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Fractional Calculus
Student: Manal AL-Ali
Dr. Abdalla Obeidat
Designation
Designation means integration and differentiation of arbitraryorder, In other expressions it means dealing with operators like
, , is arbitrary real or Complex value.
Many definitions are proposed to find the fractional derivatives
and integrals , the most common one is Riemann – Liouville Definition.
υ
υ
dx
dυ
υ
−
−
dx
d υ
Riemann Riemann Riemann Riemann –Liouville DefinitionLiouville DefinitionLiouville DefinitionLiouville Definition....
Riemann – Liouville Definition is given by the following
equation :
⟨−−Γ
≤≤−⟩
=
∫ −−
−
x
a
m
xa
m
xa
duufux
nnxfIdx
d
xfD
0Re.....)()()(
1
Re1....0Re).......()(
)(1 υ
υ
υυ
υ
υ
υ (1)
When a = -∞, eq. (1) is equivalent to Riemann's definition,
and when a = 0, we have Liouville's definition.
Example:
evaluation of the fractional derivative for the functionf(x) = xb, b >-1.
We have for m-1 ≤ ≤ m , m € N
For b=0,
*
You can see that the fractional derivative of constants is notzero!
∫ −−−
−
+−Γ
+Γ=−
−Γ=
=
x
bbmmb
x
m
x
m
x
xb
bduuux
mdx
dxD
xfIdx
dxfD
0
1
0
00
)1(
)1()(
)(
1{)(
)()()(
υυυ
υυ
υυ
)1(0
α
αυ
−Γ=
−cx
cD x
(2)
υ
MittagMittagMittagMittag----LefflerLefflerLefflerLeffler FunctionFunctionFunctionFunctionMittag-Leffler function of one parameter is denoted by
∑∞
= +Γ=
0 )1()(
k
k
k
zzE
αα (3)
A two- parameter Mittag-Leffler function is defined by the series
expansion
∑∞
= +Γ=
0
,)(
)(k
k
k
zzE
βαβα , (α > 0, β > 0) (4)
Relation to some other function
It is follows from the defection that :
∑∞
=
=+Γ
=0
1,1)1(
)(k
zk
ek
zzE ∑
∞
=
−=
+Γ=
0
2,1
1
)2()(
k
zk
z
e
k
zzE
∑∞
=
=+Γ
=0
22
1,2 )cosh()12(
)(k
k
zk
zzE
∑∞
=
=+Γ
=0
22
2,2
)sinh(
)22()(
k
k
z
z
k
zzE
∑∞
=
=+Γ
−=−
0
22
1,2 )cos()12(
)1()(
k
kk
zk
zzE
∑∞
=
=+Γ
−=−
0
22
2,2
)sin(
)22(
)1()(
k
kk
z
z
k
zzE
Laplace transformLaplace transformLaplace transformLaplace transformLaplace transform of fractional differential operator is given by
∑−
==
−−−=ℑ1
0
0
1
00 )]([)(});({n
k
t
kp
t
kpp
t tfDssFsstfD where n-1<p<n. (5)
Laplace transform of Mittag-Leffler function
∫∞
+
−−+− =±
0 1
)(
,
1
)(
!))((
k
kkpt
ap
pkdtatEte
∓α
βαα
βαβα
,(Re (p) > ) , (6)α
1
a
Fractionalization of physical problem
Fractionalization of physical problem means bringing the
tools of fractional derivatives / fractional integrals into the
theory of the problem by fractionalization of some appropriate operators. Then search for physical meanings.
The question now is how we can choose those operators ???
The problem fractional multipoles
the general problem of
electrostatic potential
of a static electric chargedistribution in free space
the general problem of
electrostatic potential
of a static electric chargedistribution in free space
point monopole
Case 2Case 1
r
1αΦ
point dipole Intermediate cases
2
1
rαΦ
υL
L
υα
r
1Φ
0
2/ ερ−=Φ∇Poisson Equation
is the volume charge density of the source ρ
)(0 rq�
δρ = )(.1 rp���
δρ ∇−=Case 1 Case 2
zq
pL
∂
∂−= )(
Then the fractional operator will be υ
υυυ
zq
pL
∂
∂= )(
The intermediate “fractional” source can be defined by the
following equation: υρ
))((0 rqz
lL�
δρρυ
υυυ
υ∂
∂== (7)
By the help of Riemann – Liouville definition of fractional
operator eq.(7) can be written as
})()()()()2(
1{)(
1
2
2
∫∞−
−−−Γ
=z
duyxuuzdx
d
q
pq δδδ
υρ υυ
υ (8)
−Γ= −υυ
υυ
δδρ 1
2
2
)2(
1)()()( zzUyx
dz
dql (9)
The scalar potential of this factional source is found to be
R
qDlLzyx z
πεψψ υυυ
υ4
),,( 0 ∞−==
)cos(4
)1(1
0
θπε
υυυ
υ
−+Γ
=+
PR
ql
Figure.2. The Fractional potential
(10)
Fractional problems using MittagFractional problems using MittagFractional problems using MittagFractional problems using Mittag----Leffler functionLeffler functionLeffler functionLeffler function
Now we list here some solved fractional physical problemsusing series solution. and then rewrite the solutions of these
problems in terms of Mittag-Leffler function.
1) The fractional LC-RC circuit
A.A.Rousan, N.Y.Ayoub, etal suggested a fractional differential
equation that combines the simple harmonic oscillation of anLC circuit with discharging of an RC circuit.
0)(1
1
1
=+ +
+
+
Qdt
Qd α
α
α
αω where 0 ≤ α ≤ 1
ω(α) = ω1αω2
1-α(11)
When α = 0, equation (11) goes over to the equation of RC circuit
0)()(
2 =+ tQdt
tdQω where ω2 = 1/RC (12)
and, when α = 1, equation (11) become the LC circuit equation
0)()(
12
2
=+ tQdt
tQdω where ω1
2 = 1/LC (13)
Apply laplace transform to eq.11,and use the following initial
conditions
00 )())(
( Qdt
tQdt
α
α
α
αω== 0))(
( 01
1
==−
−
tdt
tQdα
α
We get
αα
α
αω
αω++ +
=ℑ110
)(
)()(
sQtQ
(14)
compare eq. (14) by eq. (6), we get
))(()()(1
1,10
ααα
α ωω +++ −= tEtQtQ (15)
Now when α = 0
And when α = 1
teQtEQtQ 2
021,10 )()(ωω −=−= Solution of eq.12
)sin())(()()( 10
2
2,20 tQtiEtQtQ ωωω ==
)cos()cos()(
)( 10110 tItQdt
tdQtI ωωω === Solution of eq.13
2) Fractional Simple Harmonic Oscillator
The equation of motion that covers a dynamic system is given
by the following equation:
0)(
2
2
=++ kxdt
dxb
dt
txdm (16)
Akram. A. Rousan, Nabil.Y.Ayoub and Khetam Khasawinah
suggested a modified equation of eq. (16) using fractionalized second term as follows:
0)(
2
2
=++ kxdt
dxb
dt
txdm
α
α
α (17)
Define the damping ratio αωη
−=
22m
band
m
k=2ω
Then take laplace transform of eq.17
0)(])(
[2])(
[22
2
2
=ℑ+ℑ+ℑ −tx
dt
txd
dt
txdωαηω
α
αα
0)(}][)({2])(
[)]([)(2
01
12
00
2 =+−+−− =−
−−
== txdt
xdsxs
dt
tdxtxssxs ttt ωαηω
α
ααα
Use the initial conditions
0][ 0 ==tdt
dx0][ 0 ==t
dt
xdα
α
cdt
xdt ==−
−
01
1
][α
α
, ,
We get
0)(})({2)(22
0
2 =+−+− −txcsxssxsxs ωαηω αα
(18)
Rearranging the equation, one can write it as follows
21222
2
0
2
2)( xx
ss
csxsx +=
++
+=
−
−
ωαηω
αηωαα
α
Let
222
01
2)(
ωαηω αα ++=
− ss
sxsx
222
2
22
2)(
ωαηω
αηωαα
α
++=
−
−
ss
csx
One can rewrite x1 and x2 as follows
∑∞
=+−−
−−
+−=
0122
12
01)2(
)1()(p
p
ppp
s
sxsx
αα
αα
αηω
ω
∑∞
=+−−
−−−
+−=
0122
22
2)2(
)1(2)(p
p
ppp
s
scsx
αα
ααα
αηω
ωαηω
(19)
(20)
Comparing eq.19 and 20 by eq(6) , we get
∑∞
=
−+− −
−=
0
2
1,2
2
01))(2()(
!
)1()(
p
p
p
pp
tEtp
xtx ααα ωαηω
∑∞
=
−+−
− −−
=0
2
2,2
22
2))(2()(
!
)1(2)(
p
p
p
pp
ttEtp
ctx ααα
α ωαηωαηω
Now x(t) = x1 + x2
∑∞
=+−
−+− +
−=
0
2,2
2
1,20
2)}(2)({)(
!
)1()(
p
p
p
p
p
pp
ztEzExtp
tx ααα
αα εηωω (21)
Where and c taken to be αωαη −−= 2
)(2 tz1
0
−αωx
a) Simple harmonic oscillator ( )0=α)cos()()( 01,20 txtExtx ωω ==
b) Damped oscillator( ) 1=α
∑∞
=++ −+−
−=
0
2,11,1
2
0 )}2(2)2({)(!
)1()(
p
p
p
p
p
pp
ttEtEtp
xtx µωηωηωω
3) Fractional Domain Wall MotionWesam Al–Sharo’a employed the idea of fractionalization of the second term in the equation of motion suggested in the problem
of simple harmonic oscillator to study domain wall motionfor two cases: 1)
2)
)()( ttf δ=
0)( fconstf ==
The fractional domain wall motion equation is:
)()()(
2)( 22
2
2
tftxdt
txd
dt
txd=++ − ωαηω
α
αα
(22)
Use the same initial conditions in S.H.O problem, and follow the
Same procedure, one get :
a) Case1: )()( ttf δ=
∑∞
=+−+− ++
−=
0
2,201,20
2)}()12()({)(
!
)1()(
p
p
p
p
p
pp
ztExzExtp
tx αααα αηωω (23)
1) For α = 0 and f (t) = 0 eq.(23) reduced to
∑ ∑∞
=
∞
=
=+Γ
−=
−=
0 0
0
2
01,2
2
0 )cos()12(
!)(
!
)1()0()(
!
)1()(
p p
pp
ppp
txp
pt
pxEt
pxtx ωωω
(24)
2) For α = 1
∑∞
=++ −+−
−=
0
2,11,1
2
0 )}2(2)2({)(!
)1()(
p
p
p
p
p
pp
ttEtEtp
xtx µωηωηωω
αωαη −−= 2)(2 tz and
1
0
−= αωxc
(25)
2) Case 2 : 0)( fconstf ==
∑∞
=+−+−+− ++
−=
0
3,2
2
02,211,20
2)}()()({)(
!
)1()(
p
p
p
p
p
p
p
pp
zEtfztExzExtp
tx ααααααω (26)
Where: αωαη −−= 2
)(2 tz and αηω21 =x
1) For α = 0 eq.(25) reduced to
)cos()cos()(2
0
2
0
0 tff
txtx ωωω
ω −+=
2) For α = 1
∑∞
=+++
′+′′+′
−=
0
3,1
2
02,111,10
2)}()()({)(
!
)1()(
p
p
p
p
p
p
p
pp
zEtfztExzExtp
tx ω
(27)
(28)
)(2 tz ωη−=′Where: and ηω21 =′
x
4- Conclusion
We were able to rewrite some solved problems from JUST
which are published in scientific journals in terms of Mettag-Leffler function, which is a demand from most prestigious
journals.
5. Reference:
[1] I. Podlubny, Fractional Differential Equations, Volume 198,ACADEMIC PRESS, 1999.
[2] R.Hilfer, application of fractional calculus in physics, WORLD SCIENTIFIC, 2000.
[3] D.H.Werner and R.Mittra, Frontiers in Electromagnetic, IEEE Press, chapter 12.
[4] A.Rousan and N.Ayoub, A Fractional LC-RC Circuit, An International Journal for
Theory and Applications.
[5] A.Rousan, N.Ayoub and K.Khasawenah, Fractional Simple Harmonic Oscillator,
International Journal of Theoretical Physics.
.[6] Wesam Al–Sharo’a, Fractional Domain Wall Motion,