Generalization of derivative to fractals
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 tα . 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 (xβ , tα ). Elementary physical concepts such as velocity are redefined as follows for
fractal spacetime (xβ , tα ):
[2]
v
′
=
d
x
′
d
t
′
=
d
x
β
d
t
α
,
α
,
β
>
0
{\displaystyle v'={\frac {dx'}{dt'}}={\frac {dx^{\beta }}{dt^{\alpha }}}\,,\quad \alpha ,\beta >0}
,
where Sα,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.
[2]
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]
∂
f
(
t
)
∂
t
α
=
lim
t
1
→
t
f
(
t
1
)
−
f
(
t
)
t
1
α
−
t
α
,
α
>
0
{\displaystyle {\frac {\partial f(t)}{\partial t^{\alpha }}}=\lim _{t_{1}\rightarrow t}{\frac {f(t_{1})-f(t)}{t_{1}^{\alpha }-t^{\alpha }}}\,,\quad \alpha >0}
,
A more general definition is given by
[3]
∂
β
f
(
t
)
∂
t
α
=
lim
t
1
→
t
f
β
(
t
1
)
−
f
β
(
t
)
t
1
α
−
t
α
,
α
>
0
,
β
>
0
{\displaystyle {\frac {\partial ^{\beta }f(t)}{\partial t^{\alpha }}}=\lim _{t_{1}\rightarrow t}{\frac {f^{\beta }(t_{1})-f^{\beta }(t)}{t_{1}^{\alpha }-t^{\alpha }}}\,,\quad \alpha >0,\beta >0}
.
For a function y(t) on
F
α
{\displaystyle F^{\alpha }}
-perfect fractal set F the fractal derivative or
F
α
{\displaystyle F^{\alpha }}
-derivative of y(t) at t is defined by
D
F
α
y
(
t
)
=
{
F
−
lim
x
→
t
y
(
x
)
−
y
(
t
)
S
F
α
(
x
)
−
S
F
α
(
t
)
,
if
t
∈
F
;
0
,
otherwise
.
{\displaystyle D_{F}^{\alpha }y(t)=\left\{{\begin{array}{ll}{\underset {x\rightarrow t}{F_{-}\lim }}~{\frac {y(x)-y(t)}{S_{F}^{\alpha }(x)-S_{F}^{\alpha }(t)}},&{\text{if}}~t\in F;\\0,&{\text{otherwise}}.\end{array}}\right.}
.
Motivation
The
derivatives of a function f can be defined in terms of the coefficients ak in the
Taylor series expansion:
f
(
x
)
=
∑
k
=
1
∞
a
k
⋅
(
x
−
x
0
)
k
=
∑
k
=
1
∞
1
k
!
d
k
f
d
x
k
(
x
0
)
⋅
(
x
−
x
0
)
k
=
f
(
x
0
)
+
f
′
(
x
0
)
⋅
(
x
−
x
0
)
+
o
(
x
−
x
0
)
{\displaystyle f(x)=\sum _{k=1}^{\infty }a_{k}\cdot (x-x_{0})^{k}=\sum _{k=1}^{\infty }{1 \over k!}{d^{k}f \over dx^{k}}(x_{0})\cdot (x-x_{0})^{k}=f(x_{0})+f'(x_{0})\cdot (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
)
x
−
x
0
=
lim
x
→
x
0
f
(
x
)
−
f
(
x
0
)
x
−
x
0
{\displaystyle 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 (xα -(x0 )α )k :
f
(
x
)
=
∑
k
=
1
∞
b
k
⋅
(
x
α
−
x
0
α
)
k
=
f
(
x
0
)
+
b
1
⋅
(
x
α
−
x
0
α
)
+
o
(
x
α
−
x
0
α
)
{\displaystyle f(x)=\sum _{k=1}^{\infty }b_{k}\cdot (x^{\alpha }-x_{0}^{\alpha })^{k}=f(x_{0})+b_{1}\cdot (x^{\alpha }-x_{0}^{\alpha })+o(x^{\alpha }-x_{0}^{\alpha })}
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
→
x
0
f
(
x
)
−
f
(
x
0
)
x
α
−
x
0
α
=
d
e
f
d
f
d
x
α
(
x
0
)
{\displaystyle b_{1}=\lim _{x\to x_{0}}{f(x)-f(x_{0}) \over x^{\alpha }-x_{0}^{\alpha }}{\overset {\underset {\mathrm {def} }{}}{=}}{df \over dx^{\alpha }}(x_{0})}
The Fractal Maclaurin series of f(t) with fractal support F is as follows:
f
(
t
)
=
∑
m
=
0
∞
(
D
F
α
)
m
f
(
t
)
|
t
=
0
m
!
(
S
F
α
(
t
)
)
m
{\displaystyle f(t)=\sum _{m=0}^{\infty }{\frac {(D_{F}^{\alpha })^{m}f(t)|_{t=0}}{m!}}(S_{F}^{\alpha }(t))^{m}}
Properties
Expansion coefficients
Just like in the Taylor series expansion, the coefficients bk can be expressed in terms of the fractal derivatives of order k of f:
b
k
=
1
k
!
(
d
d
x
α
)
k
f
(
x
=
x
0
)
{\displaystyle b_{k}={1 \over k!}{\biggl (}{d \over dx^{\alpha }}{\biggr )}^{k}f(x=x_{0})}
Proof idea: Assuming
(
d
d
x
α
)
k
f
(
x
=
x
0
)
{\textstyle ({d \over dx^{\alpha }})^{k}f(x=x_{0})}
exists, bk can be written as
b
k
=
a
k
⋅
(
d
d
x
α
)
k
f
(
x
=
x
0
)
{\textstyle b_{k}=a_{k}\cdot ({d \over dx^{\alpha }})^{k}f(x=x_{0})}
One can now use
f
(
x
)
=
(
x
α
−
x
0
α
)
n
⇒
(
d
d
x
α
)
k
f
(
x
=
x
0
)
=
n
!
δ
n
k
{\textstyle f(x)=(x^{\alpha }-x_{0}^{\alpha })^{n}\Rightarrow ({d \over dx^{\alpha }})^{k}f(x=x_{0})=n!\delta _{n}^{k}}
and since
b
n
=
!
1
⇒
a
n
=
1
n
!
{\textstyle b_{n}{\overset {\underset {\mathrm {!} }{}}{=}}1\Rightarrow a_{n}={1 \over n!}}
Chain rule
If for a given function f both the derivative Df and the fractal derivative Dα f exist, one can find an analog to the chain rule:
d
f
d
x
α
=
d
f
d
x
d
x
d
x
α
=
1
α
x
1
−
α
d
f
d
x
{\displaystyle {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
d
x
d
x
α
=
(
d
x
α
d
x
)
−
1
{\displaystyle {\frac {dx}{dx^{\alpha }}}=({\frac {dx^{\alpha }}{dx}})^{-1}}
Similarly for the more general definition:
d
β
f
d
α
x
=
d
(
f
β
)
d
α
x
=
1
α
x
1
−
α
β
f
β
−
1
(
x
)
f
′
(
x
)
{\displaystyle {d^{\beta }f \over d^{\alpha }x}={d(f^{\beta }) \over d^{\alpha }x}={1 \over \alpha }x^{1-\alpha }\beta f^{\beta -1}(x)f'(x)}
Fractal derivative for function f (t ) = t , with derivative order is α ∈ (0,1]
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 process,
[3]
d
u
(
x
,
t
)
d
t
α
=
D
∂
∂
x
β
(
∂
u
(
x
,
t
)
∂
x
β
)
,
(
1
)
{\displaystyle {\frac {du(x,t)}{dt^{\alpha }}}=D{\frac {\partial }{\partial x^{\beta }}}\left({\frac {\partial u(x,t)}{\partial x^{\beta }}}\right),\quad (1)}
u
(
x
,
0
)
=
δ
(
x
)
{\displaystyle u(x,0)=\delta (x)}
where 0 < α < 2, 0 < β < 1,
x
∈
R
{\displaystyle x\in \mathbb {R} }
, and δ (x ) is the
Dirac delta function .
To obtain the
fundamental solution , we apply the transformation of variables
t
′
=
t
α
,
x
′
=
x
β
.
{\displaystyle t'=t^{\alpha }\,,\quad x'=x^{\beta }.}
then the equation (1) becomes the normal diffusion form equation, the solution of (1) has the stretched
Gaussian
kernel :
[3]
u
(
x
,
t
)
=
1
2
π
t
α
e
−
x
2
β
4
t
α
{\displaystyle u(x,t)={\frac {1}{2{\sqrt {\pi t^{\alpha }}}}}e^{-{\frac {x^{2\beta }}{4t^{\alpha }}}}}
The
mean squared displacement of above fractal derivative diffusion equation has the
asymptote :
[3]
⟨
x
2
(
t
)
⟩
∝
t
(
3
α
−
α
β
)
/
2
β
.
{\displaystyle \left\langle x^{2}(t)\right\rangle \propto t^{(3\alpha -\alpha \beta )/2\beta }.}
Fractal-fractional calculus
The fractal derivative is connected to the classical derivative if the first derivative exists. In this case,
∂
f
(
t
)
∂
t
α
=
lim
t
1
→
t
f
(
t
1
)
−
f
(
t
)
t
1
α
−
t
α
=
d
f
(
t
)
d
t
1
α
t
α
−
1
,
α
>
0
{\displaystyle {\frac {\partial f(t)}{\partial t^{\alpha }}}=\lim _{t_{1}\rightarrow t}{\frac {f(t_{1})-f(t)}{t_{1}^{\alpha }-t^{\alpha }}}\ ={\frac {df(t)}{dt}}{\frac {1}{\alpha t^{\alpha -1}}},\quad \alpha >0}
.
However, due to the differentiability property of an integral, fractional derivatives are differentiable, thus the following new concept was introduced by Prof Abdon Atangana from South Africa.
The following differential operators were introduced and applied very recently.
[4] 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:
[4]
Having power law type kernel:
F
F
P
D
0
,
t
α
,
β
(
y
(
t
)
)
=
1
Γ
(
m
−
α
)
d
d
t
β
∫
0
t
(
t
−
s
)
m
−
α
−
1
y
(
s
)
d
s
{\displaystyle ^{FFP}D_{0,t}^{\alpha ,\beta }{\Big (}y(t){\Big )}={\dfrac {1}{\Gamma (m-\alpha )}}{\dfrac {d}{dt^{\beta }}}\int _{0}^{t}(t-s)^{m-\alpha -1}y(s)ds}
Having exponentially decaying type kernel:
F
F
E
D
0
,
t
α
,
β
(
y
(
t
)
)
=
M
(
α
)
1
−
α
d
d
t
β
∫
0
t
exp
(
−
α
1
−
α
(
t
−
s
)
)
y
(
s
)
d
s
{\displaystyle ^{FFE}D_{0,t}^{\alpha ,\beta }{\Big (}y(t){\Big )}={\dfrac {M(\alpha )}{1-\alpha }}{\dfrac {d}{dt^{\beta }}}\int _{0}^{t}\exp {\Big (}-{\dfrac {\alpha }{1-\alpha }}(t-s){\Big )}y(s)ds}
,
Having generalized Mittag-Leffler type kernel:
a
F
F
M
D
t
α
f
(
t
)
=
A
B
(
α
)
1
−
α
d
d
t
β
∫
a
t
f
(
τ
)
E
α
(
−
α
(
t
−
τ
)
α
1
−
α
)
d
τ
.
{\displaystyle {}_{a}^{FFM}D_{t}^{\alpha }f(t)={\frac {AB(\alpha )}{1-\alpha }}{\frac {d}{dt^{\beta }}}\int _{a}^{t}f(\tau )E_{\alpha }\left(-\alpha {\frac {\left(t-\tau \right)^{\alpha }}{1-\alpha }}\right)\,d\tau \,.}
The above differential operators each have an associated fractal-fractional integral operator, as follows:
[4]
F
F
P
J
0
,
t
α
,
β
(
y
(
t
)
)
=
β
Γ
(
α
)
∫
0
t
(
t
−
s
)
α
−
1
s
β
−
1
y
(
s
)
d
s
{\displaystyle ^{FFP}J_{0,t}^{\alpha ,\beta }{\Big (}y(t){\Big )}={\dfrac {\beta }{\Gamma (\alpha )}}\int _{0}^{t}(t-s)^{\alpha -1}s^{\beta -1}y(s)ds}
Exponentially decaying type kernel:
F
F
E
J
0
,
t
α
,
β
(
y
(
t
)
)
=
α
β
M
(
α
)
∫
0
t
s
β
−
1
y
(
s
)
d
s
+
β
(
1
−
α
)
t
β
−
1
y
(
t
)
M
(
α
)
{\displaystyle ^{FFE}J_{0,t}^{\alpha ,\beta }{\Big (}y(t){\Big )}={\dfrac {\alpha \beta }{M(\alpha )}}\int _{0}^{t}s^{\beta -1}y(s)ds+{\dfrac {\beta (1-\alpha )t^{\beta -1}y(t)}{M(\alpha )}}}
.
Generalized Mittag-Leffler type kernel:
F
F
M
J
0
,
t
α
,
β
(
y
(
t
)
)
=
α
β
A
B
(
α
)
∫
0
t
s
β
−
1
y
(
s
)
(
t
−
s
)
α
−
1
d
s
+
β
(
1
−
α
)
t
β
−
1
y
(
t
)
A
B
(
α
)
{\displaystyle ^{FFM}J_{0,t}^{\alpha ,\beta }{\Big (}y(t){\Big )}={\dfrac {\alpha \beta }{AB(\alpha )}}\int _{0}^{t}s^{\beta -1}y(s)(t-s)^{\alpha -1}ds+{\dfrac {\beta (1-\alpha )t^{\beta -1}y(t)}{AB(\alpha )}}}
.
FFM refers 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
β
∈
R
{\displaystyle \beta \in \mathbb {R} }
of f is defined by:
[1]
x
I
b
β
f
(
x
)
=
1
Γ
(
β
)
∫
x
b
f
(
t
)
(
S
F
α
(
t
)
−
S
F
α
(
x
)
)
1
−
β
d
F
α
t
{\displaystyle {x}{\mathcal {I}}_{b}^{\beta }f(x)={\frac {1}{\Gamma (\beta )}}\int _{x}^{b}{\frac {f(t)}{(S_{F}^{\alpha }(t)-S_{F}^{\alpha }(x))^{1-\beta }}}d_{F}^{\alpha }t}
.
Fractal analogue of the left-sided Riemann-Liouville fractional integral of order
β
∈
R
{\displaystyle \beta \in \mathbb {R} }
of f is defined by:
a
I
x
β
f
(
x
)
=
1
Γ
(
β
)
∫
a
x
f
(
t
)
(
S
F
α
(
x
)
−
S
F
α
(
t
)
)
1
−
β
d
F
α
t
.
{\displaystyle {a}{\mathcal {I}}_{x}^{\beta }f(x)={\frac {1}{\Gamma (\beta )}}\int _{a}^{x}{\frac {f(t)}{(S_{F}^{\alpha }(x)-S_{F}^{\alpha }(t))^{1-\beta }}}d_{F}^{\alpha }t.}
Fractal analogue of the right-sided Riemann-Liouville fractional derivative of order
β
∈
R
{\displaystyle \beta \in \mathbb {R} }
of f is defined by:
x
D
b
β
f
(
x
)
=
1
Γ
(
n
−
β
)
(
−
D
F
α
)
n
∫
x
b
f
(
t
)
(
S
F
α
(
t
)
−
S
F
α
(
x
)
)
−
n
+
β
+
1
d
F
α
t
{\displaystyle {x}{\mathcal {D}}_{b}^{\beta }f(x)={\frac {1}{\Gamma (n-\beta )}}(-D_{F}^{\alpha })^{n}\int _{x}^{b}{\frac {f(t)}{(S_{F}^{\alpha }(t)-S_{F}^{\alpha }(x))^{-n+\beta +1}}}d_{F}^{\alpha }t}
Fractal analogue of the left-sided Riemann-Liouville fractional derivative of order
β
∈
R
{\displaystyle \beta \in \mathbb {R} }
of f is defined by:
a
D
x
β
f
(
x
)
=
1
Γ
(
n
−
β
)
(
D
F
α
)
n
∫
a
x
f
(
t
)
(
S
F
α
(
x
)
−
S
F
α
(
t
)
)
−
n
+
β
+
1
d
F
α
t
{\displaystyle {a}{\mathcal {D}}_{x}^{\beta }f(x)={\frac {1}{\Gamma (n-\beta )}}(D_{F}^{\alpha })^{n}\int _{a}^{x}{\frac {f(t)}{(S_{F}^{\alpha }(x)-S_{F}^{\alpha }(t))^{-n+\beta +1}}}d_{F}^{\alpha }t}
Fractal analogue of the right-sided Caputo fractional derivative of order
β
∈
R
{\displaystyle \beta \in \mathbb {R} }
of f is defined by:
x
C
D
b
β
f
(
x
)
=
1
Γ
(
n
−
β
)
∫
x
b
(
S
F
α
(
t
)
−
S
F
α
(
x
)
)
n
−
β
−
1
(
−
D
F
α
)
n
f
(
t
)
d
F
α
t
{\displaystyle {x}^{C}{\mathcal {D}}_{b}^{\beta }f(x)={\frac {1}{\Gamma (n-\beta )}}\int _{x}^{b}(S_{F}^{\alpha }(t)-S_{F}^{\alpha }(x))^{n-\beta -1}(-D_{F}^{\alpha })^{n}f(t)d_{F}^{\alpha }t}
Fractal analogue of the left-sided Caputo fractional derivative of order
β
∈
R
{\displaystyle \beta \in \mathbb {R} }
of f is defined by:
a
C
D
x
β
f
(
x
)
=
1
Γ
(
n
−
β
)
∫
a
x
(
S
F
α
(
x
)
−
S
F
α
(
t
)
)
n
−
β
−
1
(
D
F
α
)
n
f
(
t
)
d
F
α
t
{\displaystyle {a}^{C}{\mathcal {D}}_{x}^{\beta }f(x)={\frac {1}{\Gamma (n-\beta )}}\int _{a}^{x}(S_{F}^{\alpha }(x)-S_{F}^{\alpha }(t))^{n-\beta -1}(D_{F}^{\alpha })^{n}f(t)d_{F}^{\alpha }t}
See also
References
^
a
b Khalili Golmankhaneh, Alireza (2022).
Fractal Calculus and its Applications . Singapore: World Scientific Pub Co Inc. p. 328.
doi :
10.1142/12988 .
ISBN
978-981-126-110-7 .
S2CID
248575991 .
^
a
b Chen, Wen (May 2006). "Time-space fabric underlying anomalous diffusion". Chaos, Solitons & Fractals . 28 (4): 923–929.
arXiv :
math-ph/0505023 .
Bibcode :
2006CSF....28..923C .
doi :
10.1016/j.chaos.2005.08.199 .
^
a
b
c
d
e Chen, Wen; Sun, Hongguang; Zhang, Xiaodi; Korošak, Dean (March 2010).
"Anomalous diffusion modeling by fractal and fractional derivatives" . Computers & Mathematics with Applications . 59 (5): 1754–1758.
doi :
10.1016/j.camwa.2009.08.020 .
^
a
b
c Atangana, Abdon; Sania, Qureshi (2019). "Modeling attractors of chaotic dynamical systems with fractal–fractional operators". Chaos, Solitons & Fractals . 123 : 320–337.
Bibcode :
2019CSF...123..320A .
doi :
10.1016/j.chaos.2019.04.020 .
S2CID
145861887 .
Bibliography
Chen, Wen (2006). "Time–space fabric underlying anomalous diffusion". Chaos, Solitons and Fractals . 28 (4): 923–929.
arXiv :
math-ph/0505023 .
Bibcode :
2006CSF....28..923C .
doi :
10.1016/j.chaos.2005.08.199 .
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