For example, "x" is called position , "dx/dt" is velocity or displacement and "d2x/dt2" is the acceleration entities. But in the case of fractional order what is the meaning of "d0.9x/dt0.9". How can we consider? Is this intermediate state of velocity and the acceleration?
Fractional order derivatives are related to memory. No other physical meaning can be attached to them at present though theory is old.
Hi,
http://vectron.mathem.pub.ro/dgds/v15/D15-ta.pdf
http://www.tuke.sk/podlubny/pspdf/pifcaa_r.pdf
http://ave.dee.isep.ipp.pt/~jtm/files/D20.pdf
http://www.scielo.org.mx/pdf/rmf/v60n1/v60n1a6.pdf
http://www.amazon.in/Fractional-Order-Systems-Applications-Scientific/dp/9814304190?tag=googinhydr18418-21&kpid=9814304190&tag=googinkenshoo-21&ascsubtag=d870664e-a409-4b22-8271-411663d93887
http://www.amazon.in/Fractional-Calculus-Applications-Varsha-Daftardar-Gejji/dp/8184873336?tag=googinhydr18418-21&kpid=8184873336&tag=googinkenshoo-21&ascsubtag=d870664e-a409-4b22-8271-411663d93887
the derivatives of some order that is not integer is called fractional order. suppose we want to calculate dx/dt of order 1/2 then fractional order concept occurs. This theory is useful in studying the anomalous behaviour of dynamical systems in electrochemistry, biology and viscoelasiticy etc.
Like first and higher order derivatives there are fractional derivatives which are dealt in fractional calculus
Names like velocity,acceleration are context oriented and relative,not absolute.Like first and second order derivatives the fractional derivatives also have useful applications as specified in Rakhi Tiwari's answer above.Once there is application it means that there is physical meaning.How to name them -I think it is too small a matter.
Please check this link about geometric and physical Interpretation of fractional integration and fractional differentiation by Igor Podlubny.
http://people.tuke.sk/igor.podlubny/pspdf/pifcaa_r.pdf
Fractional order derivatives related to memory and hereditary properties of various real materials.
There is no physical meaning yet. But I heard that there will be two papers to be published in IEEE Access soon about this. Maybe the answers are there.
The derivative of a function f(x) at a point x is a local property only when a is an integer. It is not the case for non-integer power derivatives. In other words, it is not correct to say that the fractional order derivative at x of a function depends only on values of f very near x , in the way that integer-power derivatives certainly do. Therefore, it is expected that the theory involves some sort of boundary conditions, involving information on the function. 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.
Human imagination seems wired on linearity and orthogonality. For example, we struggle with fractal dimensions. So, fractional order disturbs us. Therefore, when we encounter phenomena outside our comfort, we use otherwise pathological structures as tags. And that seems the case with fractional order, which brings us comfort in the fractals of finance for example (cf. derivatives of the Black-Scholes model).
The fractional order parameter, in case of heat conduction equation, means the level of thermal conductivity. If alpha ( fractional order parameter) is one, it show that thermal conductivity of the medium is normal. If alpha is less than one, showing that weak conductivity and alpha greater than 1 means strong conductivity.
see the following papers
https://core.ac.uk/download/pdf/82545921.pdf
https://en.wikipedia.org/wiki/Fractional-order_system
https://www2.icp.uni-stuttgart.de/~hilfer/publikationen/pdfo/ZZ-2019-HFCA-47.pdf
you will find good information
The physical entities or their functions that we humans model by employing mathematical instruments (attaching humanly devised symbols to such humanly perceived elements and functions) have limits and boundaries imposed by these symbols very existence. At some scale, these boundaries and limits, as well as the utility of these symbols become gross approximations, and they lose their meaning, which they acquired under a prior envisioned scale. Nice discussion!
I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional Calculus and Applied Analysis. 5, (2002), 367-386.
Dear esteemed researchers,
I kindly request for a physical interpretation and meaning of Caputo fractional derivative that resulted into the attached figure. In the figure, each curve corresponds to a unique derivative order \alpha in the interval (0, 1].
Kind regards.
How do we explain the memory effect of fractional order derivatives and integrals?
- Badges
- Science topic
- Similar topics
- Mathematics