predict macroscale behavior from microscale observations and measurements.

Kudos to Miled on the non apologetic tone.

About 10 years ago I actually analyzed real data measurements with respect to some analytical results that I had on fractional Brownian phase motion model (a popular model for oscillators). The theoretical results did not match the measurements and I stated this in the discussion section. However, following the review process I had to drop the measurements and leave only the mathematical derivation.

In the nineties I tried to find a model for the ECG beats. The less bad model had almost 30 poles and zeroes. I put it in the waste box. A few years ago I tried and I found a fractional model with a few pseudo - poles and zeroes that was very good. Experimental results must be mandatory.

It may be helpful to mention that the main real world motivation for fractional derivative models were empirical spectrums (periodograms) which included slopes of -10*2*n, with n being non integer (Hence the term “fractional“). It turns out that in the context of oscillating signals like ECG, fractional derivative cannot explain sharp slopes such as -30 dB per decade.

Please see below analysis and real data example:

Article Analysis of Direct Spectrum Measurement of a Sinusoidal Sign...

Dear Nir

FD can be used to explain any Bode diagram, provided that a suitable derivative is used. A slope ov -30 dB per decade corresponds to a derivative of order -3/2. However, we have to use a derivative such that D^a e^{jwt} = (jw)^a e^{jwt}. This eliminates the RL and C derivatives. I send you files on the subject. On the other hand, the definition of fBm you use is not very useful. See mine. If you have any doubts, send me a mail.

Thank you for sending the files.

I could not find the ECG example mentioned earlier, did I miss it?

Specifically, I am curious about the statistics derived from the ECG signal that are aligned with the predictions of a fractional derivative based model.

Dear Nir

I left ECG some years ago. I did not publish the result that I did not validate. Meanwhile, my group obtained interesting results in the modelling of some vibrations of the uterus of pregnant women. We'll submit the paper in a few days.

I have other applications in electrochemical capacitors. See in attach.

Dear Dr.

Miled El Hajji

Greeting!

You can see the following links may be useful to answer

https://id.elsevier.com/as/58FtF/resume/as/authorization.ping

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https://www.sciencedirect.com/science/article/pii/S0898122109005343

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https://dl.acm.org/citation.cfm?id=1746954

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https://link.springer.com/article/10.1007/s11071-015-2069-2

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Methodology of Electrochemical Capacitor Quality Control with Fractional OrderModel.pdf

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https://en.wikipedia.org/wiki/Fractional-order_system

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https://www.nature.com/articles/srep03431

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https://www.mdpi.com/2227-7390/7/6/554/htm

• Fractional model of the electrochemical capacitor relaxation phenomenon.pdf
• Fractional model of an electrochemical capacitor.pdf

Best Regards