There are many works on the subject, see e.g. the seminal paper by Podlubny

https://pdfs.semanticscholar.org/d003/4719a7dce8a949e3212e759f00626c992dba.pdf

Dear Mokhtari

I think that the attached files can help you.

Regards

There are many works on the subject, see e.g.

http://vectron.mathem.pub.ro/dgds/v15/D15-ta.pdf

I think the physical interpretation can only be specified within the context of the given physical problems.

If you have interest, please have a look at our study on fractional plasticity modelling of geo-materials. In that case, we think that the fractional derivative indicates the state-dependence and plastic flow direction of the geo-material, just like the first-order derivative of a curve indicates a normal vector (loading direction).

Regards,

Yifei

Geometrical meaning of ordinary derivative: The geometrical meaning of ordinary derivative is simple and intuitive: For smooth function f which is differentiable at x, the local behavior of f around point x. A simple definition directly from geometrical meaning: One can expect that the fractional derivative could give nonlinear (power law) approximation of the local behavior of non-differentiable functions.

Physical interpretation of fractional derivatives in viscoelastic modeling is available. See, Article Linking the fractional derivative and the Lomnitz creep law ...

Article Connecting the grain-shearing mechanism of wave propagation ...

This is encouraging and nice question in the area of fractional derivatives. We still need more elaborated answers

I agree with you Sir ( Jamilu Sabi'u ).

I think you need clear physical and geometric interpretations of fractional derivatives like order derivatives, unfortunately all the works up to now, still give the complex interpretations which you cannot understand it. and that exactly one of the main reasons why we didn't teach fractional derivatives in undergraduates level.

This is not exactly the geometry or physical interpretation. It is just an example.

Note :- See the introduction part example

Source :- Millar Rose book

Thank you very much.

Dear Mokhtari,

The space fractional derivative occurs naturally when you do the continuum limit of systems such as FPU's lattices with Long-range interactions. Space-fractional derivatives manifest the presence of long-range interactions/nonlocality in complex systems ...

For more details see: Article Fermi-Pasta-Ulam chains with harmonic and anharmonic long-ra...

Regards.

Good question!

The following papers will help:

I. Podlubny, Fractional Order System and Fractional Order Controllers, UEF-03-94, Inst. Exp. Phys, Slovak Acad. Sci., Kosice, 1994

I. Podlubny, Fractional Differential Equations , Academic Press, San Diego, 1999, 62-88

In the existing leterature, it is an open Question. There is no acceptable geometrical interpretation of fractional derivatives

Dear Mokhtari,

This file can help you: https://pdfs.semanticscholar.org/db42/15b5ce6ac702e44a645eaa827ad122819699.pdf.

Regards.

Greetings.

This is very nice question which I had in my mind since last four years. I always tried to get satisfied answer for it but had got partially by Podlubny and Heymens work. I think it is benchmark question in general theory of fractional differential operators. We are still waiting for excellent work on it.

The geometrical and physical meaning of ordinary derivative is simple and intuitive: For smooth function f which is differentiable at x, the local behavior of f around point x. The ordinary derivative gives the linear approximation of smooth function. Here we expect the fractional order derivative to have the similar geometrical meaning. We hope for non-differentiable functions, the fractional order derivative could give some kind approximation of its local behavior. A simple definition directly from geometrical meaning­­­: We expect that the fractional derivative could give nonlinear (power law) approximation of the local behavior of non-differentiable functions. Fractional dynamics is the field of study in physics and mechanics investigating the behavior of objects and systems that are characterized by power law non-locality, power law long-term memory or fractal properties by using integrations and differentiation of an arbitrary real order.

One application for fractional derivatives that is quite common is the Pade' expansion of the "one-way-wave equation" one finds in electromagnetic and acoustic absorbing boundary conditions for numerical methods. Taking progressively more terms in the expansion gives you a better approximation of a perfect absorber, but becomes less local. (I did some work in this area many many years ago.)

If I remember correctly, there are problems in fractional diffusion in, e.g. porous media, that would make use of fractional derivatives.

If you search in Wikipedia there are some geometric interpretations.

There are also in Fractals.

Fractional derivatives appear in the theory of control of dynamical systems, when the controlled system or/and the controller is described by a fractional differential equations. (Podlubny, I)

Dear Colleagues

Thank you for sharing your answers. In this question, I would like to focus on geometric interpretation instead of applications of FDEs. Absolutely, fractional differential equations appear in various fields of physics and engineering but I can not find a meaningful geometric interpretation yet!!

Best.

Payam Mokhtary There is no universally accepted geometrical interpretation for fractional derivatives. This is open for investigation. May be you could contribute something to it? Meanwhile have a look at this paper and the references there in: https://arxiv.org/pdf/math/0110241.pdf

I think that the attached file can help you

Dear friends

This is an intresting work about new definition of fractional derivative.

Regrads

@ Vikash Pandey

I agree with you

@ All. One can find viscoelastic interpretation of fractional order here: Article Linking the fractional derivative and the Lomnitz creep law ...

And here, Article Connecting the grain-shearing mechanism of wave propagation ...

However universal interpretation of fractional order seems distant in future. A challenge for us all.

Dear,

I have seen it some where that fractional derivative is a rate of change subject to unknown inherited traits.

Regards

It does not have any physical or geometrical meaning also violated causality and central limit theorem. But i think we can consider as mathematical model for the process with memory and ...

Article A physically based connection between fractional calculus an...

This work might be helpful.

I could not find geometrical meaning yet!

I Will start working on fractional equations, as son as I can conceive the geometric and physical meaning. Right now, nobody knows.

I found a very helpful paper by Mehdi Rahimy

Article Applications of Fractional Differential Equations

The unique identification of geometry, physics, and quaternion algebra is considered.

Vasiliy Karpenko could you provide an English version of the paper?

Unfortunately not.

Now, we can not know. in future we will know the interpretation.

Following [Ref.1], the applications and/or activities related to fractional calculus have appeared in at least the following fields:

– Fractional control of engineering systems.

– Advancement of Calculus of Variations and Optimal Control to fractional dynamic systems.

– Analytical and numerical tools and techniques.

– Fundamental explorations of the mechanical, electrical, and thermal constitutive relations and other properties of various engineering materials such as viscoelastic polymers, foams, gels, and animal tissues, and their engineering and scientific applications.

– Fundamental understanding of wave and diffusion phenomenon, their measurements and verifications, including applications to plasma physics (such as diffusion in Tokamak). J.T. Machado et al. / Commun Nonlinear Sci Numer Simulat 16 (2011) 1140–1153 1141 – Bioengineering and biomedical applications.

– Thermal modeling of engineering systems such as brakes and machine tools.

– Image and signal processing.

[Ref.1] J. T. Machado, V. Kiryakova, F. Mainardi, Recent history of fractional calculus, Commun Nonlinear Sci Numer Simulat 16 (2011) 1140–1153.

The classical derivatives are in local nature, i.e., using classical derivatives we can describe changes in a neghberhood of a point but using fractional derivatives we can describe changes in an interval. Namely,fractional derivative is in nonlocal nature. This property makes these derivatives suitable to simulate more physical phenomena such as earthquake vibrations, polymers and etc.. To more details, I suggest "podluny" and "Diethelm" books.

It is not easy to stablish a direct relationship. Since the concept of complete derivation has been changed for its fractional nature

Direct relationships of physics, geometry, algebra are established by the equations:

1) a + b = c & a * b = z

2) a + b = c & a - b = x

3) a + b = c & a / b = y

There is no physical or geometrical meaning for a fractional derivative that is the main problem with fractional derivatives

There is no device for measuring the order of the fractional derivative in a proposed fractional model. Also, it is non-local while all measurements in physics are local.

Stop critics and give scientific comments.

Please, check our paper at Article Back to Basics: Meaning of the Parameters of Fractional Orde...

Dear Payam,

Hi. As we mentioned in our recently published conference paper,

https://www.researchgate.net/publication/343812833_Fractional-order_IMC-PID_controller_design_for_fractional-order_time_delay_processes

the geometric and physical interpretations of fractional-order integrals and derivatives are still open problems. However, to put a simple interpretation on it, we considered viscoelastic systems as an application.

This is a very good question.

From classical calculus, we know that the first derivative is the velocity of the object along the curve, the rate of change of the velocity becomes acceleration, and so on. Observe that, the rate of change is with respect to integers and the variables of the function, here the question is what if the rate of change is with respect to real numbers between 0 and 1 and the variables of the function? is it still a rate of change? right? for example, in classical calculus the rate of change of any constant function is zero, but the velocity of an object moving on the straight line might not be totally zero, fractional calculus answers this question. For example, the derivative of constant is not zero using the Reimann-Liouville definition.

What is the geometric and physical interpretations of fractional derivatives? Check ✅

see

Article Geometric and Physical Interpretation of Fractional Integrat...

Hello

Please see the following paper:

@Article Geometric and Physical Interpretation of Fractional Integrat...

See Tarasov's paper doi:10.1016/j.aop.2008.04.005 in which a fractional version of gradient is introduced.

Let F(x,y)=0 be a smooth function given implicit form. As you can see, the vector field grad^{\alpha}F could be considered a fractional version of the normal vector field gradF to the curve F(x,y)=0, but certainly different than a canonical normal field, as expected. You can also consider J(grad^{\alpha}F) is a fractional version of the tangent vector field, where J is the complex structure of plane. Analogously, it is different than a canonical tangent field, too.

Best,

Evren

Dear Payam Mokhtary . See the following useful RG link: Article Geometric and Physical Interpretation of Fractional Integrat...

Kindly see also the following useful link: https://www.nature.com/articles/srep03431

Also check please the following useful link: https://www2.icp.uni-stuttgart.de/~hilfer/publikationen/pdfo/ZZ-2019-HFCA-47.pdf

The following link is also very useful: Article Geometric Interpretation of Fractional-Order Derivative

Payam Mokhtary (Fractional derivatives has the most important roles for applying on physical problem. Especially, Riemann-Liouville Fractional derivatives can be applied on diffusion problem, describing seismic wave location. On the other hand, you can use matlab for doing geometric interpretation. I suggested Igor Podlubny fractional calculus book with you. This book mention that special functions in first part because, it will be used for defining different types of fractional derivatives and integrals. You can look these books:

1. Fractional Differential Equations, MATHEMATICS in SCIENCE and ENGINEERING Volume 198 and author: Igor Podlubny

2. FRACTIONAL CALCULUS Models and Numerical Methods Second Edition

Author: Dumitru Baleanu, Kai Diethelm, Enrico Scalas, Juan J. Trujillo)

Basic answer of this question is nothing but fractional derivative is memory based property while simple derivative is not, In addition Fractional derivative is non local and simple derivative is local.

From the book: "Fractional Calculus: An Introduction for Physicists"

As shown by the research, there is a relationship between fractional derivative of polynomials at the tangent points and the order of the fractional derivative. For more details click here Article The geometric and physical interpretation of fractional orde...

&

A new physical interpretation is suggested for the Stieltjes integral here Article Geometric and Physical Interpretation of Fractional Integrat...