Fractal derivative explained

In applied mathematics and mathematical analysis, the fractal derivative or Hausdorff derivative is a non-Newtonian generalization of the derivative dealing with the measurement of fractals, 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 generalization of standard calculus.^[1]

Physical background

Porous media, aquifers, turbulence, 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, Darcy's law, and Fourier's law) are not applicable to fractal media. To address this, concepts such as distance and velocity 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):^[2]

v' = (dx')/(= dt') (dx ^{\beta})/(dt ^{\alpha}) , \alpha,\beta> 0 ,

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 f(t) with respect to a fractal measure t has been introduced as follows:^[3]

(\partial f(t))/(\partial t ^{\alpha}) = \lim _{t _{1} → t} (f(t _{1)-f(t)})/(t ^{\alpha - t} _{1} ^{\alpha}) , \alpha> 0 ,

A more general definition is given by

(\partial ^{\beta} f(t))/(\partial t ^{\alpha}) = \lim _{t _{1} → t} (f ^{\beta} (t ^{\beta} _{1 ) - f} (t))/(t ^{\alpha - t} _{1} ^{\alpha}) , \alpha> 0 , \beta> 0 .For a function y(t) on F ^{\alpha} -perfect fractal set F the fractal derivative or F ^{\alpha} -derivative of y(t) at t is defined byD ^{\alpha} _{F} y(t) = \left\{ \begin{array}{ll} \underset{ x → t}{F _{-} \lim}~ (y(x) - y(t))/(S ^{\alpha} _{F} (x) - S ^{\alpha} _{F} (t)) , & if ~ t\in F; \\ 0 , & otherwise . \end{array} \right. .

Motivation

The derivatives of a function f can be defined in terms of the coefficients a_k in the Taylor series expansion:

f(x) = \sum ^{inf in} _{k =1} a _{k ⋅ (x-x} ^{inf in} ^{k =1} { 1 \over k!}{d ^{kf} \over dx ^{k}(x} _{0 ) ⋅ (x - x} ^{k = f(x} ^{0 ) + f'(x} _{0) ⋅ (x-x} _{0)+o(x-x} _{0)}

From this approach one can directly obtain:

f'(x _{0)={f(x)-f(x} _{0)-o(x-x} _{0)} \over x - x _{0}=\lim} ^{x \to x _{0}} {f(x) - f(x _{0)} \over x - x _{0}}

This can be generalized approximating f with functions (x^α-(x₀)^α)^k:

f(x) = \sum ^{inf in} _{k =1} b ^{\alpha)} _{0} ^{k = f(x} ^{0 ) + b} ^{\alpha) + o(x} ^{0} ^{\alpha)} ^{0}

Note that the lowest order coefficient still has to be b₀=f(x₀), since it's still the constant approximation of the function f at x₀.

Again one can directly obtain:

b _{1=\lim} ^{x \to x _{0}} {f(x) - f(x _{0)} \over x ^{\alpha}} _{0} \overset{\underset{ def }} (x_0)

The Fractal Maclaurin series of f(t) with fractal support F is as follows:

f(t) = \sum ^{inf ty} _{m =0} ((D ^{\alpha} _{F} ) ^{m} f(t)| _{t=0})/(m!) (S ^{\alpha} _{F} (t)) ^{m}

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:

b _{k={1} \over k!} l ({d \over dx ^{\alpha}r)} ^{kf(x = x} ^{0 )}

Proof idea: Assuming

$^kf(x=x\_0)$

exists, b_k can be written as

$b\_k=a\_k\; \backslash cdot^kf(x=x\_0)$

One can now use

$f(x)\; =\; (x^\backslash alpha-x\_0^\backslash alpha)^n\; \backslash Rightarrow\; ^kf(x=x\_0)=n!\backslash delta\_n^k$

and since

$b\_n\backslash overset1\; \backslash Rightarrow\; a\_n\; =$

Chain rule

If for a given function f both the derivative Df and the fractal derivative D_αf exist, one can find an analog to the chain rule:

{df \over dx ^{\alpha}} = {df \over dx}{dx \over dx ^{\alpha}={1} \over \alpha}x ^{1-\alpha} {df \over dx}

The last step is motivated by the implicit function theorem which, under appropriate conditions, gives us

(dx)/(dx ^{\alpha}) = ( (dx ^{\alpha})/(dx) ) ^{-1}

Similarly for the more general definition:

{d ^{\beta} f \over d ^{\alpha} x} = {d(f ^{\beta)} \over d ^{\alpha} x} = { 1 \over \alpha} x ^{1-\alpha} \beta f ^{\beta} (x)f'(x)

Notes and References

1. Book: Khalili Golmankhaneh, Alireza . 2022 . Fractal Calculus and its Applications . Singapore . World Scientific Pub Co Inc. 328 . 10.1142/12988 . 978-981-126-110-7 . 248575991 .
2. Chen . Wen . May 2006 . Time-space fabric underlying anomalous diffusion . Chaos, Solitons & Fractals . 28 . 4 . 923–929 . 10.1016/j.chaos.2005.08.199. math-ph/0505023 . 2006CSF....28..923C .
3. Chen . Wen . Sun . Hongguang . Zhang . Xiaodi . Korošak . Dean . March 2010 . Anomalous diffusion modeling by fractal and fractional derivatives . Computers & Mathematics with Applications . en . 59 . 5 . 1754–1758 . 10.1016/j.camwa.2009.08.020. free .