F ξ-fractional differential form is introduced such that it can help us to derive the physical equation.. Furthermore, using the F ξ-fractional differential form of Maxwell’s equations o
Trang 1Central European Journal of Physics
Research Article
Alireza K Golmankhaneh1∗, Ali K Golmankhaneh1Dumitru Baleanu2,3,4†
1 Departments of Physics, Urmia Branch,
Islamic Azad University, P.O.BOX 969, Oromiyeh, Iran
2 Çankaya University, Faculty of Art and Sciences, Department of Mathematics and Computer Sciences
Balgat 0630, Ankara, Turkey
3 Department of Chemical and Materials Engineering, Faculty of Engineering,
King Abdulaziz University, P.O Box: 80204, Jeddah, 21589, Saudi Arabia
4 Institute of Space Sciences,
P.O.BOX, MG-23, R 76900, Magurele-Bucharest, Romania
Received 13 november 2012; accepted 14 February 2013
Abstract: In this paper we have generalized F ξ-calculus for fractals embedding in R 3 F ξ-calculus is a fractional local
derivative on fractals It is an algorithm which may be used for computer programs and is more applicable
than using measure theory In this Calculus staircase functions for fractals has important role F ξ-fractional differential form is introduced such that it can help us to derive the physical equation Furthermore, using
the F ξ-fractional differential form of Maxwell’s equations on fractals has been suggested.
PACS (2008): 02.30.Cj,02.30.Sa,02.40.Ma, 41.20.-q, 41.90.+e,
Keywords: fractal integral• fractal derivative• fractal differential equations• fractional Maxwell equation
© Versita sp z o.o.
1 Introduction
Fractal patterns appear in many natural phenomena
Fractal geometry plays important roles in some branches
of science, engineering and art Fractals have
compli-cated structures thus ordinary calculus does not apply
For example, The Lebesgue-Cantor staircase function is
zero almost everywhere, therefore this function isnot a
solution of any ordinary differential equations Fractional
derivatives are non-local therefore are suitable for
frac-∗
E-mail: alireza@physics.unipune.ac.in (Corresponding Author)
†
tal functions Several authors have recognized the need
to use fractional derivatives and integrals to explore the characteristic features of fractal walks, anomalous diffu-sion, transport, etc (see references [1 35] and references therein)
During recent decades, analysis on fractals has developed and been applied in many important cases, for example, heat conduction, fractal-time diffusion equation, waves, etc.(see[30,34,35] and several references therein) Using measure theory people have introduced calculus on fractals [32, 33] It consists of defining a derivative as
an inverse of an integral with respect to a measure and defining other operators using this derivative Taking into account all the above we conclude that a simple method
Trang 2of fractional order operators on fractal sets is only a
mod-erate survey Riemann integration like procedures have
their own place and are better and more useful algorithmic
methods [38].Recently, the Maxwell fractional-order
dif-ferential calculus addressed the skin effect and developed
a new method for implementing fractional-order inductive
elements [39] In this manuscript we have generalized F
ξ
-calculus on fractal subsets of R
3
The organization of the manuscript is as follows:
We begin the Section 2 by defining the generalized
F α
-calculus In Section 3 we introduce F ξ
-differential form Section5presents to our conclusions
2. Fractional F ξ -calculus
In this section we will introduce new definitions on fractals
subsets of R
3
We begin by defining the integral staircase
function [38]
stair-case
Now, we introduce the following new definitions:
Definition 1.
F be a fractal subset of <3
The flag function Θ(F , I ) for
a fractal set F is define as [36–38]
Θ(F , I ) =
(
1 if F ∩ I 6 = ∅,
0 otherwise
(1)
where I = [a, b] × [c, d] × [e, f ] is a subset in <
3
Definition 2.
For a set F and I= [a, b] × [c, d] × [e, f ], with subdivision
P
[a,b]×[c,d]×[e,f ] , a < b, c < d, e < fwe define
Λ
ξ
[F , I ] =
n
X
i=1
(x i − x i−1)α
Γ(α + 1)
(y i − y i−1)β
Γ(β + 1)
(z i − z i−1)ε
Γ(ε + 1) Θ(F , [x i−1, x i ] × [y i−1, y i ] × [z i−1, z i]) (2)
where ξ = α + β + ε Let 0 < α ≤ 1, 0 < β ≤ 1, and 0 <
ε ≤1.
Definition 3.
mass γ ξ δ (F , a, b, c, d, e, f ) of F ∩ [a, b] × [c, d] × [e, f ] is
given by
γ δ ξ (F , a, b, c, d, e, f ) = inf
P
[a,b]×[c,d]×[e,f ] :|P |≤δ
Λ
ξ
[F , I ] (3)
where |P | = max 1≤i≤n (x i −x i−1)(y i −y i−1)(z i −z i−1) Taking
infimum over all subdivisions P of [a, b] × [c, d] × [e, f ]
satisfying |P | ≤ δ
Definition 4.
The mass function γ
ξ
(F , a, b, c, d, e, f ) is given by [38]
γ ξ
(F , a, b, c, d, e, f ) = lim
δ→0γ ξ δ (F , a, b, c, d, e, f ) (4)
Definition 5.
Let a0, b0, c0 be fixed and real numbers. The integral
staircase function S
ξ
F (x , y, z ) of order ξ for a set F is [38]
S ξ F (x , y, z ) =
γ ξ
(F , a0, b
0, c
0, x, y, z) if
x ≥ a0, y ≥ b0, z ≥ c0
−γ ξ
(F , a0, b0, c0, x, y, z) otherwise
(5)
Definition 6.
We saythat a point (x , y, z ) is a point of change of a
function f if f is not constant over any open set (a, d) × (c, d) × (e, f ) containing (x , y, z ). The set of all points of
change of f is called the set of change of f and is denoted
by S chf [38]
Definition 7.
The η-dimension of F ∩ [a, b] × [c, d] × [e, f ] denoted by
dimη (F ∩ [a, b] × [c, d] × [e, f ]) and define
dimη (F ∩ [a, b] × [c, d] × [e, f ])
= inf {ξ : γ
ξ
(F , a, b, c, d, e, f ) = 0}
= sup{ξ : γ
ξ
(F , a, b, c, d, e, f ) = ∞} (6)
Definition 8.
Let F ⊂ R3
be such that S F ξ (x , y, z ) is finite for all
(x , y, z ) ∈ R3 for ξ = dim γ F Then the S ch(S
ξ
F) is said
to be ξ -perfect (Closed and every point of S ch(S
ξ
F) is its limit point)
Definition 9.
Let F ⊂ R
3
, f : R
3→ R3
and (x , y, z ) ∈ F A number l is
Trang 3F ξ
-limit of f as (x
0 , y 0 , z 0
) → (x , y, z ) if given any ε > 0 there exists δ > 0 such that [38]
(x
0 , y 0 , z 0
) ∈ F and
| (x 0 , y 0 , z 0
) − (x , y, z )| < δ ⇒ |f (x
0 , y 0 , z 0
) − l| < ε (7)
If such a number exists then it is denoted by
l = F ξ − lim
(x 0 ,y 0 ,z 0 )→(x ,y,z )
f (x 0 , y 0 , z 0
) (8)
Definition 10.
A function f : R3 → R3
is said to beF ξ
-continues at
(x , y, z ) ∈ F if
f (x , y, z ) = F ξ − lim
(x
0 ,y 0 ,z 0 )→(x ,y,z )
f (x 0 , y 0 , z 0) (9)
Definition 11.
Let f be a bounded function on F and I be a closed ball
[38] Then
M [f , F , I ] = sup
(x ,y,z )∈F ∩I
f (x , y, z ) if F ∩ I 6= 0
= 0 otherwise (10)
and similarly
m [f , F , I ] = inf
(x ,y,z )∈F ∩I
f (x , y, z ) if F ∩ I 6= 0
= 0 otherwise (11)
Definition 12.
Let S
ξ
F (x , y, z ) be finite for (x , y, z ) ∈ [a, b] × [c, d] × [e, f ].
Let P be a subdivision of [a, b] × [c, d] × [e, f ] with points
x0, y0, z0 , x n , y n , z n The upper F ξ -sum and the lower F ξ
-sum for function f over the subdivision P are given
respec-tively by [38]
U ξ
[f , F , P ] =
n
X
i=1
M [f , F , [(x i−1, y i−1, z i−1), (x i , y i , z i)]]
(S
ξ
F (x i , y i , z i ) − S
ξ
F (x i−1, y i−1, z i−1))
(12)
and
L ξ [f , F , P ] =
n
X
i=1
m [f , F , [(x i−1, y i−1, z i−1), (x i , y i , z i)]]
(S
ξ
F (x i , y i , z i ) − S
ξ
F (x i− , y i− , z i− )) (13)
Definition 13.
If f be a bounded function on F we say that f is F
ξ
-integrable on on I = [a, b] × [c, d] × [e, f ] if [38]
Z Z Z
I
f (x , y, z )d ξ F xd ξ F yd ξ F z= sup
P
[a,b]×[c,d]×[e,f ]
L ξ [f , F , P ]
(14)
=
Z Z Z
I
f (x , y, z )d ξ F xd ξ F yd ξ F z= inf
P
[a,b]×[c,d]×[e,f ]
U ξ
[f , F , P ]
(15)
In that case the F
ξ
-integral of f on [a, b] × [c, d] × [e, f ]
de-noted by
R R R
I f (x , y, z )d F ξ xd ξ F yd F ξ zis given by the
com-mon value [38]
Definition 14.
If F is an ξ -perfect set then the F
ξ
-partial derivative of f respect to x is [38]
x D F ξ f (x , y, z ) =
F− lim
(x 0 ,y,z )→(x ,y,z )
f (x 0 , y, z ) − f (x , y, z )
S F ξ (x 0 , y, z ) − S F ξ (x , y, z )
if (x , y, z ) ∈ F
0 otherwise
(16)
if the limit exists Likewise the
y D F ξ f (x , y, z ) and
z D F ξ f (x , y, z ) can be defined.
3. F ξ -differential form
In this section we have generalized the F
ξ
-fractional cal-culus on fractals subset of R
3
A differential fractional 1-form on an set F subset of R
3
is
a expression H (x , y, z )d
α
F x + G (x , y, z )d β F y + N (x , y, z )d ε
F z
where H , G , N are functions on the open set If f (x , y, z )
is C
1
ξ function, then its F
ξ
-fractional total differential (or exterior derivative) is
d ξ F f (x , y, z ) = x D α
F f (x , y, z )d α
F x+ y D F β f (x , y, z )d β F y
+
z
D F ε f (x , y, z )d ε F z
ξ =α + β + ε (17)
In the same manner Eq (17) can generalized to a higher
Trang 43.2. Fξ- Fractional Exactness
Suppose that H d
α
F x + G d β F y + N d ε
F z is a F ξ
-fractional
differential on F with C
1
ξ coefficients We will say that
it is exact if one can find a C2
ξ function f (x , y, z ) with
d ξ F f = H d α
F x + G d β F y + N d ε
F z We will call a F ξ
-fractional differential closed if
x
D F ξ f = H
y
D F ξ f = G
z
D β F f = N (18)
Therefor H d
α
F x + G d β F y + N d ε
F zis exact if we have
y D β F N= z D ε
F G, x D α
F G= y D F β H, z D ε
F H= x D α
F N.
(19)
A F
ξ
-Fractional 2-forms is like a M (x , y, z )d
α
F x V d β
F y+
W (x , y, z )d
β
F y V d ε
F z + L(x , y, z )d
ε
F z V d α
F x where M, W
and L are functions And
V
wedge product of two F
ξ
-Fractional 1-forms with following properties
d α F x^d β F y = −d β F y^d α F x d α F x^d F α x= 0 (20)
So far we have applied d ξ F to functions to obtain F ξ
-Fractional 1-forms, and then to F
ξ
-Fractional 1-forms to get 2-forms, so that Eq (20) can been generalized as a
standard calculus One can define F
ξ
-Fractional gradient, divergence, and curl as follows, respectively:
grad
ξ
F f= ix D α
F f+ jy D F β f+ kz D ε
F f ξ = α + β + ε
(21) div
ξ
FV= x D α
F V x+
x D F β V y+
z D ε
F V z
(22) curl
ξ
FV = i(y D F β V z − z D ε
F V y) + j(z D ε
F V x − x D α
F V z)
+ k(
x D α
F V y − y D F β V x ). (23)
4 Maxwell’s equation on fractals
We obtain the fractional Maxwell’s equation on fractals
as follows
ω ξ F = (E1d α F x1+ E2d β F x2+ E3d ε F x3)^d κ F t + B1d β F x2
^
d ε
F x3+ B2d ε
F x3^d α
F x1+ B3d β F x1^d α
F x2.
(24)
where E i , B j , i, j = 1, 2, 3 are components of
electro-magnetic field Applying d
ξ
F that is
d ξ x D µ d µ x x D µ d µ x x D µ d µ x t D µ d µ t,
to Eq (24) and supposing α = β = ε = κ = µ. Since
d ξ F ω ξ F= 0, so Eq (24) leads { x1D F µ E2− x2D F µ E1}d µ F x1^d µ F x2^d µ F t
+ {
x
1D µ F E3− x
3D µ F E1}d µ F x1^d µ F x3^d µ F t
+ {
x
1D µ F E2− x2D µ F E3}d µ F x3^d µ F x2^d µ F t
= −
t D µ F B3d µ F x1^d µ F x2^d µ F t
− t D F µ B2d F µ x1^d µ F x3^d µ F t
− t D F µ B1d F µ x3^d µ F x2^d µ F t, (26)
and, (
x
1D F µ B1+ x
2D µ F B2+ x
3D F µ B3)d µ F x1d µ F x2d µ F x3= 0. (27)
In the vector notation it will be
curl
µ
F E = − t D µ F B (28)
div
µ
Now we define a fractional form as
π F µ=
J1d µ F x2^d µ F x3+ J2d µ F x3^d F µ x1
+ J3d µ F x1^d µ F x2 ^d µ F t − ρd µ F x1^d µ F x2^d µ F x3.
(30)
where J i , i = 1, 2, 3 are components of current and ρ
den-sity of charge Likewise applying d F µ to the left side of
Eq (30) and d
µ
F π F µ= 0 we have
x
1D µ F J1+x2D F µ J2+ x3D µ F J3+ t D µ F ρ d µ
F x1^d µ F x2
^
d µ F x3^d µ F t = 0. (31)
Furthermore, conservation of charge on fractals is
div
µ
FJ+ t D F µ ρ = 0. (32)
Consider the following fractional form
ζ F µ = A1d µ F x1+ A2d µ F x2+ A3d µ F x3+ φd µ F t, (33)
where A i , i = 1, 2, 3 is vector potential. Using d F µ ζ µ F= 0 one can obtain
curl
µ
FA = B gradµ F φ − t D F µA= E. (34)
Therefore, we arrive at the wave equation as following
c2
curl
µ
FB= t D µ F E, (35)
Trang 55 Conclusion
Fractal calculus is still an open problem In this
manuscript we have generalized F
ξ
-calculus for fractals embedding in R3 F
ξ
-calculus is a local derivative of frac-tals and has an algorithm which may be used in computer
programs F ξ
-fractional differential form is introduced to
derive the physical equation By using the F
ξ
-fractional differential form, the form of the Maxwell’s equation on
fractals has been suggested
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