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Research ArticleNew Approaches for Solving Fokker Planck Equation onCantor Sets within Local Fractional Operators
Hassan Kamil Jassim1,2
1Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran2Department of Mathematics, Faculty of Education for Pure Sciences, University of Thi-Qar, Nasiriyah, Iraq
Correspondence should be addressed to Hassan Kamil Jassim; [email protected]
Received 29 July 2015; Revised 30 September 2015; Accepted 12 October 2015
Academic Editor: Peter R. Massopust
Copyright Β© 2015 Hassan Kamil Jassim.This is an open access article distributed under theCreative CommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We discuss new approaches to handling Fokker Planck equation on Cantor sets within local fractional operators by using thelocal fractional Laplace decomposition and Laplace variational iteration methods based on the local fractional calculus. Thenew approaches maintain the efficiency and accuracy of the analytical methods for solving local fractional differential equations.Illustrative examples are given to show the accuracy and reliable results.
1. Introduction
The Fokker Planck equation arises in various fields in nat-ural science, including solid-state physics, quantum optics,chemical physics, theoretical biology, and circuit theory. TheFokker Planck equationwas first used by Fokker andPlank [1]to describe the Brownianmotion of particles. A FPEdescribesthe change of probability of a random function in space andtime; hence it is naturally used to describe solute transport.
The local fractional calculus was developed and appliedto the fractal phenomenon in science and engineering [2β13].Local fractional Fokker Planck equation,whichwas an analogof a diffusion equation with local fractional derivative, wassuggested in [5] as follows:
ππΌ
π’ (π₯, π‘)
ππ‘πΌ=
Ξ (1 + πΌ)
4ππΆ(π‘) +
π2
π’ (π₯, π‘)
ππ₯2. (1)
The Fokker Planck equation on a Cantor set with localfractional derivative was presented in [6, 7] as follows:
ππΌ
π’ (π₯, π‘)
ππ‘πΌ= β
ππΌ
π’ (π₯, π‘)
ππ₯πΌ+
π2πΌ
π’ (π₯, π‘)
ππ₯2πΌ, (2)
subject to the initial condition
π’ (π₯, 0) = π (π₯) . (3)
In recent years, a variety of numerical and analyticalmethods have been applied to solve the Fokker Planckequation on Cantor sets such as local fractional variationaliteration method [6] and local fractional Adomian decom-position method [7]. Our main purpose of the paper is toapply the local fractional Laplace decomposition methodand local fractional variational iteration method to solve theFokker Planck equations on a Cantor set. The paper has beenorganized as follows. In Section 2, the basic mathematicaltools are reviewed. In Section 3, we give analysis of themethods used. In Section 4, we consider several illustrativeexamples. Finally, in Section 5, we present our conclusions.
2. Mathematical Fundamentals
Definition 1. Setting π(π₯) β πΆπΌ(π, π), the local fractional
derivative of π(π₯) of order πΌ at the point π₯ = π₯0is defined
as [2, 3, 9β12]
ππΌ
ππ₯πΌπ (π₯)
π₯=π₯0
= π(πΌ)
(π₯) = limπ₯βπ₯0
ΞπΌ
(π (π₯) β π (π₯0))
(π₯ β π₯0)πΌ
,
0 < πΌ β€ 1,
(4)
where ΞπΌ(π(π₯) β π(π₯0)) β Ξ(πΌ + 1)(π(π₯) β π(π₯
0)).
Hindawi Publishing CorporationJournal of MathematicsVolume 2015, Article ID 684598, 8 pageshttp://dx.doi.org/10.1155/2015/684598
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2 Journal of Mathematics
Definition 2. Setting π(π₯) β πΆπΌ(π, π), local fractional integral
of π(π₯) of order πΌ in the interval [π, π] is defined through [2,3, 9β12]
ππΌ(πΌ)
ππ (π₯) =
1
Ξ (1 + πΌ)β«
π
π
π (π‘) (ππ‘)πΌ
=1
Ξ (1 + πΌ)limΞπ‘β0
πβ1
β
π=0
π (π‘π) (Ξπ‘π)πΌ
,
0 < πΌ β€ 1,
(5)
where the partitions of the interval [π, π] are denoted as(π‘π, π‘π+1
), with Ξπ‘π
= π‘π+1
β π‘π, π‘0
= π, π‘π
= π, and Ξπ‘ =max{Ξπ‘
0, Ξπ‘1, . . .}, π = 0, . . . , π β 1.
Definition 3. Let (1/Ξ(1 + πΌ)) β«β0
|π(π₯)|(ππ₯)πΌ
< π < β. TheYang-Laplace transform of π(π₯) is given by [2, 3]
πΏπΌ{π (π₯)} = π
πΏ,πΌ
π (π )
=1
Ξ (1 + πΌ)β«
β
0
πΈπΌ(βπ πΌ
π₯πΌ
) π (π₯) (ππ₯)πΌ
,
0 < πΌ β€ 1,
(6)
where the latter integral converges and π πΌ β π πΌ.
Definition 4. The inverse formula of the Yang-Laplace trans-forms of π(π₯) is given by [2, 3]
πΏβ1
πΌ{ππΏ,πΌ
π (π )} = π (π‘)
=1
(2π)πΌβ«
π½+ππ
π½βππ
πΈπΌ(π πΌ
π₯πΌ
) ππΏ,πΌ
π (π ) (ππ )
πΌ
,
0 < πΌ β€ 1,
(7)
where π πΌ = π½πΌ + ππΌππΌ, fractal imaginary unit ππΌ, and Re(π ) =π½ > 0.
3. Analytical Methods
In order to illustrate two analytical methods, we investigatethe local fractional partial differential equation as follows:
πΏπΌπ’ (π₯, π‘) + π
πΌπ’ (π₯, π‘) = π (π₯, π‘) , (8)
where πΏπΌ
= ππΌ
/ππ‘πΌ denotes the linear local fractional
differential operator,π πΌis the remaining linear operators, and
π(π₯, π‘) is a source term of the nondifferential functions.
3.1. Local Fractional Laplace Decomposition Method(LFLDM). Taking Yang-Laplace transform on (8), weobtain
πΏπΌ{πΏπΌπ’ (π₯, π‘)} + πΏ
πΌ{π πΌπ’ (π₯, π‘)} = πΏ
πΌ{π (π₯, π‘)} . (9)
By applying the local fractional Laplace transform differenti-ation property, we have
π πΌ
πΏπΌ{π’ (π₯, π‘)} β π’ (π₯, 0) + πΏ
πΌ{π πΌπ’ (π₯, π‘)}
= πΏπΌ{π (π₯, π‘)} ,
(10)
or
πΏπΌ{π’ (π₯, π‘)} =
1
π πΌπ’ (π₯, 0) +
1
π πΌπΏπΌ{π (π₯, π‘)}
β1
π πΌπΏ {π πΌ(π₯, π‘)} .
(11)
Taking the inverse of local fractional Laplace transform on(11), we obtain
π’ (π₯, π‘) = π’ (π₯, 0) + πΏβ1
πΌ(
1
π πΌπΏπΌ{π (π₯, π‘)})
β πΏβ1
πΌ(
1
π πΌπΏπΌ{π πΌπ’ (π₯, π‘)}) .
(12)
We are going to represent the solution in an infinite seriesgiven below:
π’ (π₯, π‘) =
β
β
π=0
π’π(π₯, π‘) . (13)
Substitute (13) into (12), which gives us this resultβ
β
π=0
π’π(π₯, π‘) = π’ (π₯, 0) + πΏ
β1
πΌ(
1
π πΌπΏπΌ{π (π₯, π‘)})
β πΏβ1
πΌ(
1
π πΌπΏπΌ{π πΌ
β
β
π=0
π’π(π₯, π‘)}) .
(14)
When we compare the left- and right-hand sides of (14), weobtain
π’0(π₯, π‘) = π’ (π₯, 0) + πΏ
β1
πΌ(
1
π πΌπΏπΌ{π (π₯, π‘)}) ,
π’1(π₯, π‘) = β πΏ
β1
πΌ(
1
π πΌπΏπΌ{π πΌπ’0(π₯, π‘)}) ,
π’2(π₯, π‘) = β πΏ
β1
πΌ(
1
π πΌπΏπΌ{π πΌπ’1(π₯, π‘)})
.
.
.
(15)
The recursive relation, in general form, is
π’0(π₯, π‘) = π’ (π₯, 0) + πΏ
β1
πΌ(
1
π πΌπΏπΌ{π (π₯, π‘)}) ,
π’π+1
(π₯, π‘) = β πΏβ1
πΌ(
1
π πΌπΏπΌ{π πΌπ’π(π₯, π‘)}) , π β₯ 0.
(16)
3.2. Local Fractional Laplace Variational Iteration Method(LFLDM). According to the rule of local fractional varia-tional iteration method, the correction local fractional func-tional for (8) is constructed as [13]
π’π+1
(π‘) = π’π(π‘)
+0πΌ(πΌ)
π‘(π (π‘ β π)
πΌ
Ξ (1 + πΌ)[πΏπΌπ’π(π) + π
πΌοΏ½ΜοΏ½π(π) β π (π)]) ,
(17)
where π(π‘ β π)πΌ/Ξ(1 + πΌ) is a fractal Lagrange multiplier.
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Journal of Mathematics 3
We now take Yang-Laplace transform of (17); namely,
πΏπΌ{π’π+1
(π‘)} = πΏπΌ{π’π(π‘)}
+ πΏπΌ{0πΌ(πΌ)
π‘(π (π‘ β π)
πΌ
Ξ (1 + πΌ)[πΏπΌπ’π(π) + π
πΌοΏ½ΜοΏ½π(π)
β π (π)])} ,
(18)
or
πΏπΌ{π’π+1
(π‘)}
= πΏπΌ{π’π(π‘)}
+ πΏπΌ{
π (π‘)πΌ
Ξ (1 + πΌ)} πΏπΌ{πΏπΌπ’π(π‘) + π
πΌοΏ½ΜοΏ½π(π‘) β π (π‘)} .
(19)
Take the local fractional variation of (19), which is given by
πΏπΌ
(πΏπΌ{π’π+1
(π‘)}) = πΏπΌ
(πΏπΌ{π’π(π‘)})
+ πΏπΌ
(πΏπΌ{
π (π‘)πΌ
Ξ (1 + πΌ)}
β πΏπΌ{πΏπΌπ’π(π‘) β π
πΌοΏ½ΜοΏ½π(π‘) β π (π‘)}) .
(20)
By using computation of (20), we get
πΏπΌ
(πΏπΌ{π’π+1
(π‘)})
= πΏπΌ
(πΏπΌ{π’π(π‘)})
+ πΏπΌ{
π (π‘)πΌ
Ξ (1 + πΌ)} πΏπΌ
(πΏπΌ{πΏπΌπ’π(π‘)}) = 0.
(21)
Hence, from (21) we get
1 + πΏπΌ{
π (π‘)πΌ
Ξ (1 + πΌ)} π πΌ
= 0, (22)
where
πΏπΌ
(πΏπΌ{πΏπΌπ’π(π‘)}) = πΏ
πΌ
(π πΌ
πΏπΌ{π’π(π‘)} β π’
π(0))
= π πΌ
πΏπΌ
(πΏπΌ{π’π(π‘)}) .
(23)
Therefore, we get
πΏπΌ{
π (π‘)πΌ
Ξ (1 + πΌ)} = β
1
π πΌ. (24)
Therefore, we have the following iteration algorithm:
πΏπΌ{π’π+1
(π‘)} = πΏπΌ{π’π(π‘)}
β1
π πΌπΏπΌ{πΏπΌπ’π(π‘) + π
πΌπ’π(π‘) β π (π‘)}
= πΏπΌ{π’π(π‘)} β
1
π πΌπΏπΌ{πΏπΌπ’π(π‘)}
β1
π πΌπΏπΌ{π πΌπ’π(π‘) β π (π‘)}
= πΏπΌ{π’π(π‘)} β
1
π πΌπΏπΌ{π πΌ
π’π(π‘) β π’ (0)}
β1
π πΌπΏπΌ{π πΌπ’π(π‘) β π (π‘)}
=1
π πΌπ’ (0) β
1
π πΌπΏπΌ{π πΌπ’π(π‘) β π (π‘)} ,
(25)
where the initial value reads as
πΏπΌ{π’0(π‘)} =
1
π πΌπ’ (0) . (26)
Thus, the local fractional series solution of (8) is
π’ (π₯, π‘) = limπββ
πΏβ1
πΌ(πΏπΌ{π’π(π₯, π‘)}) . (27)
4. Illustrative Examples
In this section three examples for Fokker Planck equationare presented in order to demonstrate the simplicity and theefficiency of the above methods.
Example 1. Let us consider the following Fokker Planckequation on Cantor sets with local fractional derivative in theform
ππΌ
π’ (π₯, π‘)
ππ‘πΌ= β
ππΌ
π’ (π₯, π‘)
ππ₯πΌ+
π2πΌ
π’ (π₯, π‘)
ππ₯2πΌ, (28)
subject to the initial value
π’ (π₯, 0) = πΈπΌ(βπ₯πΌ
) . (29)
(I) By Using LFLDM. In view of (16) and (28) the localfractional iteration algorithm can be written as follows:
π’0(π₯, π‘) = πΈ
πΌ(βπ₯πΌ
) ,
π’π+1
(π₯, π‘)
= πΏβ1
πΌ(
1
π πΌπΏπΌ{β
ππΌ
π’π(π₯, π‘)
ππ₯πΌ+
π2πΌ
π’π(π₯, π‘)
ππ₯2πΌ}) ,
π β₯ 0.
(30)
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4 Journal of Mathematics
Therefore, from (30) we give the components as follows:
π’0(π₯, π‘) = πΈ
πΌ(βπ₯πΌ
) ,
π’1(π₯, π‘)
= πΏβ1
πΌ(
1
π πΌπΏπΌ{β
ππΌ
π’0(π₯, π‘)
ππ₯πΌ+
π2πΌ
π’0(π₯, π‘)
ππ₯2πΌ})
= πΏβ1
πΌ(
2
π 2πΌπΈπΌ(βπ₯πΌ
)) =2π‘πΌ
Ξ (1 + πΌ)πΈπΌ(βπ₯πΌ
) ,
π’2(π₯, π‘)
= πΏβ1
πΌ(
1
π πΌπΏπΌ{β
ππΌ
π’1(π₯, π‘)
ππ₯πΌ+
π2πΌ
π’1(π₯, π‘)
ππ₯2πΌ})
= πΏβ1
πΌ(
4
π 3πΌπΈπΌ(βπ₯πΌ
)) =4π‘2πΌ
Ξ (1 + 2πΌ)πΈπΌ(βπ₯πΌ
) ,
π’3(π₯, π‘)
= πΏβ1
πΌ(
1
π πΌπΏπΌ{β
ππΌ
π’2(π₯, π‘)
ππ₯πΌ+
π2πΌ
π’2(π₯, π‘)
ππ₯2πΌ})
= πΏβ1
πΌ(
8
π 4πΌπΈπΌ(βπ₯πΌ
)) =8π‘3πΌ
Ξ (1 + 3πΌ)πΈπΌ(βπ₯πΌ
)
.
.
.
(31)
Finally, we can present the solution in local fractional seriesform as
π’ (π₯, π‘) = πΈπΌ(βπ₯πΌ
)
β (1 +2π‘πΌ
Ξ (1 + πΌ)+
4π‘2πΌ
Ξ (1 + 2πΌ)+
8π‘3πΌ
Ξ (1 + 3πΌ)+ β β β )
= πΈπΌ(βπ₯πΌ
) πΈπΌ(2π‘πΌ
) = πΈπΌ(2π‘πΌ
β π₯πΌ
) .
(32)
(II) By Using LFLVIM. Using relation (25) we structure theiterative relation as
πΏπΌ{π’π+1
(π₯, π‘)}
= πΏπΌ{π’π(π₯, π‘)}
β1
π πΌπΏπΌ{
ππΌ
π’π(π₯, π‘)
ππ‘πΌ+
ππΌ
π’π(π₯, π‘)
ππ₯πΌβ
π2πΌ
π’π(π₯, π‘)
ππ₯2πΌ}
= πΏπΌ{π’π(π₯, π‘)} β
1
π πΌ(π πΌ
πΏπΌ{π’π(π₯, π‘)} β π’
π(π₯, 0))
β1
π πΌπΏπΌ{
ππΌ
π’π(π₯, π‘)
ππ₯πΌβ
π2πΌ
π’π(π₯, π‘)
ππ₯2πΌ} ,
(33)
where the initial value is given by
πΏπΌ{π’0(π₯, π‘)} = πΏ
πΌ{πΈπΌ(βπ₯πΌ
)} =1
π πΌπΈπΌ(βπ₯πΌ
) . (34)
Therefore, the successive approximations areπΏπΌ{π’1(π₯, π‘)} = πΏ
πΌ{π’π(π₯, π‘)}
β1
π πΌ(π πΌ
πΏπΌ{π’π(π₯, π‘)} β π’
π(π₯, 0))
β1
π πΌπΏπΌ{
ππΌ
π’π(π₯, π‘)
ππ₯πΌβ
π2πΌ
π’π(π₯, π‘)
ππ₯2πΌ}
=1
π πΌπΈπΌ(βπ₯πΌ
) +2
π 2πΌπΈπΌ(βπ₯πΌ
) ,
πΏπΌ{π’2(π₯, π‘)} = πΏ
πΌ{π’1(π₯, π‘)}
β1
π πΌ(π πΌ
πΏπΌ{π’1(π₯, π‘)} β π’
1(π₯, 0))
β1
π πΌπΏπΌ{
ππΌ
π’1(π₯, π‘)
ππ₯πΌβ
π2πΌ
π’1(π₯, π‘)
ππ₯2πΌ}
=1
π πΌπΈπΌ(βπ₯πΌ
) +2
π 2πΌπΈπΌ(βπ₯πΌ
) +4
π 3πΌπΈπΌ(βπ₯πΌ
)
= πΈπΌ(βπ₯πΌ
) (1
π πΌ+
2
π 2πΌ+
4
π 3πΌ) ,
πΏπΌ{π’3(π₯, π‘)} = πΏ
πΌ{π’2(π₯, π‘)}
β1
π πΌ(π πΌ
πΏπΌ{π’2(π₯, π‘)} β π’
2(π₯, 0))
β1
π πΌπΏπΌ{
ππΌ
π’2(π₯, π‘)
ππ₯πΌβ
π2πΌ
π’2(π₯, π‘)
ππ₯2πΌ}
=1
π πΌπΈπΌ(βπ₯πΌ
) +2
π 2πΌπΈπΌ(βπ₯πΌ
) +4
π 3πΌπΈπΌ(βπ₯πΌ
)
+8
π 4πΌπΈπΌ(βπ₯πΌ
)
= πΈπΌ(βπ₯πΌ
) (1
π πΌ+
2
π 2πΌ+
4
π 3πΌ+
8
π 4πΌ)
.
.
.
(35)
Hence, the local fractional series solution isπ’ (π₯, π‘) = lim
πββ
πΏβ1
πΌ(πΏπΌ{π’π(π₯, π‘)})
= limπββ
πΏβ1
πΌ(πΈπΌ(βπ₯πΌ
)
π
β
π=0
2π
π (π+1)πΌ)
= πΈπΌ(βπ₯πΌ
)
β
β
π=0
2π
π‘ππΌ
Ξ (1 + ππΌ)= πΈπΌ(2π‘πΌ
β π₯πΌ
) .
(36)
Example 2. We present the Fokker Planck equation on aCantor set with local fractional derivative
ππΌ
π’ (π₯, π‘)
ππ‘πΌ= β
ππΌ
π’ (π₯, π‘)
ππ₯πΌ+
π2πΌ
π’ (π₯, π‘)
ππ₯2πΌ, (37)
and the initial condition is
π’ (π₯, 0) =π₯2πΌ
Ξ (1 + 2πΌ). (38)
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Journal of Mathematics 5
(I) By Using LFLDM. In view of (16) and (37) the localfractional iteration algorithm can be written as follows:
π’0(π₯, π‘) =
π₯2πΌ
Ξ (1 + 2πΌ),
π’π+1
(π₯, π‘)
= πΏβ1
πΌ(
1
π πΌπΏπΌ{β
ππΌ
π’π(π₯, π‘)
ππ₯πΌ+
π2πΌ
π’π(π₯, π‘)
ππ₯2πΌ}) ,
π β₯ 0.
(39)
Therefore, from (39) we give the components as follows:
π’0(π₯, π‘) =
π₯2πΌ
Ξ (1 + 2πΌ),
π’1(π₯, π‘)
= πΏβ1
πΌ(
1
π πΌπΏπΌ{β
ππΌ
π’0(π₯, π‘)
ππ₯πΌ+
π2πΌ
π’0(π₯, π‘)
ππ₯2πΌ})
= πΏβ1
πΌ(β
1
π 2πΌ
π₯πΌ
Ξ (1 + πΌ)+
1
π 2πΌ)
= βπ‘πΌ
Ξ (1 + πΌ)
π₯πΌ
Ξ (1 + πΌ)+
π‘πΌ
Ξ (1 + πΌ),
π’2(π₯, π‘)
= πΏβ1
πΌ(
1
π πΌπΏπΌ{β
ππΌ
π’1(π₯, π‘)
ππ₯πΌ+
π2πΌ
π’1(π₯, π‘)
ππ₯2πΌ})
= πΏβ1
πΌ(
1
π 3πΌ) =
π‘2πΌ
Ξ (1 + 2πΌ),
π’3(π₯, π‘)
= πΏβ1
πΌ(
1
π πΌπΏπΌ{β
ππΌ
π’2(π₯, π‘)
ππ₯πΌ+
π2πΌ
π’2(π₯, π‘)
ππ₯2πΌ})
= 0,
π’4(π₯, π‘)
= πΏβ1
πΌ(
1
π πΌπΏπΌ{β
ππΌ
π’3(π₯, π‘)
ππ₯πΌ+
π2πΌ
π’3(π₯, π‘)
ππ₯2πΌ}) = 0
.
.
.
(40)
Finally, we can present the solution in local fractional seriesform as
π’ (π₯, π‘) =π₯2πΌ
Ξ (1 + 2πΌ)+
π‘2πΌ
Ξ (1 + 2πΌ)
βπ₯πΌ
Ξ (1 + πΌ)
π‘πΌ
Ξ (1 + πΌ)+
π‘πΌ
Ξ (1 + πΌ).
(41)
(II) By Using LFLVIM. Using relation (25) we structure theiterative relation as
πΏπΌ{π’π+1
(π₯, π‘)} = πΏπΌ{π’π(π₯, π‘)}
β1
π πΌπΏπΌ{
ππΌ
π’π(π₯, π‘)
ππ‘πΌ+
ππΌ
π’π(π₯, π‘)
ππ₯πΌβ
π2πΌ
π’π(π₯, π‘)
ππ₯2πΌ}
= πΏπΌ{π’π(π₯, π‘)} β
1
π πΌ(π πΌ
πΏπΌ{π’π(π₯, π‘)} β π’
π(π₯, 0))
β1
π πΌπΏπΌ{
ππΌ
π’π(π₯, π‘)
ππ₯πΌβ
π2πΌ
π’π(π₯, π‘)
ππ₯2πΌ} ,
(42)
where the initial value is given by
πΏπΌ{π’0(π₯, π‘)} = πΏ
πΌ{
π₯2πΌ
Ξ (1 + 2πΌ)} =
1
π πΌ
π₯2πΌ
Ξ (1 + 2πΌ). (43)
Therefore, the successive approximations are
πΏπΌ{π’1(π₯, π‘)} = πΏ
πΌ{π’0(π₯, π‘)}
β1
π πΌ(π πΌ
πΏπΌ{π’0(π₯, π‘)} β π’
0(π₯, 0))
β1
π πΌπΏπΌ{
ππΌ
π’0(π₯, π‘)
ππ₯πΌβ
π2πΌ
π’0(π₯, π‘)
ππ₯2πΌ}
=1
π πΌ
π₯2πΌ
Ξ (1 + 2πΌ)β
1
π 2πΌ
π₯πΌ
Ξ (1 + πΌ)+
1
π 2πΌ,
πΏπΌ{π’2(π₯, π‘)} = πΏ
πΌ{π’1(π₯, π‘)}
β1
π πΌ(π πΌ
πΏπΌ{π’1(π₯, π‘)} β π’
1(π₯, 0))
β1
π πΌπΏπΌ{
ππΌ
π’1(π₯, π‘)
ππ₯πΌβ
π2πΌ
π’1(π₯, π‘)
ππ₯2πΌ}
=1
π πΌ
π₯2πΌ
Ξ (1 + 2πΌ)β
1
π 2πΌ
π₯πΌ
Ξ (1 + πΌ)+
1
π 3πΌ+
1
π 2πΌ,
πΏπΌ{π’3(π₯, π‘)} = πΏ
πΌ{π’2(π₯, π‘)}
β1
π πΌ(π πΌ
πΏπΌ{π’2(π₯, π‘)} β π’
2(π₯, 0))
β1
π πΌπΏπΌ{
ππΌ
π’2(π₯, π‘)
ππ₯πΌβ
π2πΌ
π’2(π₯, π‘)
ππ₯2πΌ}
=1
π πΌ
π₯2πΌ
Ξ (1 + 2πΌ)β
1
π 2πΌ
π₯πΌ
Ξ (1 + πΌ)+
1
π 3πΌ+
1
π 2πΌ
.
.
.
(44)
-
6 Journal of Mathematics
Hence, the local fractional series solution is
π’ (π₯, π‘) = limπββ
πΏβ1
πΌ(πΏπΌ{π’π(π₯, π‘)})
= limπββ
πΏβ1
πΌ(
1
π πΌ
π₯2πΌ
Ξ (1 + 2πΌ)+
1
π 3πΌβ
1
π 2πΌ
π₯πΌ
Ξ (1 + πΌ)
+1
π 2πΌ) =
π₯2πΌ
Ξ (1 + 2πΌ)+
π‘2πΌ
Ξ (1 + 2πΌ)β
π₯πΌ
Ξ (1 + πΌ)
β π‘πΌ
Ξ (1 + πΌ)+
π‘πΌ
Ξ (1 + πΌ).
(45)
Example 3. We consider the Fokker Planck equation on aCantor set with local fractional derivative
ππΌ
π’ (π₯, π‘)
ππ‘πΌ= β
ππΌ
π’ (π₯, π‘)
ππ₯πΌ+
π2πΌ
π’ (π₯, π‘)
ππ₯2πΌ, (46)
and the initial condition is
π’ (π₯, 0) = βπ₯3πΌ
Ξ (1 + 3πΌ). (47)
(I) By Using LFLDM. In view of (16) and (46) the localfractional iteration algorithm can be written as follows:
π’0(π₯, π‘) = β
π₯3πΌ
Ξ (1 + 3πΌ),
π’π+1
(π₯, π‘)
= πΏβ1
πΌ(
1
π πΌπΏπΌ{β
ππΌ
π’π(π₯, π‘)
ππ₯πΌ+
π2πΌ
π’π(π₯, π‘)
ππ₯2πΌ}) ,
π β₯ 0.
(48)
Therefore, from (30) we give the components as follows:
π’0(π₯, π‘) = β
π₯3πΌ
Ξ (1 + 3πΌ),
π’1(π₯, π‘)
= πΏβ1
πΌ(
1
π πΌπΏπΌ{β
ππΌ
π’0(π₯, π‘)
ππ₯πΌ+
π2πΌ
π’0(π₯, π‘)
ππ₯2πΌ})
= πΏβ1
πΌ(
1
π 2πΌ
π₯2πΌ
Ξ (1 + 2πΌ)β
1
π 2πΌ
π₯πΌ
Ξ (1 + πΌ))
=π₯2πΌ
Ξ (1 + 2πΌ)
π‘πΌ
Ξ (1 + πΌ)β
π₯πΌ
Ξ (1 + πΌ)
π‘πΌ
Ξ (1 + πΌ),
π’2(π₯, π‘)
= πΏβ1
πΌ(
1
π πΌπΏπΌ{β
ππΌ
π’1(π₯, π‘)
ππ₯πΌ+
π2πΌ
π’1(π₯, π‘)
ππ₯2πΌ})
= πΏβ1
πΌ(β
1
π 3πΌ
π₯πΌ
Ξ (1 + πΌ)+
2
π 3πΌ)
= βπ₯πΌ
Ξ (1 + πΌ)
π‘2πΌ
Ξ (1 + 2πΌ)+
2π‘2πΌ
Ξ (1 + 2πΌ),
π’3(π₯, π‘)
= πΏβ1
πΌ(
1
π πΌπΏπΌ{β
ππΌ
π’2(π₯, π‘)
ππ₯πΌ+
π2πΌ
π’2(π₯, π‘)
ππ₯2πΌ})
= πΏβ1
πΌ(
1
π 4πΌ) =
π‘3πΌ
Ξ (1 + 3πΌ),
π’4(π₯, π‘)
= πΏβ1
πΌ(
1
π πΌπΏπΌ{β
ππΌ
π’3(π₯, π‘)
ππ₯πΌ+
π2πΌ
π’3(π₯, π‘)
ππ₯2πΌ}) = 0
.
.
.
(49)
Finally, we can present the solution in local fractional seriesform as
π’ (π₯, π‘) = βπ₯3πΌ
Ξ (1 + 3πΌ)+
π₯2πΌ
Ξ (1 + 2πΌ)
π‘πΌ
Ξ (1 + πΌ)
βπ₯πΌ
Ξ (1 + πΌ)
π‘πΌ
Ξ (1 + πΌ)
βπ₯πΌ
Ξ (1 + πΌ)
π‘2πΌ
Ξ (1 + 2πΌ)+
2π‘2πΌ
Ξ (1 + 2πΌ)
+π‘3πΌ
Ξ (1 + 3πΌ).
(50)
(II) By Using LFLVIM. Using relation (18) we structure theiterative relation as
πΏπΌ{π’π+1
(π₯, π‘)}
= πΏπΌ{π’π(π₯, π‘)}
β1
π πΌπΏπΌ{
ππΌ
π’π(π₯, π‘)
ππ‘πΌ+
ππΌ
π’π(π₯, π‘)
ππ₯πΌβ
π2πΌ
π’π(π₯, π‘)
ππ₯2πΌ}
= πΏπΌ{π’π(π₯, π‘)} β
1
π πΌ(π πΌ
πΏπΌ{π’π(π₯, π‘)} β π’
π(π₯, 0))
β1
π πΌπΏπΌ{
ππΌ
π’π(π₯, π‘)
ππ₯πΌβ
π2πΌ
π’π(π₯, π‘)
ππ₯2πΌ} ,
(51)
where the initial value is given by
πΏπΌ{π’0(π₯, π‘)} = πΏ
πΌ{β
π₯3πΌ
Ξ (1 + 3πΌ)} = β
1
π πΌ
π₯3πΌ
Ξ (1 + 3πΌ). (52)
Therefore, the successive approximations are
πΏπΌ{π’1(π₯, π‘)} = πΏ
πΌ{π’0(π₯, π‘)}
β1
π πΌ(π πΌ
πΏπΌ{π’0(π₯, π‘)} β π’
π(π₯, 0))
β1
π πΌπΏπΌ{
ππΌ
π’0(π₯, π‘)
ππ₯πΌβ
π2πΌ
π’0(π₯, π‘)
ππ₯2πΌ}
-
Journal of Mathematics 7
= β1
π πΌ
π₯3πΌ
Ξ (1 + 3πΌ)+
1
π 2πΌ
π₯2πΌ
Ξ (1 + 2πΌ)β
1
π 2πΌ
π₯πΌ
Ξ (1 + πΌ),
πΏπΌ{π’2(π₯, π‘)} = πΏ
πΌ{π’1(π₯, π‘)}
β1
π πΌ(π πΌ
πΏπΌ{π’1(π₯, π‘)} β π’
1(π₯, 0))
β1
π πΌπΏπΌ{
ππΌ
π’1(π₯, π‘)
ππ₯πΌβ
π2πΌ
π’1(π₯, π‘)
ππ₯2πΌ}
= β1
π πΌ
π₯3πΌ
Ξ (1 + 3πΌ)+
1
π 2πΌ
π₯2πΌ
Ξ (1 + 2πΌ)β
1
π 2πΌ
π₯πΌ
Ξ (1 + πΌ)
β1
π 3πΌ
π₯πΌ
Ξ (1 + 3πΌ)+
2
π 3πΌ,
πΏπΌ{π’3(π₯, π‘)} = πΏ
πΌ{π’2(π₯, π‘)}
β1
π πΌ(π πΌ
πΏπΌ{π’2(π₯, π‘)} β π’
2(π₯, 0))
β1
π πΌπΏπΌ{
ππΌ
π’2(π₯, π‘)
ππ₯πΌβ
π2πΌ
π’2(π₯, π‘)
ππ₯2πΌ}
= β1
π πΌ
π₯3πΌ
Ξ (1 + 3πΌ)+
1
π 2πΌ
π₯2πΌ
Ξ (1 + 2πΌ)β
1
π 2πΌ
π₯πΌ
Ξ (1 + πΌ)
β1
π 3πΌ
π₯πΌ
Ξ (1 + 3πΌ)+
2
π 3πΌ+
1
π 4πΌ,
πΏπΌ{π’4(π₯, π‘)} = πΏ
πΌ{π’3(π₯, π‘)}
β1
π πΌ(π πΌ
πΏπΌ{π’3(π₯, π‘)} β π’
3(π₯, 0))
β1
π πΌπΏπΌ{
ππΌ
π’3(π₯, π‘)
ππ₯πΌβ
π2πΌ
π’3(π₯, π‘)
ππ₯2πΌ}
= β1
π πΌ
π₯3πΌ
Ξ (1 + 3πΌ)+
1
π 2πΌ
π₯2πΌ
Ξ (1 + 2πΌ)β
1
π 2πΌ
π₯πΌ
Ξ (1 + πΌ)
β1
π 3πΌ
π₯πΌ
Ξ (1 + 3πΌ)+
2
π 3πΌ+
1
π 4πΌ
.
.
.
(53)Hence, the local fractional series solution is
π’ (π₯, π‘) = limπββ
πΏβ1
πΌ(πΏπΌ{π’π(π₯, π‘)})
= βπ₯3πΌ
Ξ (1 + 3πΌ)+
π₯2πΌ
Ξ (1 + 2πΌ)
π‘πΌ
Ξ (1 + πΌ)
βπ₯πΌ
Ξ (1 + πΌ)
π‘πΌ
Ξ (1 + πΌ)
βπ₯πΌ
Ξ (1 + πΌ)
π‘2πΌ
Ξ (1 + 2πΌ)+
2π‘2πΌ
Ξ (1 + 2πΌ)
+π‘3πΌ
Ξ (1 + 3πΌ).
(54)
5. Conclusions
In this work solving Fokker Planck equation by using thelocal fractional Laplace decomposition method and localfractional Laplace variational iteration method with localfractional operators is discussed in detail. Three examples ofapplications of these methods are investigated. The reliableobtained results are complementary to the ones presented inthe literature.
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper.
Acknowledgment
Hassan Kamil Jassim acknowledges Ministry of Higher Edu-cation and Scientific Research in Iraq for its support to thiswork.
References
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