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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 © 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 [2–13].Local
fractional Fokker Planck equation,whichwas an analogof a diffusion
equation with local fractional derivative, wassuggested in [5] as
follows:
𝜕𝛼
𝑢 (𝑥, 𝑡)
𝜕𝑡𝛼=
Γ (1 + 𝛼)
4𝜒𝐶(𝑡) +
𝜕2
𝑢 (𝑥, 𝑡)
𝜕𝑥2. (1)
The Fokker Planck equation on a Cantor set with localfractional
derivative was presented in [6, 7] as follows:
𝜕𝛼
𝑢 (𝑥, 𝑡)
𝜕𝑡𝛼= −
𝜕𝛼
𝑢 (𝑥, 𝑡)
𝜕𝑥𝛼+
𝜕2𝛼
𝑢 (𝑥, 𝑡)
𝜕𝑥2𝛼, (2)
subject to the initial condition
𝑢 (𝑥, 0) = 𝑓 (𝑥) . (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 𝑓(𝑥) ∈ 𝐶𝛼(𝑎, 𝑏), the local fractional
derivative of 𝑓(𝑥) of order 𝛼 at the point 𝑥 = 𝑥0is defined
as [2, 3, 9–12]
𝑑𝛼
𝑑𝑥𝛼𝑓 (𝑥)
𝑥=𝑥0
= 𝑓(𝛼)
(𝑥) = lim𝑥→𝑥0
Δ𝛼
(𝑓 (𝑥) − 𝑓 (𝑥0))
(𝑥 − 𝑥0)𝛼
,
0 < 𝛼 ≤ 1,
(4)
where Δ𝛼(𝑓(𝑥) − 𝑓(𝑥0)) ≅ Γ(𝛼 + 1)(𝑓(𝑥) − 𝑓(𝑥
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 𝑓(𝑥) ∈ 𝐶𝛼(𝑎, 𝑏), local fractional
integral
of 𝑓(𝑥) of order 𝛼 in the interval [𝑎, 𝑏] is defined through
[2,3, 9–12]
𝑎𝐼(𝛼)
𝑏𝑓 (𝑥) =
1
Γ (1 + 𝛼)∫
𝑏
𝑎
𝑓 (𝑡) (𝑑𝑡)𝛼
=1
Γ (1 + 𝛼)limΔ𝑡→0
𝑁−1
∑
𝑗=0
𝑓 (𝑡𝑗) (Δ𝑡𝑗)𝛼
,
0 < 𝛼 ≤ 1,
(5)
where the partitions of the interval [𝑎, 𝑏] are denoted as(𝑡𝑗,
𝑡𝑗+1
), with Δ𝑡𝑗
= 𝑡𝑗+1
− 𝑡𝑗, 𝑡0
= 𝑎, 𝑡𝑁
= 𝑏, and Δ𝑡 =max{Δ𝑡
0, Δ𝑡1, . . .}, 𝑗 = 0, . . . , 𝑁 − 1.
Definition 3. Let (1/Γ(1 + 𝛼)) ∫∞0
|𝑓(𝑥)|(𝑑𝑥)𝛼
< 𝑘 < ∞. TheYang-Laplace transform of 𝑓(𝑥) is given by [2,
3]
𝐿𝛼{𝑓 (𝑥)} = 𝑓
𝐿,𝛼
𝑠(𝑠)
=1
Γ (1 + 𝛼)∫
∞
0
𝐸𝛼(−𝑠𝛼
𝑥𝛼
) 𝑓 (𝑥) (𝑑𝑥)𝛼
,
0 < 𝛼 ≤ 1,
(6)
where the latter integral converges and 𝑠𝛼 ∈ 𝑅𝛼.
Definition 4. The inverse formula of the Yang-Laplace
trans-forms of 𝑓(𝑥) is given by [2, 3]
𝐿−1
𝛼{𝑓𝐿,𝛼
𝑠(𝑠)} = 𝑓 (𝑡)
=1
(2𝜋)𝛼∫
𝛽+𝑖𝜔
𝛽−𝑖𝜔
𝐸𝛼(𝑠𝛼
𝑥𝛼
) 𝑓𝐿,𝛼
𝑠(𝑠) (𝑑𝑠)
𝛼
,
0 < 𝛼 ≤ 1,
(7)
where 𝑠𝛼 = 𝛽𝛼 + 𝑖𝛼𝜔𝛼, fractal imaginary unit 𝑖𝛼, and Re(𝑠) =𝛽
> 0.
3. Analytical Methods
In order to illustrate two analytical methods, we investigatethe
local fractional partial differential equation as follows:
𝐿𝛼𝑢 (𝑥, 𝑡) + 𝑅
𝛼𝑢 (𝑥, 𝑡) = 𝑔 (𝑥, 𝑡) , (8)
where 𝐿𝛼
= 𝜕𝛼
/𝜕𝑡𝛼 denotes the linear local fractional
differential operator,𝑅𝛼is the remaining linear operators,
and
𝑔(𝑥, 𝑡) is a source term of the nondifferential functions.
3.1. Local Fractional Laplace Decomposition Method(LFLDM).
Taking Yang-Laplace transform on (8), weobtain
𝐿𝛼{𝐿𝛼𝑢 (𝑥, 𝑡)} + 𝐿
𝛼{𝑅𝛼𝑢 (𝑥, 𝑡)} = 𝐿
𝛼{𝑔 (𝑥, 𝑡)} . (9)
By applying the local fractional Laplace transform
differenti-ation property, we have
𝑠𝛼
𝐿𝛼{𝑢 (𝑥, 𝑡)} − 𝑢 (𝑥, 0) + 𝐿
𝛼{𝑅𝛼𝑢 (𝑥, 𝑡)}
= 𝐿𝛼{𝑔 (𝑥, 𝑡)} ,
(10)
or
𝐿𝛼{𝑢 (𝑥, 𝑡)} =
1
𝑠𝛼𝑢 (𝑥, 0) +
1
𝑠𝛼𝐿𝛼{𝑔 (𝑥, 𝑡)}
−1
𝑠𝛼𝐿 {𝑅𝛼(𝑥, 𝑡)} .
(11)
Taking the inverse of local fractional Laplace transform on(11),
we obtain
𝑢 (𝑥, 𝑡) = 𝑢 (𝑥, 0) + 𝐿−1
𝛼(
1
𝑠𝛼𝐿𝛼{𝑔 (𝑥, 𝑡)})
− 𝐿−1
𝛼(
1
𝑠𝛼𝐿𝛼{𝑅𝛼𝑢 (𝑥, 𝑡)}) .
(12)
We are going to represent the solution in an infinite
seriesgiven below:
𝑢 (𝑥, 𝑡) =
∞
∑
𝑛=0
𝑢𝑛(𝑥, 𝑡) . (13)
Substitute (13) into (12), which gives us this result∞
∑
𝑛=0
𝑢𝑛(𝑥, 𝑡) = 𝑢 (𝑥, 0) + 𝐿
−1
𝛼(
1
𝑠𝛼𝐿𝛼{𝑔 (𝑥, 𝑡)})
− 𝐿−1
𝛼(
1
𝑠𝛼𝐿𝛼{𝑅𝛼
∞
∑
𝑛=0
𝑢𝑛(𝑥, 𝑡)}) .
(14)
When we compare the left- and right-hand sides of (14),
weobtain
𝑢0(𝑥, 𝑡) = 𝑢 (𝑥, 0) + 𝐿
−1
𝛼(
1
𝑠𝛼𝐿𝛼{𝑔 (𝑥, 𝑡)}) ,
𝑢1(𝑥, 𝑡) = − 𝐿
−1
𝛼(
1
𝑠𝛼𝐿𝛼{𝑅𝛼𝑢0(𝑥, 𝑡)}) ,
𝑢2(𝑥, 𝑡) = − 𝐿
−1
𝛼(
1
𝑠𝛼𝐿𝛼{𝑅𝛼𝑢1(𝑥, 𝑡)})
.
.
.
(15)
The recursive relation, in general form, is
𝑢0(𝑥, 𝑡) = 𝑢 (𝑥, 0) + 𝐿
−1
𝛼(
1
𝑠𝛼𝐿𝛼{𝑔 (𝑥, 𝑡)}) ,
𝑢𝑛+1
(𝑥, 𝑡) = − 𝐿−1
𝛼(
1
𝑠𝛼𝐿𝛼{𝑅𝛼𝑢𝑛(𝑥, 𝑡)}) , 𝑛 ≥ 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]
𝑢𝑛+1
(𝑡) = 𝑢𝑛(𝑡)
+0𝐼(𝛼)
𝑡(𝜆 (𝑡 − 𝜉)
𝛼
Γ (1 + 𝛼)[𝐿𝛼𝑢𝑛(𝜉) + 𝑅
𝛼�̃�𝑛(𝜉) − 𝑔 (𝜉)]) ,
(17)
where 𝜆(𝑡 − 𝜉)𝛼/Γ(1 + 𝛼) is a fractal Lagrange multiplier.
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Journal of Mathematics 3
We now take Yang-Laplace transform of (17); namely,
𝐿𝛼{𝑢𝑛+1
(𝑡)} = 𝐿𝛼{𝑢𝑛(𝑡)}
+ 𝐿𝛼{0𝐼(𝛼)
𝑡(𝜆 (𝑡 − 𝜉)
𝛼
Γ (1 + 𝛼)[𝐿𝛼𝑢𝑛(𝜉) + 𝑅
𝛼�̃�𝑛(𝜉)
− 𝑔 (𝜉)])} ,
(18)
or
𝐿𝛼{𝑢𝑛+1
(𝑡)}
= 𝐿𝛼{𝑢𝑛(𝑡)}
+ 𝐿𝛼{
𝜆 (𝑡)𝛼
Γ (1 + 𝛼)} 𝐿𝛼{𝐿𝛼𝑢𝑛(𝑡) + 𝑅
𝛼�̃�𝑛(𝑡) − 𝑔 (𝑡)} .
(19)
Take the local fractional variation of (19), which is given
by
𝛿𝛼
(𝐿𝛼{𝑢𝑛+1
(𝑡)}) = 𝛿𝛼
(𝐿𝛼{𝑢𝑛(𝑡)})
+ 𝛿𝛼
(𝐿𝛼{
𝜆 (𝑡)𝛼
Γ (1 + 𝛼)}
⋅ 𝐿𝛼{𝐿𝛼𝑢𝑛(𝑡) − 𝑅
𝛼�̃�𝑛(𝑡) − 𝑔 (𝑡)}) .
(20)
By using computation of (20), we get
𝛿𝛼
(𝐿𝛼{𝑢𝑛+1
(𝑡)})
= 𝛿𝛼
(𝐿𝛼{𝑢𝑛(𝑡)})
+ 𝐿𝛼{
𝜆 (𝑡)𝛼
Γ (1 + 𝛼)} 𝛿𝛼
(𝐿𝛼{𝐿𝛼𝑢𝑛(𝑡)}) = 0.
(21)
Hence, from (21) we get
1 + 𝐿𝛼{
𝜆 (𝑡)𝛼
Γ (1 + 𝛼)} 𝑠𝛼
= 0, (22)
where
𝛿𝛼
(𝐿𝛼{𝐿𝛼𝑢𝑛(𝑡)}) = 𝛿
𝛼
(𝑠𝛼
𝐿𝛼{𝑢𝑛(𝑡)} − 𝑢
𝑛(0))
= 𝑠𝛼
𝛿𝛼
(𝐿𝛼{𝑢𝑛(𝑡)}) .
(23)
Therefore, we get
𝐿𝛼{
𝜆 (𝑡)𝛼
Γ (1 + 𝛼)} = −
1
𝑠𝛼. (24)
Therefore, we have the following iteration algorithm:
𝐿𝛼{𝑢𝑛+1
(𝑡)} = 𝐿𝛼{𝑢𝑛(𝑡)}
−1
𝑠𝛼𝐿𝛼{𝐿𝛼𝑢𝑛(𝑡) + 𝑅
𝛼𝑢𝑛(𝑡) − 𝑔 (𝑡)}
= 𝐿𝛼{𝑢𝑛(𝑡)} −
1
𝑠𝛼𝐿𝛼{𝐿𝛼𝑢𝑛(𝑡)}
−1
𝑠𝛼𝐿𝛼{𝑅𝛼𝑢𝑛(𝑡) − 𝑔 (𝑡)}
= 𝐿𝛼{𝑢𝑛(𝑡)} −
1
𝑠𝛼𝐿𝛼{𝑠𝛼
𝑢𝑛(𝑡) − 𝑢 (0)}
−1
𝑠𝛼𝐿𝛼{𝑅𝛼𝑢𝑛(𝑡) − 𝑔 (𝑡)}
=1
𝑠𝛼𝑢 (0) −
1
𝑠𝛼𝐿𝛼{𝑅𝛼𝑢𝑛(𝑡) − 𝑔 (𝑡)} ,
(25)
where the initial value reads as
𝐿𝛼{𝑢0(𝑡)} =
1
𝑠𝛼𝑢 (0) . (26)
Thus, the local fractional series solution of (8) is
𝑢 (𝑥, 𝑡) = lim𝑛→∞
𝐿−1
𝛼(𝐿𝛼{𝑢𝑛(𝑥, 𝑡)}) . (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
𝜕𝛼
𝑢 (𝑥, 𝑡)
𝜕𝑡𝛼= −
𝜕𝛼
𝑢 (𝑥, 𝑡)
𝜕𝑥𝛼+
𝜕2𝛼
𝑢 (𝑥, 𝑡)
𝜕𝑥2𝛼, (28)
subject to the initial value
𝑢 (𝑥, 0) = 𝐸𝛼(−𝑥𝛼
) . (29)
(I) By Using LFLDM. In view of (16) and (28) the localfractional
iteration algorithm can be written as follows:
𝑢0(𝑥, 𝑡) = 𝐸
𝛼(−𝑥𝛼
) ,
𝑢𝑛+1
(𝑥, 𝑡)
= 𝐿−1
𝛼(
1
𝑠𝛼𝐿𝛼{−
𝜕𝛼
𝑢𝑛(𝑥, 𝑡)
𝜕𝑥𝛼+
𝜕2𝛼
𝑢𝑛(𝑥, 𝑡)
𝜕𝑥2𝛼}) ,
𝑛 ≥ 0.
(30)
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4 Journal of Mathematics
Therefore, from (30) we give the components as follows:
𝑢0(𝑥, 𝑡) = 𝐸
𝛼(−𝑥𝛼
) ,
𝑢1(𝑥, 𝑡)
= 𝐿−1
𝛼(
1
𝑠𝛼𝐿𝛼{−
𝜕𝛼
𝑢0(𝑥, 𝑡)
𝜕𝑥𝛼+
𝜕2𝛼
𝑢0(𝑥, 𝑡)
𝜕𝑥2𝛼})
= 𝐿−1
𝛼(
2
𝑠2𝛼𝐸𝛼(−𝑥𝛼
)) =2𝑡𝛼
Γ (1 + 𝛼)𝐸𝛼(−𝑥𝛼
) ,
𝑢2(𝑥, 𝑡)
= 𝐿−1
𝛼(
1
𝑠𝛼𝐿𝛼{−
𝜕𝛼
𝑢1(𝑥, 𝑡)
𝜕𝑥𝛼+
𝜕2𝛼
𝑢1(𝑥, 𝑡)
𝜕𝑥2𝛼})
= 𝐿−1
𝛼(
4
𝑠3𝛼𝐸𝛼(−𝑥𝛼
)) =4𝑡2𝛼
Γ (1 + 2𝛼)𝐸𝛼(−𝑥𝛼
) ,
𝑢3(𝑥, 𝑡)
= 𝐿−1
𝛼(
1
𝑠𝛼𝐿𝛼{−
𝜕𝛼
𝑢2(𝑥, 𝑡)
𝜕𝑥𝛼+
𝜕2𝛼
𝑢2(𝑥, 𝑡)
𝜕𝑥2𝛼})
= 𝐿−1
𝛼(
8
𝑠4𝛼𝐸𝛼(−𝑥𝛼
)) =8𝑡3𝛼
Γ (1 + 3𝛼)𝐸𝛼(−𝑥𝛼
)
.
.
.
(31)
Finally, we can present the solution in local fractional
seriesform as
𝑢 (𝑥, 𝑡) = 𝐸𝛼(−𝑥𝛼
)
⋅ (1 +2𝑡𝛼
Γ (1 + 𝛼)+
4𝑡2𝛼
Γ (1 + 2𝛼)+
8𝑡3𝛼
Γ (1 + 3𝛼)+ ⋅ ⋅ ⋅)
= 𝐸𝛼(−𝑥𝛼
) 𝐸𝛼(2𝑡𝛼
) = 𝐸𝛼(2𝑡𝛼
− 𝑥𝛼
) .
(32)
(II) By Using LFLVIM. Using relation (25) we structure
theiterative relation as
𝐿𝛼{𝑢𝑛+1
(𝑥, 𝑡)}
= 𝐿𝛼{𝑢𝑛(𝑥, 𝑡)}
−1
𝑠𝛼𝐿𝛼{
𝜕𝛼
𝑢𝑛(𝑥, 𝑡)
𝜕𝑡𝛼+
𝜕𝛼
𝑢𝑛(𝑥, 𝑡)
𝜕𝑥𝛼−
𝜕2𝛼
𝑢𝑛(𝑥, 𝑡)
𝜕𝑥2𝛼}
= 𝐿𝛼{𝑢𝑛(𝑥, 𝑡)} −
1
𝑠𝛼(𝑠𝛼
𝐿𝛼{𝑢𝑛(𝑥, 𝑡)} − 𝑢
𝑛(𝑥, 0))
−1
𝑠𝛼𝐿𝛼{
𝜕𝛼
𝑢𝑛(𝑥, 𝑡)
𝜕𝑥𝛼−
𝜕2𝛼
𝑢𝑛(𝑥, 𝑡)
𝜕𝑥2𝛼} ,
(33)
where the initial value is given by
𝐿𝛼{𝑢0(𝑥, 𝑡)} = 𝐿
𝛼{𝐸𝛼(−𝑥𝛼
)} =1
𝑠𝛼𝐸𝛼(−𝑥𝛼
) . (34)
Therefore, the successive approximations are𝐿𝛼{𝑢1(𝑥, 𝑡)} = 𝐿
𝛼{𝑢𝑛(𝑥, 𝑡)}
−1
𝑠𝛼(𝑠𝛼
𝐿𝛼{𝑢𝑛(𝑥, 𝑡)} − 𝑢
𝑛(𝑥, 0))
−1
𝑠𝛼𝐿𝛼{
𝜕𝛼
𝑢𝑛(𝑥, 𝑡)
𝜕𝑥𝛼−
𝜕2𝛼
𝑢𝑛(𝑥, 𝑡)
𝜕𝑥2𝛼}
=1
𝑠𝛼𝐸𝛼(−𝑥𝛼
) +2
𝑠2𝛼𝐸𝛼(−𝑥𝛼
) ,
𝐿𝛼{𝑢2(𝑥, 𝑡)} = 𝐿
𝛼{𝑢1(𝑥, 𝑡)}
−1
𝑠𝛼(𝑠𝛼
𝐿𝛼{𝑢1(𝑥, 𝑡)} − 𝑢
1(𝑥, 0))
−1
𝑠𝛼𝐿𝛼{
𝜕𝛼
𝑢1(𝑥, 𝑡)
𝜕𝑥𝛼−
𝜕2𝛼
𝑢1(𝑥, 𝑡)
𝜕𝑥2𝛼}
=1
𝑠𝛼𝐸𝛼(−𝑥𝛼
) +2
𝑠2𝛼𝐸𝛼(−𝑥𝛼
) +4
𝑠3𝛼𝐸𝛼(−𝑥𝛼
)
= 𝐸𝛼(−𝑥𝛼
) (1
𝑠𝛼+
2
𝑠2𝛼+
4
𝑠3𝛼) ,
𝐿𝛼{𝑢3(𝑥, 𝑡)} = 𝐿
𝛼{𝑢2(𝑥, 𝑡)}
−1
𝑠𝛼(𝑠𝛼
𝐿𝛼{𝑢2(𝑥, 𝑡)} − 𝑢
2(𝑥, 0))
−1
𝑠𝛼𝐿𝛼{
𝜕𝛼
𝑢2(𝑥, 𝑡)
𝜕𝑥𝛼−
𝜕2𝛼
𝑢2(𝑥, 𝑡)
𝜕𝑥2𝛼}
=1
𝑠𝛼𝐸𝛼(−𝑥𝛼
) +2
𝑠2𝛼𝐸𝛼(−𝑥𝛼
) +4
𝑠3𝛼𝐸𝛼(−𝑥𝛼
)
+8
𝑠4𝛼𝐸𝛼(−𝑥𝛼
)
= 𝐸𝛼(−𝑥𝛼
) (1
𝑠𝛼+
2
𝑠2𝛼+
4
𝑠3𝛼+
8
𝑠4𝛼)
.
.
.
(35)
Hence, the local fractional series solution is𝑢 (𝑥, 𝑡) = lim
𝑛→∞
𝐿−1
𝛼(𝐿𝛼{𝑢𝑛(𝑥, 𝑡)})
= lim𝑛→∞
𝐿−1
𝛼(𝐸𝛼(−𝑥𝛼
)
𝑛
∑
𝑘=0
2𝑘
𝑠(𝑘+1)𝛼)
= 𝐸𝛼(−𝑥𝛼
)
∞
∑
𝑘=0
2𝑘
𝑡𝑘𝛼
Γ (1 + 𝑘𝛼)= 𝐸𝛼(2𝑡𝛼
− 𝑥𝛼
) .
(36)
Example 2. We present the Fokker Planck equation on aCantor set
with local fractional derivative
𝜕𝛼
𝑢 (𝑥, 𝑡)
𝜕𝑡𝛼= −
𝜕𝛼
𝑢 (𝑥, 𝑡)
𝜕𝑥𝛼+
𝜕2𝛼
𝑢 (𝑥, 𝑡)
𝜕𝑥2𝛼, (37)
and the initial condition is
𝑢 (𝑥, 0) =𝑥2𝛼
Γ (1 + 2𝛼). (38)
-
Journal of Mathematics 5
(I) By Using LFLDM. In view of (16) and (37) the localfractional
iteration algorithm can be written as follows:
𝑢0(𝑥, 𝑡) =
𝑥2𝛼
Γ (1 + 2𝛼),
𝑢𝑛+1
(𝑥, 𝑡)
= 𝐿−1
𝛼(
1
𝑠𝛼𝐿𝛼{−
𝜕𝛼
𝑢𝑛(𝑥, 𝑡)
𝜕𝑥𝛼+
𝜕2𝛼
𝑢𝑛(𝑥, 𝑡)
𝜕𝑥2𝛼}) ,
𝑛 ≥ 0.
(39)
Therefore, from (39) we give the components as follows:
𝑢0(𝑥, 𝑡) =
𝑥2𝛼
Γ (1 + 2𝛼),
𝑢1(𝑥, 𝑡)
= 𝐿−1
𝛼(
1
𝑠𝛼𝐿𝛼{−
𝜕𝛼
𝑢0(𝑥, 𝑡)
𝜕𝑥𝛼+
𝜕2𝛼
𝑢0(𝑥, 𝑡)
𝜕𝑥2𝛼})
= 𝐿−1
𝛼(−
1
𝑠2𝛼
𝑥𝛼
Γ (1 + 𝛼)+
1
𝑠2𝛼)
= −𝑡𝛼
Γ (1 + 𝛼)
𝑥𝛼
Γ (1 + 𝛼)+
𝑡𝛼
Γ (1 + 𝛼),
𝑢2(𝑥, 𝑡)
= 𝐿−1
𝛼(
1
𝑠𝛼𝐿𝛼{−
𝜕𝛼
𝑢1(𝑥, 𝑡)
𝜕𝑥𝛼+
𝜕2𝛼
𝑢1(𝑥, 𝑡)
𝜕𝑥2𝛼})
= 𝐿−1
𝛼(
1
𝑠3𝛼) =
𝑡2𝛼
Γ (1 + 2𝛼),
𝑢3(𝑥, 𝑡)
= 𝐿−1
𝛼(
1
𝑠𝛼𝐿𝛼{−
𝜕𝛼
𝑢2(𝑥, 𝑡)
𝜕𝑥𝛼+
𝜕2𝛼
𝑢2(𝑥, 𝑡)
𝜕𝑥2𝛼})
= 0,
𝑢4(𝑥, 𝑡)
= 𝐿−1
𝛼(
1
𝑠𝛼𝐿𝛼{−
𝜕𝛼
𝑢3(𝑥, 𝑡)
𝜕𝑥𝛼+
𝜕2𝛼
𝑢3(𝑥, 𝑡)
𝜕𝑥2𝛼}) = 0
.
.
.
(40)
Finally, we can present the solution in local fractional
seriesform as
𝑢 (𝑥, 𝑡) =𝑥2𝛼
Γ (1 + 2𝛼)+
𝑡2𝛼
Γ (1 + 2𝛼)
−𝑥𝛼
Γ (1 + 𝛼)
𝑡𝛼
Γ (1 + 𝛼)+
𝑡𝛼
Γ (1 + 𝛼).
(41)
(II) By Using LFLVIM. Using relation (25) we structure
theiterative relation as
𝐿𝛼{𝑢𝑛+1
(𝑥, 𝑡)} = 𝐿𝛼{𝑢𝑛(𝑥, 𝑡)}
−1
𝑠𝛼𝐿𝛼{
𝜕𝛼
𝑢𝑛(𝑥, 𝑡)
𝜕𝑡𝛼+
𝜕𝛼
𝑢𝑛(𝑥, 𝑡)
𝜕𝑥𝛼−
𝜕2𝛼
𝑢𝑛(𝑥, 𝑡)
𝜕𝑥2𝛼}
= 𝐿𝛼{𝑢𝑛(𝑥, 𝑡)} −
1
𝑠𝛼(𝑠𝛼
𝐿𝛼{𝑢𝑛(𝑥, 𝑡)} − 𝑢
𝑛(𝑥, 0))
−1
𝑠𝛼𝐿𝛼{
𝜕𝛼
𝑢𝑛(𝑥, 𝑡)
𝜕𝑥𝛼−
𝜕2𝛼
𝑢𝑛(𝑥, 𝑡)
𝜕𝑥2𝛼} ,
(42)
where the initial value is given by
𝐿𝛼{𝑢0(𝑥, 𝑡)} = 𝐿
𝛼{
𝑥2𝛼
Γ (1 + 2𝛼)} =
1
𝑠𝛼
𝑥2𝛼
Γ (1 + 2𝛼). (43)
Therefore, the successive approximations are
𝐿𝛼{𝑢1(𝑥, 𝑡)} = 𝐿
𝛼{𝑢0(𝑥, 𝑡)}
−1
𝑠𝛼(𝑠𝛼
𝐿𝛼{𝑢0(𝑥, 𝑡)} − 𝑢
0(𝑥, 0))
−1
𝑠𝛼𝐿𝛼{
𝜕𝛼
𝑢0(𝑥, 𝑡)
𝜕𝑥𝛼−
𝜕2𝛼
𝑢0(𝑥, 𝑡)
𝜕𝑥2𝛼}
=1
𝑠𝛼
𝑥2𝛼
Γ (1 + 2𝛼)−
1
𝑠2𝛼
𝑥𝛼
Γ (1 + 𝛼)+
1
𝑠2𝛼,
𝐿𝛼{𝑢2(𝑥, 𝑡)} = 𝐿
𝛼{𝑢1(𝑥, 𝑡)}
−1
𝑠𝛼(𝑠𝛼
𝐿𝛼{𝑢1(𝑥, 𝑡)} − 𝑢
1(𝑥, 0))
−1
𝑠𝛼𝐿𝛼{
𝜕𝛼
𝑢1(𝑥, 𝑡)
𝜕𝑥𝛼−
𝜕2𝛼
𝑢1(𝑥, 𝑡)
𝜕𝑥2𝛼}
=1
𝑠𝛼
𝑥2𝛼
Γ (1 + 2𝛼)−
1
𝑠2𝛼
𝑥𝛼
Γ (1 + 𝛼)+
1
𝑠3𝛼+
1
𝑠2𝛼,
𝐿𝛼{𝑢3(𝑥, 𝑡)} = 𝐿
𝛼{𝑢2(𝑥, 𝑡)}
−1
𝑠𝛼(𝑠𝛼
𝐿𝛼{𝑢2(𝑥, 𝑡)} − 𝑢
2(𝑥, 0))
−1
𝑠𝛼𝐿𝛼{
𝜕𝛼
𝑢2(𝑥, 𝑡)
𝜕𝑥𝛼−
𝜕2𝛼
𝑢2(𝑥, 𝑡)
𝜕𝑥2𝛼}
=1
𝑠𝛼
𝑥2𝛼
Γ (1 + 2𝛼)−
1
𝑠2𝛼
𝑥𝛼
Γ (1 + 𝛼)+
1
𝑠3𝛼+
1
𝑠2𝛼
.
.
.
(44)
-
6 Journal of Mathematics
Hence, the local fractional series solution is
𝑢 (𝑥, 𝑡) = lim𝑛→∞
𝐿−1
𝛼(𝐿𝛼{𝑢𝑛(𝑥, 𝑡)})
= lim𝑛→∞
𝐿−1
𝛼(
1
𝑠𝛼
𝑥2𝛼
Γ (1 + 2𝛼)+
1
𝑠3𝛼−
1
𝑠2𝛼
𝑥𝛼
Γ (1 + 𝛼)
+1
𝑠2𝛼) =
𝑥2𝛼
Γ (1 + 2𝛼)+
𝑡2𝛼
Γ (1 + 2𝛼)−
𝑥𝛼
Γ (1 + 𝛼)
⋅𝑡𝛼
Γ (1 + 𝛼)+
𝑡𝛼
Γ (1 + 𝛼).
(45)
Example 3. We consider the Fokker Planck equation on aCantor set
with local fractional derivative
𝜕𝛼
𝑢 (𝑥, 𝑡)
𝜕𝑡𝛼= −
𝜕𝛼
𝑢 (𝑥, 𝑡)
𝜕𝑥𝛼+
𝜕2𝛼
𝑢 (𝑥, 𝑡)
𝜕𝑥2𝛼, (46)
and the initial condition is
𝑢 (𝑥, 0) = −𝑥3𝛼
Γ (1 + 3𝛼). (47)
(I) By Using LFLDM. In view of (16) and (46) the localfractional
iteration algorithm can be written as follows:
𝑢0(𝑥, 𝑡) = −
𝑥3𝛼
Γ (1 + 3𝛼),
𝑢𝑛+1
(𝑥, 𝑡)
= 𝐿−1
𝛼(
1
𝑠𝛼𝐿𝛼{−
𝜕𝛼
𝑢𝑛(𝑥, 𝑡)
𝜕𝑥𝛼+
𝜕2𝛼
𝑢𝑛(𝑥, 𝑡)
𝜕𝑥2𝛼}) ,
𝑛 ≥ 0.
(48)
Therefore, from (30) we give the components as follows:
𝑢0(𝑥, 𝑡) = −
𝑥3𝛼
Γ (1 + 3𝛼),
𝑢1(𝑥, 𝑡)
= 𝐿−1
𝛼(
1
𝑠𝛼𝐿𝛼{−
𝜕𝛼
𝑢0(𝑥, 𝑡)
𝜕𝑥𝛼+
𝜕2𝛼
𝑢0(𝑥, 𝑡)
𝜕𝑥2𝛼})
= 𝐿−1
𝛼(
1
𝑠2𝛼
𝑥2𝛼
Γ (1 + 2𝛼)−
1
𝑠2𝛼
𝑥𝛼
Γ (1 + 𝛼))
=𝑥2𝛼
Γ (1 + 2𝛼)
𝑡𝛼
Γ (1 + 𝛼)−
𝑥𝛼
Γ (1 + 𝛼)
𝑡𝛼
Γ (1 + 𝛼),
𝑢2(𝑥, 𝑡)
= 𝐿−1
𝛼(
1
𝑠𝛼𝐿𝛼{−
𝜕𝛼
𝑢1(𝑥, 𝑡)
𝜕𝑥𝛼+
𝜕2𝛼
𝑢1(𝑥, 𝑡)
𝜕𝑥2𝛼})
= 𝐿−1
𝛼(−
1
𝑠3𝛼
𝑥𝛼
Γ (1 + 𝛼)+
2
𝑠3𝛼)
= −𝑥𝛼
Γ (1 + 𝛼)
𝑡2𝛼
Γ (1 + 2𝛼)+
2𝑡2𝛼
Γ (1 + 2𝛼),
𝑢3(𝑥, 𝑡)
= 𝐿−1
𝛼(
1
𝑠𝛼𝐿𝛼{−
𝜕𝛼
𝑢2(𝑥, 𝑡)
𝜕𝑥𝛼+
𝜕2𝛼
𝑢2(𝑥, 𝑡)
𝜕𝑥2𝛼})
= 𝐿−1
𝛼(
1
𝑠4𝛼) =
𝑡3𝛼
Γ (1 + 3𝛼),
𝑢4(𝑥, 𝑡)
= 𝐿−1
𝛼(
1
𝑠𝛼𝐿𝛼{−
𝜕𝛼
𝑢3(𝑥, 𝑡)
𝜕𝑥𝛼+
𝜕2𝛼
𝑢3(𝑥, 𝑡)
𝜕𝑥2𝛼}) = 0
.
.
.
(49)
Finally, we can present the solution in local fractional
seriesform as
𝑢 (𝑥, 𝑡) = −𝑥3𝛼
Γ (1 + 3𝛼)+
𝑥2𝛼
Γ (1 + 2𝛼)
𝑡𝛼
Γ (1 + 𝛼)
−𝑥𝛼
Γ (1 + 𝛼)
𝑡𝛼
Γ (1 + 𝛼)
−𝑥𝛼
Γ (1 + 𝛼)
𝑡2𝛼
Γ (1 + 2𝛼)+
2𝑡2𝛼
Γ (1 + 2𝛼)
+𝑡3𝛼
Γ (1 + 3𝛼).
(50)
(II) By Using LFLVIM. Using relation (18) we structure
theiterative relation as
𝐿𝛼{𝑢𝑛+1
(𝑥, 𝑡)}
= 𝐿𝛼{𝑢𝑛(𝑥, 𝑡)}
−1
𝑠𝛼𝐿𝛼{
𝜕𝛼
𝑢𝑛(𝑥, 𝑡)
𝜕𝑡𝛼+
𝜕𝛼
𝑢𝑛(𝑥, 𝑡)
𝜕𝑥𝛼−
𝜕2𝛼
𝑢𝑛(𝑥, 𝑡)
𝜕𝑥2𝛼}
= 𝐿𝛼{𝑢𝑛(𝑥, 𝑡)} −
1
𝑠𝛼(𝑠𝛼
𝐿𝛼{𝑢𝑛(𝑥, 𝑡)} − 𝑢
𝑛(𝑥, 0))
−1
𝑠𝛼𝐿𝛼{
𝜕𝛼
𝑢𝑛(𝑥, 𝑡)
𝜕𝑥𝛼−
𝜕2𝛼
𝑢𝑛(𝑥, 𝑡)
𝜕𝑥2𝛼} ,
(51)
where the initial value is given by
𝐿𝛼{𝑢0(𝑥, 𝑡)} = 𝐿
𝛼{−
𝑥3𝛼
Γ (1 + 3𝛼)} = −
1
𝑠𝛼
𝑥3𝛼
Γ (1 + 3𝛼). (52)
Therefore, the successive approximations are
𝐿𝛼{𝑢1(𝑥, 𝑡)} = 𝐿
𝛼{𝑢0(𝑥, 𝑡)}
−1
𝑠𝛼(𝑠𝛼
𝐿𝛼{𝑢0(𝑥, 𝑡)} − 𝑢
𝑛(𝑥, 0))
−1
𝑠𝛼𝐿𝛼{
𝜕𝛼
𝑢0(𝑥, 𝑡)
𝜕𝑥𝛼−
𝜕2𝛼
𝑢0(𝑥, 𝑡)
𝜕𝑥2𝛼}
-
Journal of Mathematics 7
= −1
𝑠𝛼
𝑥3𝛼
Γ (1 + 3𝛼)+
1
𝑠2𝛼
𝑥2𝛼
Γ (1 + 2𝛼)−
1
𝑠2𝛼
𝑥𝛼
Γ (1 + 𝛼),
𝐿𝛼{𝑢2(𝑥, 𝑡)} = 𝐿
𝛼{𝑢1(𝑥, 𝑡)}
−1
𝑠𝛼(𝑠𝛼
𝐿𝛼{𝑢1(𝑥, 𝑡)} − 𝑢
1(𝑥, 0))
−1
𝑠𝛼𝐿𝛼{
𝜕𝛼
𝑢1(𝑥, 𝑡)
𝜕𝑥𝛼−
𝜕2𝛼
𝑢1(𝑥, 𝑡)
𝜕𝑥2𝛼}
= −1
𝑠𝛼
𝑥3𝛼
Γ (1 + 3𝛼)+
1
𝑠2𝛼
𝑥2𝛼
Γ (1 + 2𝛼)−
1
𝑠2𝛼
𝑥𝛼
Γ (1 + 𝛼)
−1
𝑠3𝛼
𝑥𝛼
Γ (1 + 3𝛼)+
2
𝑠3𝛼,
𝐿𝛼{𝑢3(𝑥, 𝑡)} = 𝐿
𝛼{𝑢2(𝑥, 𝑡)}
−1
𝑠𝛼(𝑠𝛼
𝐿𝛼{𝑢2(𝑥, 𝑡)} − 𝑢
2(𝑥, 0))
−1
𝑠𝛼𝐿𝛼{
𝜕𝛼
𝑢2(𝑥, 𝑡)
𝜕𝑥𝛼−
𝜕2𝛼
𝑢2(𝑥, 𝑡)
𝜕𝑥2𝛼}
= −1
𝑠𝛼
𝑥3𝛼
Γ (1 + 3𝛼)+
1
𝑠2𝛼
𝑥2𝛼
Γ (1 + 2𝛼)−
1
𝑠2𝛼
𝑥𝛼
Γ (1 + 𝛼)
−1
𝑠3𝛼
𝑥𝛼
Γ (1 + 3𝛼)+
2
𝑠3𝛼+
1
𝑠4𝛼,
𝐿𝛼{𝑢4(𝑥, 𝑡)} = 𝐿
𝛼{𝑢3(𝑥, 𝑡)}
−1
𝑠𝛼(𝑠𝛼
𝐿𝛼{𝑢3(𝑥, 𝑡)} − 𝑢
3(𝑥, 0))
−1
𝑠𝛼𝐿𝛼{
𝜕𝛼
𝑢3(𝑥, 𝑡)
𝜕𝑥𝛼−
𝜕2𝛼
𝑢3(𝑥, 𝑡)
𝜕𝑥2𝛼}
= −1
𝑠𝛼
𝑥3𝛼
Γ (1 + 3𝛼)+
1
𝑠2𝛼
𝑥2𝛼
Γ (1 + 2𝛼)−
1
𝑠2𝛼
𝑥𝛼
Γ (1 + 𝛼)
−1
𝑠3𝛼
𝑥𝛼
Γ (1 + 3𝛼)+
2
𝑠3𝛼+
1
𝑠4𝛼
.
.
.
(53)Hence, the local fractional series solution is
𝑢 (𝑥, 𝑡) = lim𝑛→∞
𝐿−1
𝛼(𝐿𝛼{𝑢𝑛(𝑥, 𝑡)})
= −𝑥3𝛼
Γ (1 + 3𝛼)+
𝑥2𝛼
Γ (1 + 2𝛼)
𝑡𝛼
Γ (1 + 𝛼)
−𝑥𝛼
Γ (1 + 𝛼)
𝑡𝛼
Γ (1 + 𝛼)
−𝑥𝛼
Γ (1 + 𝛼)
𝑡2𝛼
Γ (1 + 2𝛼)+
2𝑡2𝛼
Γ (1 + 2𝛼)
+𝑡3𝛼
Γ (1 + 3𝛼).
(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
[1] H. Risken,The Fokker Planck Equation: Method of Solution
andApplications, Springer, Berlin, Germany, 1989.
[2] X. J. Yang, Advanced Local Fractional Calculus and Its
Applica-tions, World Science Publisher, New York, NY, USA,
2012.
[3] X. J. Yang, Local Fractional Functional Analysis and Its
Applica-tions, Asian Academic Publisher, Hong Kong, 2011.
[4] X.-J. Yang, H. M. Srivastava, J.-H. He, and D. Baleanu,
“Cantor-type cylindrical-coordinate method for differential
equationswith local fractional derivatives,” Physics Letters,
Section A:General, Atomic and Solid State Physics, vol. 377, no.
28–30, pp.1696–1700, 2013.
[5] K. M. Kolwankar and A. D. Gangal, “Local fractional
Fokker-Planck equation,” Physical Review Letters, vol. 80, no. 2,
pp. 214–217, 1998.
[6] X. J. Yang andD. Baleanu, “Local fractional variational
iterationmethod for Fokker-Planck equation on a Cantor set,”
ActaUniversitaria, vol. 23, no. 2, pp. 3–8, 2013.
[7] S.-H. Yan, X.-H. Chen, G.-N. Xie, C. Cattani, and X.-J.
Yang,“Solving Fokker-Planck equations on Cantor sets using
localfractional decomposition method,” Abstract and Applied
Anal-ysis, vol. 2014, Article ID 396469, 6 pages, 2014.
[8] H. Jafari and H. K. Jassim, “A coupling method of
localfractional variational iteration method and yang-laplace
trans-form for solving laplace equation on cantor sets,”
InternationalJournal of Pure and Applied Sciences and Technology,
vol. 26, no.1, pp. 24–33, 2015.
[9] S.-Q. Wang, Y.-J. Yang, and H. K. Jassim, “Local
fractionalfunction decomposition method for solving
inhomogeneouswave equations with local fractional derivative,”
Abstract andApplied Analysis, vol. 2014, Article ID 176395, 7
pages, 2014.
[10] S.-P. Yan, H. Jafari, and H. K. Jassim, “Local fractional
ado-mian decomposition and function decomposition methods
forlaplace equation within local fractional operators,” Advances
inMathematical Physics, vol. 2014, Article ID 161580, 7 pages,
2014.
[11] H. Jafari and H. K. Jassim, “Local fractional series
expansionmethod for solving Laplace and Schrödinger equations
oncantor sets within local fractional operators,” International
-
8 Journal of Mathematics
Journal of Mathematics and Computer Research, vol. 2, pp.
736–744, 2014.
[12] H. Jafari and H. K. Jassim, “Local fractional adomian
decom-position method for solving two dimensional heat
conductionequations within local fractional operators,” Journal of
Advancein Mathematics, vol. 9, no. 4, pp. 2574–2582, 2014.
[13] D. Baleanu, J. A. T. Machado, C. Cattani, M. C. Baleanu,and
X.-J. Yang, “Local fractional variational iteration
anddecompositionmethods forwave equation on cantor setswithinlocal
fractional operators,” Abstract and Applied Analysis, vol.2014,
Article ID 535048, 6 pages, 2014.
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