Fractional Calculus is a branch of calculus which generalizes the derivative of a function to non-integer order. As I know, you can use this concept for conservation of mass, advection dispersion equation, some structural damping models, Schrödinger equation (in quantum theory), acoustical wave equations, time or space fractional diffusion equation models, and much more. But I'm not sure that it can be usefull in all branches of science.

The subject of fractional calculus deals with investigation of integrals and derivatives of arbitrary real or complex order. Fractional calculus is thus an extension of integer order calculus, which is applicable in many areas of science and engineering, such as fluid flow, rheology, electric network, viscoelasticity, electrochemistry of corrosion etc.

I would like to mention some of the books to

• Oldham, K. B. and Spanier, J., Fractional Calculus: Theory and Applications, Differentiation and Integration to Arbitrary Order, New York: Academic Press. (1974). Chapter 11 Applications To Diffusion problems

• Podlubny, I., Fractional Differential Equations, an Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications, Academic Press. (1999).Chapter 10 Survey of Applications of the Fractional Calculus

• Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., Theory and Application of Fractional Differential Equations. North Holland Mathematics Studies, 204, (2006). Chapter 8 FURTHER APPLICATIONS OF FRACTIONAL MODELS

• Francesca Biagini, Yaozhong Hu, Bernt Øksendal ,Tusheng Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications, Springer-Verlag, London (2008).

• F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity

An Introduction to Mathematical Models, Imperial College Press, London, (2010).

Fractional Calculus is very useful in many applications not only in sciences, but also in engineering. The mere fact that there is an exponential increase in the number of publications on the application of fractional calculus in various fields prove that the topic indeed is very promising.

I specialize in control theory and engineering which I find fractional calculus very useful.

The fraction, in algebric term represents a class modulo an equivalence relation. Of course, the fractions inherit of the structure of the entire parent when the relationship is compatible with stucture laws. Therefore the fractional calculus is applied every time the fractional model is solicited to respond specific needs of a situation . Those needs can appear in any science in particular mathematics, physic, biology, astronomy, ect ....

Let us consider for example that x(t) is the displacement of an object. From classical physics, we know that x'(t) [first-order derivative] is the velocity, while x''(t) [2nd-order derivative] is the acceleration, and x'''(t) [3rd-order derivative] is the jerk of the object. Therefore, as long as x(t) is given, anyone will be able to get the velocity, acceleration, and jerk equations easily.

Fractional calculus as mentioned by other responders in this thread deals with derivative and integrals having non-integer orders, e.g. 1/2-order. If you get, for example, the 1/2-order derivative of x(t), then it will result in x^{0.5}(t). Getting another 1/2-order derivative of x^{0.5}(t) results in x'(t) which is velocity. But the basic questions are:

1) How do you call it if you get 1/2-derivative of displacement x(t)?

2) Is it something between displacement and velocity that could be measured physically using available instruments?

3) What characteristics can x^{1/2}(t) provide that cannot be seen in x(t) and x'(t)?

These are just examples on how fractional calculus is applied. Is it useful? Maybe yes maybe not, but it has been seen in various fields in science and engineering that fractional calculus is ubiquitious.

Hope this helps.

Of course. It has many applications in different fields of the science: nowadays its applications in economy, signal processing and image processing are among the most interesting. Fractional calculus can be applied also to other fields of Mathematics: for instance, I used it to investigate in the theory of special function. If you are interested, you can read my attached article on it.

Article Fractional derivative of the Hurwitz ζ-function and chaotic ...

There are some interesting trials to explain the physical meaning of Fractional calculus related to the memory of the variable. 

The definition of fractional derivative involves an integration which is non-local operator (as it is defined on an interval) so fractional derivative is a non-local operator. In other words, calculating time-fractional derivative of a function  at some time  requires all the previous history, i.e. all from .Fractional derivatives have the unique property of capturing the history of the variable, that is, they have memory. This cannot be easily done by means of the integer order derivatives.

Check these references: 

[1]  M. Du, Z. Wang and H. Hu, Measuring memory with the order of fractional derivative. Sci. Rep. 3(2013).

[2]  T. Sardar, S. Rana, S. Bhattacharya, K. Al-Khaled, J. Chattopadhyay, A generic model for a single strain mosquito-transmitted disease with memory on the host and the vector, Math. Biosci. 263 (2015) 18–36.

Thanks for your kind replay.  Please let me know how we can explain the graphical representation of fractional derivatives . 

Recently some improvements have been developed for the theory of differential equations involving fractional (in time) derivatives and operators "of fractional order". In particular, please refer to the following two (of my) contributions to the matter.

Book Fractional-in-Time Semilinear Parabolic Equations and Applications

Preprint Well-posedness results for a class of semi-linear super-diff...

Any feedback is welcomed, of course. Many thanks.

Thanks for your response.

Of course the answer is a big YES. The question one really ought to ask: How come Fractional Calculus is not part of standard undergraduate/graduate mathematics curriculum in most Universities?

Sure, Calculus is so important, especially, for engineering sciences and physics.

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.

Fractional derivative is used in different Mathematical/Physical modeling to capture the memory effect on the system, but it's physical meaning or significance is not clear to me. I am also interested to know the physical meaning of fractional order derivative.

When we want to keep memory effect in a system, we may use memory dependent derivative instead of fractional derivative, for better geometrical meaning of memory dependent derivative.

A colleage of mine, Matthew Harker , is working in the field of fractional calculus.

Maybe you find some interesting insights in his papers, e.g.:

• Harker, M. and O’Leary, P. (2017) ‘Numerical Solution of the Anomalous Diffusion Equation in a Rectangular Domain via Hypermatrix Equations’, IFAC-PapersOnLine. Elsevier B.V., 50(1), pp. 9730–9735. doi: 10.1016/j.ifacol.2017.08.2176.
• Harker, M. (2018) ‘Runge-Heun-Kutta Methods for Nonlinear Fractional Order Differential Equations’, in Proceedings of International Conference on Fractional Differentiation and its Applications (ICFDA) 2018. SSRN, p. 6. doi: 10.2139/ssrn.3281668.
• Harker, M. (2018) ‘Discrete Variational Method for Sturm-Liouville Problems with Fractional Order Derivatives’, in Proceedings of International Conference on Fractional Differentiation and its Applications (ICFDA) 2018. SSRN, p. 5. doi: 10.2139/ssrn.3277684.
• Harker, M. and O’Leary, P. (2018) ‘Regularized Fractional Order Integration and Differentiation Via Discrete Orthogonal Polynomials’, in Proceedings of International Conference on Fractional Differentiation and its Applications (ICFDA) 2018. SSRN, p. 6. doi: 10.2139/ssrn.3286040.
• Harker, M. and O’Leary, P. (2014) ‘Sylvester Equations and the numerical solution of partial fractional differential equations’, Journal of Computational Physics. Elsevier Inc., 293, pp. 370–384. doi: 10.1016/j.jcp.2014.12.033.

No doubt, it is very important due to its far reaching applications in other practical sciences. The subject of fractional calculus (that is, calculus of integrals and derivatives of fractional order) has emerged as a powerful and efficient mathematical instrument during the past six decades, mainly due to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. See for details : https://www.elsevier.com/books/theory-and-applications-of-fractional-differential-equations/kilbas/978-0-444-51832-3?countrycode=AU&format=print&campaign_source=google_ads&campaign_medium=paid_search&campaign_name=Australia_shopping&gclid=EAIaIQobChMI7eTK3cWO6AIVkomPCh2i5wkZEAYYASABEgKAtPD_BwE

Dear Gunawat,

Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator, the practical use of fractional differential operators most common in electrical transmission line analysis is most common. In applied mathematics and mathematical analysis, a fractional derivative is a derivative of any arbitrary order, real or complex. Unlike classical Newtonian derivatives, a fractional derivative is defined via a fractional integral. In the context of functional analysis, functions f (D) more general than powers are studied in the functional calculus of spectral theory. The theory of pseudo-differential operators also allows one to consider powers of D. it can be applied in Fractional conservation of mass, Groundwater flow problem, Time-space fractional diffusion equation models, Fractional advection dispersion equation, PID controllers, Acoustical wave equations for complex media, in physics(Fractional Schrödinger equation in quantum theory, Variable-order fractional Schrödinger equation) etc.

I would say, it is used in mathematics, physics and utmost use in electrical engineering. This is not used in all branch of science. Even in mechanical science or civil science some part (not all parts) off course can be analyzed using functional calculus. In spectral theory it is very useful.

Hope it is helpful to you.

Ashish

The subject of fractional calculus has applications in diverse and widespread fields of engineering and science such as electromagnetics, viscoelasticity, fluid mechanics, electrochemistry, biological population models, optics, and signals processing.

yes, see a snapshot, Article A new collection of real world applications of fractional ca...

Fractional calculus is a generalization of the traditional integer order integration and differentiation to non-integer order. It initially appeared as theoretical development in mathematical analysis. In effect, fractional differential equations (FDEs) play an effective role in the modeling of anomalous relaxation and diffusion processes. The fact that fractional derivatives introduce a convolution integral with a power-law memory kernel makes the FDEs an important one to describe memory effects in complex systems. This nonlocal property means that the next state of the considered system depends not only upon its current state but also upon all of its historical states. The increasing interest of FDEs is motivated by their wide applications in various fields of science such as physics, finance, chemistry, electricity, medicine, control theory and so on. (what we mentioned above is concerning the fractional calculus in the general case i.e., the fractional order is constant).

On the other hand, despite the constant order fractional calculus can address some very relevant physical problems, it cannot provide always a best description of the complex phenomena in the nature and cannot capture important classes of physical phenomena where the fractional order itself is a function of either dependent or independent variables. As an example, the reaction kinetics of proteins has been found to show relaxation mechanisms that are properly described by a temperature-dependent fractional order. Thus, the underlying physics of the reaction kinetics (captured by the order of the relaxation mechanism) changes with temperature. Hence, it is reasonable to think that a di erential equation with operators that update their order as a function of temperature will better describe the protein kinetics. This example is enough to convince us that there exist classes of physical problems that would be better described by variable order operators.

Hope this brief overview can help you!

A priori the answer is yes, like any mathematical tool. The question is where does it provide "best" help? In those local processes (velocity, acceleration, ...) ordinary calculus, together with the new local differential operators, provides better results; On the other hand, fractional calculus is better in processes in which past history is significant (population growth, epidemic, tumors, ...). I must clarify something: fractional calculus IS NOT A GENERALIZATION of ordinary calculus. On several occasions I have clarified it because it seems to be a recurring error, there is no way in which the fractional derivative is reduced to the ordinary one, therefore it is not a generalization, IT IS AN EXTENSION. Some of the new local differential operators, IF THEY ARE A GENERALIZATION of the ordinary derivative, can be for a value of the order, for example, in the Khalil conformable derivative if α is 1, it reduces to the ordinary derivative, the same thing happens with our derivative N, if the kernel is 1, we have the ordinary derivative.

Fractional calculus isn't necessarily a go-to tool for every scientific field, but it's becoming increasingly relevant across many disciplines.