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

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.


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.
Local fractional Fokker Planck equation, which was an analog of a diffusion equation with local fractional derivative, was suggested in [5] as follows: The Fokker Planck equation on a Cantor set with local fractional derivative was presented in [6,7] as follows: subject to the initial condition  (, 0) =  () . ( 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.
where the latter integral converges and   ∈   .

Analytical Methods
In order to illustrate two analytical methods, we investigate the local fractional partial differential equation as follows: where   =   /  denotes the linear local fractional differential operator,   is the remaining linear operators, and (, ) is a source term of the nondifferential functions.
Taking the inverse of local fractional Laplace transform on (11), we obtain We are going to represent the solution in an infinite series given below: Substitute ( 13) into (12), which gives us this result When we compare the left-and right-hand sides of ( 14), we obtain The recursive relation, in general form, is (16)

Journal of Mathematics 3
We now take Yang-Laplace transform of (17); namely, or Take the local fractional variation of (19), which is given by By using computation of (20), we get Hence, from (21) we get where Therefore, we get Therefore, we have the following iteration algorithm: where the initial value reads as Thus, the local fractional series solution of ( 8) is

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 subject to the initial value (I) By Using LFLDM.In view of ( 16) and (28) the local fractional iteration algorithm can be written as follows: Therefore, from (30) we give the components as follows: Finally, we can present the solution in local fractional series form as (II) By Using LFLVIM.Using relation (25) we structure the iterative relation as where the initial value is given by Therefore, the successive approximations are Hence, the local fractional series solution is and the initial condition is (I) By Using LFLDM.In view of ( 16) and (37) the local fractional iteration algorithm can be written as follows: Therefore, from (39) we give the components as follows: Finally, we can present the solution in local fractional series form as (II) By Using LFLVIM.Using relation (25) we structure the iterative relation as where the initial value is given by Therefore, the successive approximations are Hence, the local fractional series solution is and the initial condition is (I) By Using LFLDM.In view of ( 16) and ( 46) the local fractional iteration algorithm can be written as follows: Therefore, from (30) we give the components as follows: Finally, we can present the solution in local fractional series form as (II) By Using LFLVIM.Using relation (18) we structure the iterative relation as where the initial value is given by Therefore, the successive approximations are  (54)

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.