thank you Dr Mohamed Herzallah , but I need more

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.

thank you Dr Mourad Chamekh

I have some problems with this type of "fractional" derivative: it is just the first-order derivative, multiplied by a continuous function. See

Article A remark on local fractional calculus and ordinary derivatives

Best regards Ricardo

Hello,

I just saw the question, to the best of my knowledge, each of fractional formula has some sort of meaning, but how we must interpret them would be different case by case (it depends on the question on which we are doing research), as FC is a particular form of convolution integral with a specific weight function.

Please read the introduction of the attached article; you might find information that you are looking for.

With all good wishes,

Mohammad

Article The Concepts and Applications of Fractional Order Differenti...

Dear colleague Tarig

A physical interpretation for conformable derivatives is similar to the one we present in "THE LOCAL NON CONFORMABLE DERIVATIVE AND MITTAG LEFFER FUNCTION" for non conformable derivatives.

Although the local derivatives, conformable and non-conformable, have received criticism from different authors, I believe it is a new tool that can be used in applications. Do not there exist different notions of derivability in the ordinary case? apart from this, I want to point out that the global derivatives also have a large number of drawbacks (which even make a physical interpretation impossible) and yet they are accepted. Will it be because of their antiquity? To cite only four of these deficiencies:

1) Not always the derivative of a constant is zero.

2) The known rules of the derivative of a product and a quotient are not exist.

3) The Rule of the Chain does not make sense, whereby any analytical method that includes the derivative of a compound function, can not be used (for example, the Second Lyapunov Method or integral transformations).

4) It is known that in systems of differential equations of integer order, that satisfy the conditions of existence and uniqueness, two different attitudies do not intercect each other in nite time, however, fractional systems do not satisfy this property.

That is, both are useful tools to solve problems, one should not be minimized in terms of the other, let alone compare them, they are mathematical objects of a different nature¡¡¡¡

Article General conformable fractional derivative and its physical i...

Abstract :

Fractional calculus is a powerful and effective tool for modelling nonlinear systems. In this paper, we introduce a class of new fractional derivative named general conformable fractional derivative (GCFD) to describe the physical world. The GCFD is generalized from the concept of conformable fractional derivative (CFD) proposed by Khalil. We point out that the term \(t^{1-\alpha }\) in CFD definition is not essential and it is only a kind of “fractional conformable function”. We also give physical and geometrical interpretations of GCFD which thus indicate potential applications in physics and engineering. It is easy to demonstrate that CFD is a special case of GCFD, then to the authors’ knowledge, so far we first give the physical and geometrical interpretations of CFD. The above work is done by a new framework named Extended Gâteaux derivative and Linear Extended Gâteaux derivative which are natural extensions of Gâteaux derivative. As an application, we discuss a scheme for solving fractional differential equations of GCFD.

I would recommend the recent paper that might help:

Roshdi Khalil , Mohammed AL Horani, Mamon Abu Hammad

Geometric meaning of conformable derivative via fractional cords, J. Math. Computer Sci., 19 (2019), 241–245

Thanks, Juan Napoles , Miled El Hajji and Jehad Alzabut

This question is very interesting. The physical and geometric meanings of conformal fractional derivatives are very important to us.

The attached link is related to the physical meaning of conformable fractional derivatives:

Article General conformable fractional derivative and its physical i...

It is just the first-order derivative, you can find important notes and details about classical and fractional calculus in this paper DOI: 10.1515/math-2016-0104

I understand that each person wants to link their name to a new concept or result, but this is not the case with the concept of conformable fractional derivative. The concept of conformable fractional derivative brings nothing new, neither mathematically nor physically.

In fact, as you can see from the attached note, each person can define his/her own fractional derivative.

Interesting thread Dear Prof. Elzaki Tarig ,

I would like to ask Prof. Juan Eduardo Nápoles Valdes, what is the physical interpretaion of the local non conformable derivative & "ML function"? what does it make ML function special for a physical interpretation?

I agree with Prof. Elzaki Tarig that to give a physical interpretation of a derivative is crucial to understand the derivative itself. I would like to refer to the words spoken by the legendary Mathematician Acad. Vladimir Arnold: "Mathematics is a part of physics"[1]

[1] https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html

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).

Thank you Prof. A. S. Kravchuk

It is very interesting question.

A very interesting question. Thanks to all for the answer.

If you prefer, you can refer to the physical interpretation of a local generalized derivative in my work THE LOCAL NON CONFORMABLE DERIVATIVE AND MITTAG LEFFLER FUNCTION.

Keep in touch.

Greetings

Thank you Prof. Juan Napoles

It is interesting.

the effects show that the resolved method is efficient and can be applied to find the general solutions of all cases related to the conformable fractional differential equations.

It is very interesting question

Thank you Dr Bachir

Please check:

http://www.iosrjournals.org/iosr-jm/papers/Vol13-issue2/Version-2/M1302028187.pdf

and

Article A new definition of conformable fractional derivative on arb...

La primera derivada significa desde el punto de vista físico la velocidad y la segunda derivada significa la aceleración.

It is known that if a function f is differentiable at a point t, then its corresponding derivative exists at t and is equal to t^{1-alpha} times the derivative of f. So, I think the physical meaning of the corresponding fractional derivative could be the slope of the graph times t^{1-alpha}.

Can anyone explain how to compute the kernel and weight in the Milne-type inequality by using conformable fractional integrals with twice differentiable function. For example in this paper

Error estimates for perturbed Milne-type

inequalities by twice-differentiable functions

using conformable fractional integrals.

Is there any general method to compute the kernel and weight?