Variational Iteration Algorithm-I with an Auxiliary Parameter for Solving Fokker-Planck Equation

In this paper, variational iteration algorithm-I with an auxiliary parameter is implemented to investigate Fokker-Planck equations. To show the accuracy and reliability of the technique comparisons are made between the variational iteration algorithm-I with an auxiliary parameter and classic variational iteration algorithm-I. The comparison shows that variational iteration algorithm-I with an auxiliary parameter is more powerful and suitable method for solving Fokker-Planck equations. Furthermore, the proposed algorithm can successfully be applied to a large class of nonlinear and linear problems.


Introduction
The aim of this work is to apply the variational iteration algorithm-I [1] with an auxiliary parameter for the analytical treatment of the Fokker-Planck equation.The method is able to provide analytical results for nonlinear and linear problems, in a direct way very conveniently.One of the main characteristics of this method is that approximate solution of great accuracy can be obtained by only a few iterations.This method has a simple procedure, acceptable results and above all, this method can successfully be applied to a large class of linear and nonlinear problems [2]- [6].

Variational Iteration Algorithm-I
Consider a general differential equation the terms represent the linear and nonlinear term respectively, while ( ) is the inhomogeneous source term.Constructing a correction function for Eq.
(1) as, where λ is a parameter, which is not known and called the Lagrange multiplier [7].
Taking the variation δ on the one side as well as the other side of Eq. ( 2) with respect to ( ), where ( ) k u η is considered as a restricted term which means Using optimality conditions, the value of Lagrange multiplier ( ) η λ can be identified.
An exact solution obtains when .
In short, the formula for equation (1) is,

Insertion of an Auxiliary Parameter in Variational Iteration Algorithm-I
In VIA-I, an auxiliary parameter h can be inserted.The optimal choice of unknown h improves the correctness, precision and effectiveness of the technique.After inserting h, equation (5) will become approximat initial e appropriat an is 0 1 This technique is known as VIA-I with AP.Actually, this technique is simple, has a lesser size of calculation, not difficult to analyze and have the ability to approximate the solution precisely in solution domain of wide range.

The Fokker-Planck Equation [2]
In this section, the general form of Fokker-Planck equation which is also called forward Kolmogorov equation is with conditions: It is the equation for the motion of concentration field ( ).

, t x u
The backward Kolmogorov equation can be written in the following form Let the initial conditions, ( ) ( ) Then equation ( 8) becomes First, we solve this example by VIA-I.
Constructing the correction function for equation ( 9) as, ( ) ( ) Taking the variation δ on the one side as well as the other side with respect to ( ) Ignoring the restricted terms The stationary conditions are: we get the value of ( ) η λ which is ( ) Using this value of ( ) η λ in equation ( 10) results in the below iterative scheme: ( ) ( ) other approximations by using the scheme (11),  The absolute error of ( ) can be seen in Figure 1.Using VIA-I with AP, the recurrence relation for equation ( 9) is Other approximations can be get by using the recurrence relation ( 12), ( , , The following residual function is defined ( The square of residual function for 10th-order approximation with respect to h for  Comparing Figure 1 and Figure 2, it is clear that VIA-I with AP gives better results as compared to VIA-I.Numerical comparison betwixt the exact and approximate solutions of both methods is given in the table below.

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The above table shows that VIA-I with AP is better for a large domain of t as compared to VIA-I.

Conclusions
In this paper, variational iteration algorithm-I with an auxiliary parameter has been used in a way that accomplished the desired aim for solving Fokker-Planck equation.This work has made sure that the variational iteration algorithm-I with an auxiliary parameter offers noteworthy advantages in terms of its easy applicability, its computational success, and its adequacy to solve a wide class of differential equations.Graphical and numerical results reveal that this modification of variational iteration algorithm-I is suitable for all linear and nonlinear problems arise in physical sciences and engineering, superior to the variational iteration algorithm-I.

Figure 1 .
Figure 1.Absolute error betwixt the exact and approximate solutions by VIA-I.
minimum value of above square residual function occurs at .solution domain ( ) [ ] [ ], the exact and approximate solutions can be seen in Figure2.

Figure 2 .
Figure 2. Absolute error betwixt the approximate and exact solutions by VIA-I with AP.

Table 1 .
Comparison of absolute errors for 6th order approximation by VIA-I and VIA-I with AP.