In
applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
and
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
These theories are usually studied ...
, the fractal derivative or Hausdorff derivative is a non-Newtonian generalization of the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
dealing with the measurement of
fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as ill ...
s, defined in fractal geometry. Fractal derivatives were created for the study of anomalous diffusion, by which traditional approaches fail to factor in the fractal nature of the media. A
fractal measure ''t'' is scaled according to ''t
α''. Such a derivative is local, in contrast to the similarly applied
fractional derivative. Fractal calculus is formulated as a generalized of standard calculus
Physical background
Porous media
A porous medium or a porous material is a material containing pores (voids). The skeletal portion of the material is often called the "matrix" or "frame". The pores are typically filled with a fluid (liquid or gas). The skeletal material is usu ...
,
aquifer
An aquifer is an underground layer of water-bearing, permeable rock, rock fractures, or unconsolidated materials ( gravel, sand, or silt). Groundwater from aquifers can be extracted using a water well. Aquifers vary greatly in their characte ...
s,
turbulence
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between ...
, and other media usually exhibit fractal properties. Classical diffusion or dispersion laws based on random walks in free space (essentially the same result variously known as
Fick's laws of diffusion
Fick's laws of diffusion describe diffusion and were derived by Adolf Fick in 1855. They can be used to solve for the diffusion coefficient, . Fick's first law can be used to derive his second law which in turn is identical to the diffusion equ ...
,
Darcy's law
Darcy's law is an equation that describes the flow of a fluid through a porous medium. The law was formulated by Henry Darcy based on results of experiments on the flow of water through beds of sand, forming the basis of hydrogeology, a branch of ...
, and
Fourier's law
Conduction is the process by which heat is transferred from the hotter end to the colder end of an object. The ability of the object to conduct heat is known as its '' thermal conductivity'', and is denoted .
Heat spontaneously flows along a t ...
) are not applicable to fractal media. To address this, concepts such as
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
and
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
must be redefined for fractal media; in particular, scales for space and time are to be transformed according to (''x
β'', ''t
α''). Elementary physical concepts such as velocity are redefined as follows for
fractal spacetime (''x
β'', ''t
α''):
:
,
where ''S
α,β'' represents the fractal spacetime with scaling indices ''α'' and ''β''. The traditional definition of velocity makes no sense in the non-differentiable fractal spacetime.
Definition
Based on above discussion, the concept of the fractal derivative of a function ''u''(''t'') with respect to a
fractal measure ''t'' has been introduced as follows:
:
,
A more general definition is given by
:
.
For a function y(t) on
-perfect fractal set F the fractal derivative or
-derivative of at t, is defined by
:
.
Motivation
The
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
s of a function f can be defined in terms of the coefficients a
k in the
Taylor series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
expansion:
From this approach one can directly obtain:
This can be generalized approximating f with functions (x
α-(x
0)α)
k:
note: the lowest order coefficient still has to be b
0=f(x
0), since it's still the constant approximation of the function f at x
0.
Again one can directly obtain:
*The Fractal Maclaurin series of f(t) with fractal support F is as follows:
Properties
Expansion coefficients
Just like in the Taylor series expansion, the coefficients b
k can be expressed in terms of the fractal derivatives of order k of f:
''Proof idea: assuming
exists, b
k can be written as
''
''one can now use
and since
''
Connection with Derivative
If for a given function f both the derivative Df and the fractal derivative D
αf exists, one can find an analog to the chain rule:
The last step is motivated by the
Implicit function theorem which, under appropriate conditions, gives us dx/dx
α = (dx
α/dx)
−1
Similarly for the more general definition:
Application in anomalous diffusion
As an alternative modeling approach to the classical Fick's second law, the fractal derivative is used to derive a linear anomalous transport-diffusion equation underlying
anomalous diffusion
Anomalous diffusion is a diffusion process with a non-linear relationship between the mean squared displacement (MSD), \langle r^(\tau )\rangle , and time. This behavior is in stark contrast to Brownian motion, the typical diffusion process descri ...
process,
:
:
where 0 < ''α'' < 2, 0 < ''β'' < 1, and ''δ''(''x'') is the
Dirac delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...
.
In order to obtain the
fundamental solution
In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not a ...
, we apply the transformation of variables
:
then the equation (1) becomes the normal diffusion form equation, the solution of (1) has the stretched
Gaussian
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponym ...
form:
:
The
mean squared displacement
In statistical mechanics, the mean squared displacement (MSD, also mean square displacement, average squared displacement, or mean square fluctuation) is a measure of the deviation of the position of a particle with respect to a reference positio ...
of above fractal derivative diffusion equation has the
asymptote
In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related context ...
:
:
Fractal-fractional calculus
The fractal derivative is connected to the classical derivative if the first derivative of the function under investigation exists. In this case,
:
.
However, due to the differentiability property of an integral, fractional derivatives are differentiable, thus the following new concept was introduced
The following differential operators were introduced and applied very recently.
Supposing that y(t) be continuous and fractal differentiable on (a, b) with order ''β'', several definitions of a fractal–fractional derivative of y(t) hold with order α in the Riemann–Liouville sense:
*Having power law type kernel:
*Having exponentially decaying type kernel:
,
*Having generalized Mittag-Leffler type kernel:
The above differential operators each have an associated fractal-fractional integral operator, as follows:
*Power law type kernel:
*Exponentially decaying type kernel:
.
*Generalized Mittag-Leffler type kernel:
.
FFM is refereed to fractal-fractional with the generalized Mittag-Leffler kernel.
Fractal non-local calculus
*Fractal analogue of the right-sided Riemann-Liouville fractional integral of order
of f is defined by:
.
*Fractal analogue of the left-sided Riemann-Liouville fractional integral of order
of f is defined by:
*Fractal analogue of the right-sided Riemann-Liouville fractional derivative of order
of f is defined by:
*Fractal analogue of the left-sided Riemann-Liouville fractional derivative of order
of f is defined by:
*Fractal analogue of the right-sided Caputo fractional derivative of order
of f is defined by:
*Fractal analogue of the left-sided Caputo fractional derivative of order
of f is defined by:
See also
*
Fractional calculus
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 D
:D f(x) = \frac f(x)\,,
and of the integration ...
*
Fractional-order system
In the fields of dynamical systems and control theory, a fractional-order system is a dynamical system that can be modeled by a fractional differential equation containing derivatives of non-integer order. Such systems are said to have ''fraction ...
*
Multifractal system
References
*
*
*
*
*
*
{{refend
External links
Power Law & Fractional DynamicsNon-Newtonian calculus website
Fractals
Applied mathematics
Non-Newtonian calculus
Mathematical analysis