# New Approaches for Solving Fokker Planck Equation on Cantor Sets within Local Fractional Operators

Authors: Hassan Kamil Jassim
Journal: Journal of Mathematics
 (2015)

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## Abstract

We discuss new approaches to handling Fokker Planck equation on Cantor sets within local fractional operators by using the local fractional Laplace decomposition and Laplace variational iteration methods based on the local fractional calculus. The new 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.


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## Body

## 1. Introduction

The Fokker Planck equation arises in various fields in natural science, including solid-state physics, quantum optics, chemical physics, theoretical biology, and circuit theory. The Fokker Planck equation was first used by Fokker and Plank [1] to describe the Brownian motion of particles. A FPE describes the change of probability of a random function in space and time; hence it is naturally used to describe solute transport.The local fractional calculus was developed and applied to the fractal phenomenon in science and engineering [2–13]. Local fractional Fokker Planck equation, which was an analog of a diffusion equation with local fractional derivative, was suggested in [5] as follows:(1)∂αux,t∂tα=Γ1+α4χCt+∂2ux,t∂x2.The Fokker Planck equation on a Cantor set with local fractional derivative was presented in [6, 7] as follows:(2)∂αux,t∂tα=-∂αux,t∂xα+∂2αux,t∂x2α,subject to the initial condition(3)ux,0=fx.In recent years, a variety of numerical and analytical methods have been applied to solve the Fokker Planck equation on Cantor sets such as local fractional variational iteration method [6] and local fractional Adomian decomposition method [7]. Our main purpose of the paper is to apply the local fractional Laplace decomposition method and local fractional variational iteration method to solve the Fokker Planck equations on a Cantor set. The paper has been organized as follows. In Section 2, the basic mathematical tools are reviewed. In Section 3, we give analysis of the methods used. In Section 4, we consider several illustrative examples. Finally, in Section 5, we present our conclusions.


## 2. Mathematical Fundamentals

Definition 1.
Settingf(x)∈Cα(a,b), the local fractional derivative of f(x) of order α at the point x=x0 is defined as [2, 3, 9–12] (4)dαdxαfxx=x0=fαx=limx→x0Δαfx-fx0x-x0α,0<α≤1,where Δα(f(x)-f(x0))≅Γ(α+1)(f(x)-f(x0)).Definition 2.
Settingf(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](5)Iabαfx=1Γ1+α∫abftdtα=1Γ1+αlimΔt→0∑j=0N-1ftjΔtjα,0<α≤1,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{Δt0,Δt1,…},j=0,…,N-1.Definition 3.
Let1/Γ(1+α)∫0∞f(x)(dx)α<k<∞. The Yang-Laplace transform of f(x) is given by [2, 3] (6)Lαfx=fsL,αs=1Γ1+α∫0∞Eα-sαxαfxdxα,0<α≤1,where the latter integral converges and sα∈Rα.Definition 4.
The inverse formula of the Yang-Laplace transforms off(x) is given by [2, 3](7)Lα-1fsL,αs=ft=12πα∫β-iωβ+iωEαsαxαfsL,αsdsα,0<α≤1,where sα=βα+iαωα, fractal imaginary unit iα, and Re(s)=β>0.


## 3. Analytical Methods

In order to illustrate two analytical methods, we investigate the local fractional partial differential equation as follows:(8)Lαux,t+Rαux,t=gx,t,where Lα=∂α/∂tα denotes the linear local fractional differential operator, Rα is 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), we obtain(9)LαLαux,t+LαRαux,t=Lαgx,t.By applying the local fractional Laplace transform differentiation property, we have(10)sαLαux,t-ux,0+LαRαux,t=Lαgx,t,or (11)Lαux,t=1sαux,0+1sαLαgx,t-1sαLRαx,t.Taking the inverse of local fractional Laplace transform on (11), we obtain(12)ux,t=ux,0+Lα-11sαLαgx,t-Lα-11sαLαRαux,t.We are going to represent the solution in an infinite series given below:(13)ux,t=∑n=0∞unx,t.Substitute (13) into (12), which gives us this result(14)∑n=0∞unx,t=ux,0+Lα-11sαLαgx,t-Lα-11sαLαRα∑n=0∞unx,t.When we compare the left- and right-hand sides of (14), we obtain(15)u0x,t=ux,0+Lα-11sαLαgx,t,u1x,t=-Lα-11sαLαRαu0x,t,u2x,t=-Lα-11sαLαRαu1x,t⋮The recursive relation, in general form, is (16)u0x,t=ux,0+Lα-11sαLαgx,t,un+1x,t=-Lα-11sαLαRαunx,t,n≥0.


### 3.2. Local Fractional Laplace Variational Iteration Method (LFLDM)

According to the rule of local fractional variational iteration method, the correction local fractional functional for (8) is constructed as [13](17)un+1t=unt+I0tαλt-ξαΓ1+αLαunξ+Rαu~nξ-gξ,where λ(t-ξ)α/Γ(1+α) is a fractal Lagrange multiplier.We now take Yang-Laplace transform of (17); namely,(18)Lαun+1t=Lαunt+LαI0tαλt-ξαΓ1+αLαunξ+Rαu~nξ-gξ,or (19)Lαun+1t=Lαunt+LαλtαΓ1+αLαLαunt+Rαu~nt-gt.Take the local fractional variation of (19), which is given by(20)δαLαun+1t=δαLαunt+δαLαλtαΓ1+αLαLαunt-Rαu~nt-gt.By using computation of (20), we get(21)δαLαun+1t=δαLαunt+LαλtαΓ1+αδαLαLαunt=0.Hence, from (21) we get(22)1+LαλtαΓ1+αsα=0,where(23)δαLαLαunt=δαsαLαunt-un0=sαδαLαunt.Therefore, we get(24)LαλtαΓ1+α=-1sα.Therefore, we have the following iteration algorithm: (25)Lαun+1t=Lαunt-1sαLαLαunt+Rαunt-gt=Lαunt-1sαLαLαunt-1sαLαRαunt-gt=Lαunt-1sαLαsαunt-u0-1sαLαRαunt-gt=1sαu0-1sαLαRαunt-gt,where the initial value reads as(26)Lαu0t=1sαu0.Thus, the local fractional series solution of (8) is(27)ux,t=limn→∞Lα-1Lαunx,t.


## 3.1. Local Fractional Laplace Decomposition Method (LFLDM)

Taking Yang-Laplace transform on (8), we obtain(9)LαLαux,t+LαRαux,t=Lαgx,t.By applying the local fractional Laplace transform differentiation property, we have(10)sαLαux,t-ux,0+LαRαux,t=Lαgx,t,or (11)Lαux,t=1sαux,0+1sαLαgx,t-1sαLRαx,t.Taking the inverse of local fractional Laplace transform on (11), we obtain(12)ux,t=ux,0+Lα-11sαLαgx,t-Lα-11sαLαRαux,t.We are going to represent the solution in an infinite series given below:(13)ux,t=∑n=0∞unx,t.Substitute (13) into (12), which gives us this result(14)∑n=0∞unx,t=ux,0+Lα-11sαLαgx,t-Lα-11sαLαRα∑n=0∞unx,t.When we compare the left- and right-hand sides of (14), we obtain(15)u0x,t=ux,0+Lα-11sαLαgx,t,u1x,t=-Lα-11sαLαRαu0x,t,u2x,t=-Lα-11sαLαRαu1x,t⋮The recursive relation, in general form, is (16)u0x,t=ux,0+Lα-11sαLαgx,t,un+1x,t=-Lα-11sαLαRαunx,t,n≥0.


## 3.2. Local Fractional Laplace Variational Iteration Method (LFLDM)

According to the rule of local fractional variational iteration method, the correction local fractional functional for (8) is constructed as [13](17)un+1t=unt+I0tαλt-ξαΓ1+αLαunξ+Rαu~nξ-gξ,where λ(t-ξ)α/Γ(1+α) is a fractal Lagrange multiplier.We now take Yang-Laplace transform of (17); namely,(18)Lαun+1t=Lαunt+LαI0tαλt-ξαΓ1+αLαunξ+Rαu~nξ-gξ,or (19)Lαun+1t=Lαunt+LαλtαΓ1+αLαLαunt+Rαu~nt-gt.Take the local fractional variation of (19), which is given by(20)δαLαun+1t=δαLαunt+δαLαλtαΓ1+αLαLαunt-Rαu~nt-gt.By using computation of (20), we get(21)δαLαun+1t=δαLαunt+LαλtαΓ1+αδαLαLαunt=0.Hence, from (21) we get(22)1+LαλtαΓ1+αsα=0,where(23)δαLαLαunt=δαsαLαunt-un0=sαδαLαunt.Therefore, we get(24)LαλtαΓ1+α=-1sα.Therefore, we have the following iteration algorithm: (25)Lαun+1t=Lαunt-1sαLαLαunt+Rαunt-gt=Lαunt-1sαLαLαunt-1sαLαRαunt-gt=Lαunt-1sαLαsαunt-u0-1sαLαRαunt-gt=1sαu0-1sαLαRαunt-gt,where the initial value reads as(26)Lαu0t=1sαu0.Thus, the local fractional series solution of (8) is(27)ux,t=limn→∞Lα-1Lαunx,t.


## 4. Illustrative Examples

In this section three examples for Fokker Planck equation are presented in order to demonstrate the simplicity and the efficiency of the above methods.Example 1.
Let us consider the following Fokker Planck equation on Cantor sets with local fractional derivative in the form(28)∂αux,t∂tα=-∂αux,t∂xα+∂2αux,t∂x2α,subject to the initial value(29)ux,0=Eα-xα.(I) By Using LFLDM. In view of (16) and (28) the local fractional iteration algorithm can be written as follows:(30)u0x,t=Eα-xα,un+1x,t=Lα-11sαLα-∂αunx,t∂xα+∂2αunx,t∂x2α,n≥0.Therefore, from (30) we give the components as follows:(31)u0x,t=Eα-xα,u1x,t=Lα-11sαLα-∂αu0x,t∂xα+∂2αu0x,t∂x2α=Lα-12s2αEα-xα=2tαΓ1+αEα-xα,u2x,t=Lα-11sαLα-∂αu1x,t∂xα+∂2αu1x,t∂x2α=Lα-14s3αEα-xα=4t2αΓ1+2αEα-xα,u3x,t=Lα-11sαLα-∂αu2x,t∂xα+∂2αu2x,t∂x2α=Lα-18s4αEα-xα=8t3αΓ1+3αEα-xα⋮Finally, we can present the solution in local fractional series form as(32)ux,t=Eα-xα1+2tαΓ1+α+4t2αΓ1+2α+8t3αΓ1+3α+⋯=Eα-xαEα2tα=Eα2tα-xα.(II) By Using LFLVIM. Using relation (25) we structure the iterative relation as(33)Lαun+1x,t=Lαunx,t-1sαLα∂αunx,t∂tα+∂αunx,t∂xα-∂2αunx,t∂x2α=Lαunx,t-1sαsαLαunx,t-unx,0-1sαLα∂αunx,t∂xα-∂2αunx,t∂x2α,where the initial value is given by(34)Lαu0x,t=LαEα-xα=1sαEα-xα.Therefore, the successive approximations are(35)Lαu1x,t=Lαunx,t-1sαsαLαunx,t-unx,0-1sαLα∂αunx,t∂xα-∂2αunx,t∂x2α=1sαEα-xα+2s2αEα-xα,Lαu2x,t=Lαu1x,t-1sαsαLαu1x,t-u1x,0-1sαLα∂αu1x,t∂xα-∂2αu1x,t∂x2α=1sαEα-xα+2s2αEα-xα+4s3αEα-xα=Eα-xα1sα+2s2α+4s3α,Lαu3x,t=Lαu2x,t-1sαsαLαu2x,t-u2x,0-1sαLα∂αu2x,t∂xα-∂2αu2x,t∂x2α=1sαEα-xα+2s2αEα-xα+4s3αEα-xα+8s4αEα-xα=Eα-xα1sα+2s2α+4s3α+8s4α⋮Hence, the local fractional series solution is(36)ux,t=limn→∞Lα-1Lαunx,t=limn→∞Lα-1Eα-xα∑k=0n2ksk+1α=Eα-xα∑k=0∞2ktkαΓ1+kα=Eα2tα-xα.Example 2.
We present the Fokker Planck equation on a Cantor set with local fractional derivative(37)∂αux,t∂tα=-∂αux,t∂xα+∂2αux,t∂x2α,and the initial condition is(38)ux,0=x2αΓ1+2α.(I) By Using LFLDM. In view of (16) and (37) the local fractional iteration algorithm can be written as follows:(39)u0x,t=x2αΓ1+2α,un+1x,t=Lα-11sαLα-∂αunx,t∂xα+∂2αunx,t∂x2α,n≥0.Therefore, from (39) we give the components as follows: (40)u0x,t=x2αΓ1+2α,u1x,t=Lα-11sαLα-∂αu0x,t∂xα+∂2αu0x,t∂x2α=Lα-1-1s2αxαΓ1+α+1s2α=-tαΓ1+αxαΓ1+α+tαΓ1+α,u2x,t=Lα-11sαLα-∂αu1x,t∂xα+∂2αu1x,t∂x2α=Lα-11s3α=t2αΓ1+2α,u3x,t=Lα-11sαLα-∂αu2x,t∂xα+∂2αu2x,t∂x2α=0,u4x,t=Lα-11sαLα-∂αu3x,t∂xα+∂2αu3x,t∂x2α=0⋮Finally, we can present the solution in local fractional series form as (41)ux,t=x2αΓ1+2α+t2αΓ1+2α-xαΓ1+αtαΓ1+α+tαΓ1+α.(II) By Using LFLVIM. Using relation (25) we structure the iterative relation as(42)Lαun+1x,t=Lαunx,t-1sαLα∂αunx,t∂tα+∂αunx,t∂xα-∂2αunx,t∂x2α=Lαunx,t-1sαsαLαunx,t-unx,0-1sαLα∂αunx,t∂xα-∂2αunx,t∂x2α,where the initial value is given by(43)Lαu0x,t=Lαx2αΓ1+2α=1sαx2αΓ1+2α.Therefore, the successive approximations are(44)Lαu1x,t=Lαu0x,t-1sαsαLαu0x,t-u0x,0-1sαLα∂αu0x,t∂xα-∂2αu0x,t∂x2α=1sαx2αΓ1+2α-1s2αxαΓ1+α+1s2α,Lαu2x,t=Lαu1x,t-1sαsαLαu1x,t-u1x,0-1sαLα∂αu1x,t∂xα-∂2αu1x,t∂x2α=1sαx2αΓ1+2α-1s2αxαΓ1+α+1s3α+1s2α,Lαu3x,t=Lαu2x,t-1sαsαLαu2x,t-u2x,0-1sαLα∂αu2x,t∂xα-∂2αu2x,t∂x2α=1sαx2αΓ1+2α-1s2αxαΓ1+α+1s3α+1s2α⋮Hence, the local fractional series solution is(45)ux,t=limn→∞Lα-1Lαunx,t=limn→∞Lα-11sαx2αΓ1+2α+1s3α-1s2αxαΓ1+α+1s2α=x2αΓ1+2α+t2αΓ1+2α-xαΓ1+αtαΓ1+α+tαΓ1+α.Example 3.
We consider the Fokker Planck equation on a Cantor set with local fractional derivative(46)∂αux,t∂tα=-∂αux,t∂xα+∂2αux,t∂x2α,and the initial condition is(47)ux,0=-x3αΓ1+3α.(I) By Using LFLDM. In view of (16) and (46) the local fractional iteration algorithm can be written as follows:(48)u0x,t=-x3αΓ1+3α,un+1x,t=Lα-11sαLα-∂αunx,t∂xα+∂2αunx,t∂x2α,n≥0.Therefore, from (30) we give the components as follows: (49)u0x,t=-x3αΓ1+3α,u1x,t=Lα-11sαLα-∂αu0x,t∂xα+∂2αu0x,t∂x2α=Lα-11s2αx2αΓ1+2α-1s2αxαΓ1+α=x2αΓ1+2αtαΓ1+α-xαΓ1+αtαΓ1+α,u2x,t=Lα-11sαLα-∂αu1x,t∂xα+∂2αu1x,t∂x2α=Lα-1-1s3αxαΓ1+α+2s3α=-xαΓ1+αt2αΓ1+2α+2t2αΓ1+2α,u3x,t=Lα-11sαLα-∂αu2x,t∂xα+∂2αu2x,t∂x2α=Lα-11s4α=t3αΓ1+3α,u4x,t=Lα-11sαLα-∂αu3x,t∂xα+∂2αu3x,t∂x2α=0⋮Finally, we can present the solution in local fractional series form as(50)ux,t=-x3αΓ1+3α+x2αΓ1+2αtαΓ1+α-xαΓ1+αtαΓ1+α-xαΓ1+αt2αΓ1+2α+2t2αΓ1+2α+t3αΓ1+3α.(II) By Using LFLVIM. Using relation (18) we structure the iterative relation as(51)Lαun+1x,t=Lαunx,t-1sαLα∂αunx,t∂tα+∂αunx,t∂xα-∂2αunx,t∂x2α=Lαunx,t-1sαsαLαunx,t-unx,0-1sαLα∂αunx,t∂xα-∂2αunx,t∂x2α,where the initial value is given by (52)Lαu0x,t=Lα-x3αΓ1+3α=-1sαx3αΓ1+3α.Therefore, the successive approximations are (53)Lαu1x,t=Lαu0x,t-1sαsαLαu0x,t-unx,0-1sαLα∂αu0x,t∂xα-∂2αu0x,t∂x2α=-1sαx3αΓ1+3α+1s2αx2αΓ1+2α-1s2αxαΓ1+α,Lαu2x,t=Lαu1x,t-1sαsαLαu1x,t-u1x,0-1sαLα∂αu1x,t∂xα-∂2αu1x,t∂x2α=-1sαx3αΓ1+3α+1s2αx2αΓ1+2α-1s2αxαΓ1+α-1s3αxαΓ1+3α+2s3α,Lαu3x,t=Lαu2x,t-1sαsαLαu2x,t-u2x,0-1sαLα∂αu2x,t∂xα-∂2αu2x,t∂x2α=-1sαx3αΓ1+3α+1s2αx2αΓ1+2α-1s2αxαΓ1+α-1s3αxαΓ1+3α+2s3α+1s4α,Lαu4x,t=Lαu3x,t-1sαsαLαu3x,t-u3x,0-1sαLα∂αu3x,t∂xα-∂2αu3x,t∂x2α=-1sαx3αΓ1+3α+1s2αx2αΓ1+2α-1s2αxαΓ1+α-1s3αxαΓ1+3α+2s3α+1s4α⋮Hence, the local fractional series solution is(54)ux,t=limn→∞Lα-1Lαunx,t=-x3αΓ1+3α+x2αΓ1+2αtαΓ1+α-xαΓ1+αtαΓ1+α-xαΓ1+αt2αΓ1+2α+2t2αΓ1+2α+t3αΓ1+3α.


## 5. Conclusions

In this work solving Fokker Planck equation by using the local fractional Laplace decomposition method and local fractional Laplace variational iteration method with local fractional operators is discussed in detail. Three examples of applications of these methods are investigated. The reliable obtained results are complementary to the ones presented in the literature.



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