Dear Mohammed K A Kaabar ,

Applications are everywhere in many branches of sciences.

See for example:

Article Some Applications of Fractional Calculus in Engineering

Article A new collection of real world applications of fractional ca...

etc.

Best regards

Dear Mohammed K A Kaabar ,

Applications are everywhere in many branches of sciences.

See for example:

Article Some Applications of Fractional Calculus in Engineering

Article A new collection of real world applications of fractional ca...

etc.

Best regards

Thank you Dr. Issam Kaddoura very much for your answer and your suggested articles!!

In modeling of Anomalous transport (non-Fickian transport) of heat, or any pollution in a fluid they use fractional calculus. Basically were you have a very shape gradient and Fick's low is not valid anymore

Thank you Dr. Kaveh Zamani very much for your answer!! I greatly appreciate it!!

The fractional order derivative comes into the picture when there is a shape changing space. In terms of mechanical engineering, it is same as when there are visco-elastic or plasto-elastic material. These materials are purely nonlinear in nature and there mathematical modeling involves the generalized version of ordinary differentiation (integer order differentiation), called as fractional order derivatives.

One of the best physical applications is related to human respiratory airways. The airway tubes are visco-elastic in nature and the complete tracheo-bronchial tree is bifurcating , dichotomous and self similar in nature . The mathematical model to study the pressure variations due to dynamic breathing patterns will involve fractional order derivative itself.

Thank you Dr. Miled El Hajji very much for your detailed explanation and your suggested applications!! I greatly appreciate that!!

The fractional calculation is that branch of mathematical analysis that deals with the generalization of all linear operators based on the concept of derivative (derivative, integral, Laplace transforms, Fourier and Mellin etc.) by introducing the derivative operator fractional, that is real powers (or complex) of the derivative operator.

Fractional calculus was rigorously developed by mathematicians in the mid-19th century, and finds its application in the field of physics and engineering only a century later.

Leibniz in 1695 writes to de l'Hôpital:

Thus it follows that d^(1/2)x will be equal to x √dx:x,

an apparent paradox, from which one day useful consequences will be drawn.

In physics it is now known to be used in the theory of Hamiltonian chaotic particles systems and biophysics.

One of the probable major uses is in rheology, where it is widely used in problems affecting polymers and therefore viscoelastic damping,as well as the mechanics of damage and fracture.

A really interesting approach is also that of the generalization of models of differential transport equations through a differential equation of fractional order.

There are many books to recommend, but they depend on your field of study and where you want to go to do research.

Certainly, as you probably know when switching from integers to reals,the Gamma function appears,therefore a good previous knowledge of special functions is recommended.

Among the general books that i can recommend are:

1) Fractional Calculus for Scientists and Engineers,Manuel Duarte Ortigueira,Springer Science & Business Media, 2 giu 2011

2)The Analysis of Fractional Differential Equations,Kai Diethelm,Springer Science & Business Media, 3 set 2010

3)Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Igor Podlubny,Elsevier, 27 ott 1998

4)Generalized Fractional Calculus and Applications,Virginia S Kiryakova,1993

5)Fractional Calculus & Integral Transforms of Generalized Functions,Adam C. McBride and A. C. McBride,Pitman Advanced Publishing Program, 1979

Good work!

Thank you Christian Iandiorio for your answer!!

Dear Mohammed

I am not sure about what you mean by “Fractional Geometric Calculus”. I presume that you are thinking in geometric formulation, not geometric interpretation. In the first case, the study of lines, curvatures, … I only know the works of A. and K. Lazopoulos. Although, they work in the context of the Caputo derivative. I do not believe that we can obtain great results working on R+.

I made some work trying to generalise classic vectorial calculus. I did it for rectangular domains. I’m trying to generalise for “curved” spaces. For it, I need to define fractional line and surface integrals. Nobody did it. I have ideas, but it may happen that other people can have a better one.

Let me do some other comments:

a) The supposed letter from L’Hopital to Leibniz is a myth. Nobody found it. I suggest people to read the S. Dugowson’s thesis and the book of Samko, Kilbas and Marichev. The first time such question appeared was in a letter to Bernoulli.

b) In the last years some “new” operators claiming to be “fractional derivatives” were proposed. I call them “fractional fraud”. Some papers showed that they are useless. In particular, the called AB operator is very poor.

I’m always available to talk with civilised people, really interested in changing ideas without offensive words. For any interested

[email protected]

Thank you very much Dr. Manuel Duarte Ortigueira for your detailed answer and discussion!! I greatly appreciate that! Yes, that's true! I was asking about the geometric formulation for fractional derivatives.

It's a field where there are almost no publications. See in attach mine on fractional vectorial calculus.

Thank you Dr. Manuel Duarte Ortigueira for sharing your publications with us!!

There seems to be a rash of questions on fractional derivatives as people point out applications. It's been around a long time. In the '50's and 60's there was a lot of work in fractional powers of closed densely defined linear operators (which the derivative is one) by Kato and his students. This construct is also closely related to the theory of semigroups of operators. Koto's book covers this nicely.

https://www.amazon.com/Perturbation-Theory-Operators-Classics-Mathematics/dp/354058661X

Of course if the operator is symmetric and one defines the domain correctly, it will give rise to a self adjoint operator so that the spectral theorem can be used and the calculus of the spectral resolution can be used to define the fractional power. In case of partial differential equations, the expansion to pseudo differential operators gives rise to fractional powers through the symbol calculus. There are many equivalent ways to skin the same cat.

The one property that is given up with fractional powers of differential operators is "locality." That is if a function is zero in the neighborhood of a point x0, then the value of the a differential operator applied to that function is also zero in a neighborhood of x0. That is replaced by the weaker property that the preservation of the singular support.

Professor Ortigueira thank's for posting your papers. I've from time to time wondered how the classical definitions of fractional derivatives fit into the wider area of pseudo differential operators but didn't have the time to pursue such questions. Since I have some time now, I'm going to pursue it.

Dear Truman

Thanks for pointing the operators side. Unfortunately, my knowledge on the subject is very small. However, in my mind there is a fight: is the fractional power of the laplacian the correct operator for generalising the wave equation. And Schrodinger's?

The designation "pseudo-differential" operators has been assigned to operators that I consider true derivatives. See my paper at

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

I have a continuation, but it is not published yet.

Manuel,

A differential operator can be viewed as a inverse Fourier transform of the multiplication operator of the operator symbol and the Fourier transform of the function - in the correct function space. Today the correct function space to start is the space of Schwartz functions and Tempered distributions and restrictions of such. The psiDO is an operator defined similarly but with symbols satisfying that don't grow too rapidly in the the "frequency domain." The symbol of of the square root of the Laplacian is simply the square root of the symbol for the Laplacian ) -(w1^2 + w2^2 + ... + wn^2).

There are lot of good introduction text on distributions and psiDO's available today. PsiDO's are the bread and butter in the study of elliptic (and hypo elliptic PDE's) because the PsiDO includes both the elliptic PDO's and their inverses (singular integral operators) in the class. The class of elliptic psiDO gives rise to a parametrix which is an approximate fundamental solution that can be easily constructed.

The Laplacian on the appropriate domain of definition is self adjoint so there is a spectral decomposition. The square root of the Laplacian can be realized using that resolution of the identity by using the function f(lambda) = lambda^(1/2). On the right domain the Laplacian has compact resolvent so that the Laplacian has an eigenfunction expansion. The square root of the Laplacian also has an eigenfunction expansion with the same eigenfunctions but the eigenvalues are the square root of the eigenvalues of the Laplacian.

For any closed operator, the fractional power is defined using the resolvent operator similarly to the way the "fractional derivative is defined in your paper.

Here is an one of the earlier papers.

https://msp.org/pjm/1966/19-2/pjm-v19-n2-p11-p.pdf

These concepts are all semi-groups of operators and semi-groups underwent tremendous progress in the 50's and 60's.

So I have to believe that there is a a fundamental relationship between all these concepts although they are being developed in different settings. For example electrical engineers often use the Laplace transform - while mathematics develop similar concepts using the Fourier transform.

Truman

Thank you Dr. Truman Prevatt for your answer with full details of explanation!! I greatly appreciate your help!!

Dear Truman

You did not understand what I wanted to say. See my paper in attach. I am not sure about the generalisation of the laplacian in the wave or Schrodinger equation. Mathematicians like to work with powers of operators, but I'm not sure this is the correct option in such cases.

I questioned the use of the designation "psudo-differential" in cases where we know a definite expression for the derivative. For example, my two-sided derivative implements the Riesz-Feller derivative. So. the "pseudo" must be removed.

About the Fourier and Laplace transforms. In 25 years working in fractional calculus I was very surprised with some things: a) mathematicians know very well the Fourier transform, but have small idea about its use; b) they are fully ignorant of the two-sided Laplace transform; c) they are "grasped like limpet to a rock" concerning the one-sided Laplace transform.

Since the eighties, mainly after the publication of the book "Signals and Systems" by Oppenheim and Wilsky, the two-sided LT became the standard in dealing with analysis of systems. Meanwhile, the Fourier transform was the base for dealing with signals. But, as the FT is a special case of the FT, we use them in parallel and use "s" or "i omega" simultaneously. On the other hand, in Electrical engineering, people discovered, a long time ago, that the initial conditions of the one-sided LT are frequently incorrect (in FC they almost always). This transform was modified by people from Control, but the problems in FC remain. Therefore, it will be put in the waste box of mathematical curiosities in the future.

Dear Manuel,

After spending time as boat a "pure mathematician" and an "applied mathematician," I have been blamed for using the quote often in frustration, "mathematicians and engineers talk a lot about the same things but give them different names - so more grad students can have thesis topics." Sometimes one often needs a translator. I get what you are saying about pseudo - but PsiDO's are embedded in mathematics since the '50's and the tool has led to tremendous progress in many diverse areas, e.g., the Atiyah Singer Index theorem in differential topology.

You are correct, today the Laplace transform is only really talked about in mathematics courses in complex analysis if then. The Laplace transform has its maximal utility in engineering in the analysis of causal systems and control theory. Since the advent of digital signal processing gives the ability to implement non-causal systems, with all the advantages of linear phase even EE's coming out of college today often have not seen any development of the Laplace transform. With the advent of distribution theory, the Fourier transform has become the major tool for micro-local analysis and partial differential equations where pseudo-differential operations and Fourier integral operators the fundamental objects. The predominance of the Fourier transform arises naturally as the isomorphism between the functions on a topological group and it's dual group. For example the Fourier series is the Fourier transform on the group of integers to the dual group of the circle. It just so happens the dual group of the real numbers is the real numbers.

Like a lot of concepts in mathematics, mathematicians have abstracted them to the point that they may seem unrecognizable too many. Often times expansion of the language and tools leads to better solutions to solutions to new and problems and easier descriptions and proofs of standard problems and allows one to see patterns more easily. However, this abstraction can run counter to the need of the scientist or engineer in focusing on a specific problem set. After all an engineer or scientist does not deal with signals that are infinite in time in both directions but mostly with a small snippet of time. .

For example the special functions of the early 1900's, many that arise as solutions of the Strum-Louisville differential equations today are often developed using a group representation theory approach.

https://www.amazon.com/Special-Functions-Theory-Group-Representations/dp/0821815725

A beautiful theory that drills down to the underlying fundamental concepts but on the other hand may not help someone trying to apply special functions to a specific problem. Mathematics is developed in waves often new theories at times obliterating the the beauty of the theories they are abstracting. The classical analysis of Courant, Hilbert, Hardy, Littlewood, etc. giving rise to the new school of Krein, Gelfand, Banach, Schwartz, Zygmund, Calderon, Hormander, Stein, Fefferman and as the power of functional analysis swept over the classical analysis that arose in the mathematical physics of the late 19th and early 20th century as weak derivatives replaced derivatives and then distributions (functionals ) replaced functions give us amazing tools for not only mathematicians but to put a firm logical underpinning under much of the scientists and engineering (the Dirac delta was used in physics and engineering before there was a firm mathematical definition). However, pure mathematics and the sciences are different and it often takes years before the utility of a trend in mathematics finds it way into applications.

Fractional Geometric Calculus is proposed normally, which intends to give a binding together language for arithmetic, material science and study of intricacy of the 21st century. Further, Geometric algebra based math gave a ground-breaking scientific structure in which the most progressive ideas present day physic, for example, quantum mechanics, relativity, electromagnetism, and so on

Dear Muzammal

It will be good, but for now we are very far from such objectve.

Dear Truman

I was in another "war" that I suggest to have a look

Article DISCUSSION PAPER THE FAILURE OF CERTAIN FRACTIONAL CALCULUS ...

Returning to our talk I would like to complement my ideas with something else. I'm happy for having a collaboration with a great mathematician: Juan Trujillo. With him I could really understand what such designation means. And I could understand the difficulties of many mathematicians in understanding what I call "mathematical engineering". The main problem is in the "little will" of learning. I have now collaborations with other two mathematicians that manifested clearly desire of learning how to use Mathematics in Engineering. Frequently, I remember a paper I caught some years ago:

Mathematics is applied by everyone except applied mathematicians , by David Wilson,

Applied Mathematics Letters 22 (2009) 636–637

Best

Thank you very much Dr. Muzammal Hameed Tariq for your answer!! I greatly appreciate that!!

There are several applications of this mathematical phenomenon in mechanics, physics, chemistry, control theory and so on.

Thank you Dr. Doufnoune RACHIDA Rachida for your answer!!

check the links

(1)https://booksc.xyz/book/13734865/33f846

Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus

Ji-Huan He, S.K. Elagan, Z.B. Li

(2)https://booksc.xyz/book/5794675/180b5c

Some geometric aspects of the calculus of fractions

Jean Bénabou

best luck