research article new approaches for solving fokker planck...

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

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.

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


𝜕𝑥2𝛼}) ,

𝑛 ≥ 0.

(30)


Therefore, from (30) we give the components as follows:

𝑢0(𝑥, 𝑡) = 𝐸

𝛼(−𝑥𝛼

) ,

𝑢1(𝑥, 𝑡)

= 𝐿−1

𝛼(

1


𝜕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


𝜕𝛼

𝑢1(𝑥, 𝑡)

𝜕𝑥𝛼+

𝜕2𝛼

𝑢1(𝑥, 𝑡)

𝜕𝑥2𝛼})

= 𝐿−1

𝛼(

4


)) =4𝑡2𝛼

Γ (1 + 2𝛼)𝐸𝛼(−𝑥𝛼

) ,

𝑢3(𝑥, 𝑡)

= 𝐿−1

𝛼(

1


𝜕𝛼

𝑢2(𝑥, 𝑡)

𝜕𝑥𝛼+

𝜕2𝛼

𝑢2(𝑥, 𝑡)

𝜕𝑥2𝛼})

= 𝐿−1

𝛼(

8


)) =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


𝜕𝑡𝛼+

𝜕𝛼


𝜕𝑥𝛼−

𝜕2𝛼


𝜕𝑥2𝛼}

= 𝐿𝛼{𝑢𝑛(𝑥, 𝑡)} −

1

𝑠𝛼(𝑠𝛼

𝐿𝛼{𝑢𝑛(𝑥, 𝑡)} − 𝑢

𝑛(𝑥, 0))

−1

𝑠𝛼𝐿𝛼{

𝜕𝛼


𝜕𝑥𝛼−

𝜕2𝛼


𝜕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


𝑛(𝑥, 0))

−1

𝑠𝛼𝐿𝛼{

𝜕𝛼


𝜕𝑥𝛼−

𝜕2𝛼


𝜕𝑥2𝛼}

=1


) +2


) ,

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


) +2


) +4


)


) (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


) +2


) +4


)

+8


)


) (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)


)

∞

∑

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)



𝑢0(𝑥, 𝑡) =

𝑥2𝛼

Γ (1 + 2𝛼),

𝑢𝑛+1

(𝑥, 𝑡)

= 𝐿−1

𝛼(

1


𝜕𝛼


𝜕𝑥𝛼+

𝜕2𝛼


𝜕𝑥2𝛼}) ,

𝑛 ≥ 0.

(39)


𝑢0(𝑥, 𝑡) =

𝑥2𝛼

Γ (1 + 2𝛼),

𝑢1(𝑥, 𝑡)

= 𝐿−1

𝛼(

1


𝜕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


𝜕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


𝜕𝛼

𝑢2(𝑥, 𝑡)

𝜕𝑥𝛼+

𝜕2𝛼

𝑢2(𝑥, 𝑡)

𝜕𝑥2𝛼})

= 0,

𝑢4(𝑥, 𝑡)

= 𝐿−1

𝛼(

1


𝜕𝛼

𝑢3(𝑥, 𝑡)

𝜕𝑥𝛼+

𝜕2𝛼

𝑢3(𝑥, 𝑡)

𝜕𝑥2𝛼}) = 0

.

.

.

(40)


𝑢 (𝑥, 𝑡) =𝑥2𝛼

Γ (1 + 2𝛼)+

𝑡2𝛼

Γ (1 + 2𝛼)

−𝑥𝛼

Γ (1 + 𝛼)

𝑡𝛼

Γ (1 + 𝛼)+

𝑡𝛼

Γ (1 + 𝛼).

(41)


𝐿𝛼{𝑢𝑛+1

(𝑥, 𝑡)} = 𝐿𝛼{𝑢𝑛(𝑥, 𝑡)}

−1

𝑠𝛼𝐿𝛼{

𝜕𝛼


𝜕𝑡𝛼+

𝜕𝛼


𝜕𝑥𝛼−

𝜕2𝛼


𝜕𝑥2𝛼}

= 𝐿𝛼{𝑢𝑛(𝑥, 𝑡)} −

1

𝑠𝛼(𝑠𝛼


𝑛(𝑥, 0))

−1

𝑠𝛼𝐿𝛼{

𝜕𝛼


𝜕𝑥𝛼−

𝜕2𝛼


𝜕𝑥2𝛼} ,

(42)


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)


Hence, the local fractional series solution is


𝐿−1


= 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)


𝑢0(𝑥, 𝑡) = −

𝑥3𝛼

Γ (1 + 3𝛼),

𝑢𝑛+1

(𝑥, 𝑡)

= 𝐿−1

𝛼(

1


𝜕𝛼


𝜕𝑥𝛼+

𝜕2𝛼


𝜕𝑥2𝛼}) ,

𝑛 ≥ 0.

(48)


𝑢0(𝑥, 𝑡) = −

𝑥3𝛼

Γ (1 + 3𝛼),

𝑢1(𝑥, 𝑡)

= 𝐿−1

𝛼(

1


𝜕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


𝜕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


𝜕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


𝜕𝛼

𝑢3(𝑥, 𝑡)

𝜕𝑥𝛼+

𝜕2𝛼

𝑢3(𝑥, 𝑡)

𝜕𝑥2𝛼}) = 0

.

.

.

(49)


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)


𝐿𝛼{𝑢𝑛+1

(𝑥, 𝑡)}

= 𝐿𝛼{𝑢𝑛(𝑥, 𝑡)}

−1

𝑠𝛼𝐿𝛼{

𝜕𝛼


𝜕𝑡𝛼+

𝜕𝛼


𝜕𝑥𝛼−

𝜕2𝛼


𝜕𝑥2𝛼}

= 𝐿𝛼{𝑢𝑛(𝑥, 𝑡)} −

1

𝑠𝛼(𝑠𝛼


𝑛(𝑥, 0))

−1

𝑠𝛼𝐿𝛼{

𝜕𝛼


𝜕𝑥𝛼−

𝜕2𝛼


𝜕𝑥2𝛼} ,

(51)


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}


= −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


𝐿−1


= −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


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.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

research article new approaches for solving fokker planck...

Documents