for example in dynamics first derivative is rate or velocity: dx/dt
or the second derivative is acceleration: d2x/dt2
but in some cases we see fractional differential equations such as( d^0.8x/dt^0.8). I mean the order is fractional and 0.8 can be another fractional number. in mathematics there is no problem with this but what is the physical meaning? this equations are frequently used in anomalous diffusion.
The attached article is a thesis. It explains quite well, beginning on page 51---application of fractional differential equation in 4 cases. Cheers.
The is an old Dover book called 'the Fractional Calculus' by Oldham and Spannier. They describe many applications of the calculus.
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.
Thank you for answers. I know about fractional derivatives but I want to know what should I say when someone asked me: "what does mean (d^0.8x/dt^0.8)?" how can I explain it??
fractional derivatives are non-local so the 1/2 derivative can not have a local meaning like tangent or curvature but would have to take into account the properties of the curve over a large extent (boundary conditions).
Fractional differential equations can explain time delay, fractal properties etc...
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.
The meaning for fractional (in time) derivative may change from one definition to the next. In the case of Riemann-Louiville and Caputo like fractional derivatives, the differential equations that involve them arise as models (in the limit) of a variety of stochastic processes with time delay (see Appendix C, III), but there are also the consequence of some formalism in heat transfer phenomena that was proposed back in the 1960s (see Appendix C, II). Appendix C is found in the attached link/paper.
Book Fractional-in-Time Semilinear Parabolic Equations and Applications
the main answer and physical meaning of fractional is "Next Dimension"
in normal diffusion, the least square displacement of particles describes by a linear correlation which can be characterized by classical calculus. A good example would be the heat conduction based on Fourier's law. in this case, it is assumed that the thermal disturbance would diffuse by Fourier law, or in other words by diffusion process. however, in many case, for instance heat transfer at short temporal scales, porous media especially those non-linear behaviors, the Fourier's law is not valid anymore, and thus such processes can be described by ODEs based on fractional (non-integer) order. what is more, such processes are known as anomalous in which the correlation between time and properties is defined by a power law with non-integer order. additionally, in these cases, the memory effects of system are rather important to being considered, meaning that the previous states are so essential to calculate the current states. this objective only can be achieved by use fractional calculus
fractional derivatives are non-local so the 1/2 derivative can not have a local meaning like tangent or curvature but would have to take into account the properties of the curve over a large extent (boundary conditions)
One physical method for determining the fractional derivative (a ratio or tangent or speed=distance/time) is, for maximal boundary conditions, surround the object of measure with an equilateral triangle, each side equated to length 1 and midpoints or 1/2, halfway points connected at the base, iteratively infinitely forming equilateral triangles to the minimal limit of 0, or a dimensionless point, physically referred to as the Planck length h or a Big Bang and Black Hole equivalent, as the fastest physical phenomenon in the universe -- to which all else must therefore be relative, namely the linear (1D) speed of light, c=1, versus orthogonal electromagnetism or parabolic-curved c²=+/-1 -- travels, as the fastest, the shortest path, in mathematical physics, an ellipse two-dimensionally or geodesic three-dimensionally or in a fractal universe, a rugby ball shape (cf. Grigory Perelman's solution to the Poincare conjecture in 3D), two formulaic cones facing each other.
With this absolute, maximal attribute of the universe established and equateable to 1, all other objects may then be measured in relation to this maximal physical boundary as again a ratio or fractal derivative or rate of speed=distance/time. It's almost like, if not actually, inverting the sum of the areas of right triangles (1/2 a square) UNDER a smooth curve to summing the areas of triangles (in 3D, volumes of cones) surrounding jagged or roughly-curved space boundaries.
As, discussed above that Fractional derivatives are non-local (i.g;), but if we talk about say, half derivative (d^0.5y/dt^0.5) have pretty good connection with the time scales (i.g; independent variable scales), say if we have "dy/dt" the rate of change scale is homogenous, but for half derivative is like Time-slowing => rate of change of y is faster at lower value's of "t" then higher values of "t",
the similar behavior can be obtained from "Tautochrone Problem'' using Riemann-Louiville definition for half derivative
Ahmad Sharifi-viand
Fractional derivative is non-local. It is used to simulate those systems/ models that need accurate modeling.
Recall the relationship
between stress and strain in a material under the influence of external forces.
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