Applications of OHAM and MOHAM for Time-Fractional Klein-Fock-Gordon Equation

In this paper the optimal homotopy asymptotic method (OHAM) and multistage optimal homotopy asymptotic method (MOHAM) are applied to obtain an analytic approximate solution to a time-fractional Klein-Fock-Gordon (FKFG) equation. The FKFG equation plays an important role in characterizing the relativistic electrons. The MOHAM relies on OHAM to obtain analytic approximate solutions, it actually applies OHAM in each subinterval and we show that it achieves better results than OHAM over the large intervals; this is one of the advantages of this method which can be used for large intervals and to obtain good results. The convergence of the method is also addressed.


Introduction
Fractional calculus (FC) is a part of mathematical analysis that studies the derivation of integrals and derivatives of rational orders [1]. The concept of fractional calculus (integrals and derivatives of any rational order) is established over 300 years ago, and nowadays is a very important subject. Gradually, researchers in different fields of sciences have discovered that fractional differential models have much better descriptors for different phenomena. Fractional calculus has widespread applications of in physics, chemistry, economics, dynamic systems, medical engineering, biological sciences, imaging, etc. On the other hand, physicists Klein, Fock, x u x f x and u x g x = = (2) Where a and b are real constants and n is a positive integer.
OHAM results in to satisfactory solutions on short domains, but when the interval becomes longer, the accuracy of the method decreases, so a new approach was proposed by Anakira, et al. which is called multistage optimal homotopy asymptotic method (MOHAM) that suitable for analytic approximate solutions for large intervals [18].
Finally, the approximate solutions obtained from both methods are compared with the exact solution.

Basic Definitions of Fractional Calculus
In this section, some basic definitions of fractional calculus are explained briefly [1,19]. µ > − is defined as follows By considering f C µ ∈ , 1 µ > − , , 0 α β > and 1 γ > − , the main properties of the operator J α are listed as follows Definition 4.3: The fractional-order derivative of ( ), f t in Caputo sense, is defined as follows and 1 µ ≥ − , the following properties will be valid.

Basic Principles of the Proposed Techniques
A short introduction to the methods that will be used in this research.

OHAM
Let's consider the following fractional equation with the boundary conditions ( ) Where ξ is an independent variable, , , i i A deformation equation of zero-order as the following Where p is an embedding parameter in the interval [0,1], ( , ) H p ξ , is an auxiliary function with non-zero and zero outputs for 0 p ≠ and 0 p = , respectively and 0 ( ) u ξ represents the initial condition of ( ) u ξ , and ( , ) p ϕ ξ is an unknown function. By inserting 0 p = and 1 into Eq. (13), the following functions are obtained Therefore, ( , ) p ϕ ξ will change continuously from the initial guess to the solution, 0 ( ) u ξ , to ( ) u ξ when p increase from 0 up to 1.
By putting = 0 p into Eq. (13), the initial solution 0 ( ) u ξ is determined as a solution for the problem Next, choose an auxiliary function ( ) H p in the following form By putting Eqs. (14)(15)(16)(17)(18) into (13), and equating the coefficients of the terms with identical powers of p , one will obtain the governing equation of the initial approximation 0 ( ) u ξ , given by Eq. (16), and then the governing equation of the first order problem is defined as And the governing equation of the m th order is defined as  The approximate solution of Eq. (11) can be calculated as can be determined based on the following conditions In order to get an analytic approximate solution at the level m , the obtained optimal coefficients will be substituted in Equation (23).

MOHAM
Although the OHAM is used to obtain approximate solutions of nonlinear problems. It has some disadvantage in nonlinear problems with large domain. To control this drawback, we introduce in this section a multistage OHAM to obtain the nonlinear problem with long of the domain. A simple way to confirm the validity of the approximate solutions of large T is by dividing the interval [0, ] T in to subinterval as ( , , ,..., ) ( ( , , ,..., )) ( ) ( ( , , ,..., )).
If 0 i R = , then i y will be the exact solution. Generally, such a case will not arise for nonlinear problems, but we can minimize the function Thus, the analytic approximate solution will be as follows By this way, we successfully gain the solution of the initial value problem for a large interval T . It should be noted that if 0 J = the MOHAM expresses the OHAM. One of the benefits of MOHAM is that it provides a simple way to control convergence and regulate convergence region and adjust the convergence region though the auxiliary function ( ) In general, this method eliminates the difficulty of finding approximate solutions in large ranges.

Proof:
Since the series According to (38) and the limit, we have Applying the linear operator Eq.(40) can be written as following So by choosing the optimal , 1, 2,3,... k c k = , Eq.(41) is converted to the following Which is the exact solution of the problem.

Solution of the Fractional KFG Equation
We consider one of the nonlinear cases of FKFG equation, for

Solution of FKFG equation by OHAM
Having the linear operator t L D α = and nonlinear operator The First-order problem: The Second-order problem:  , and 0 0 t = up to 2 1 t T = = as in Table 2.
Approximate solution for 2 α = is as the following form    Figure 1, Figure 2, Figure 3 and Figure 4 one can see that the solutions obtained by OHAM and MOHAM are nearly identical with the exact solution Table 3.

Conclusion
In this study OHAM and MOHAM are used to derive an analytic approximate solution for the time-frac-tional Klein-Fock-Gordon (FKFG) equation. The results obtained from these methods show that MOHAM converges better than OHAM. One observes that the results agree very well with the exact solution. The MOHAM by dividing the interval [ , ] o T can obtain better solution than OHAM. As far as the authors are aware, MOHAM has not been used to solve fractional partial differential equations, so far, the method has been tested on fractional-PDE and yields to satisfactory results. The Figures and Tables expose that good results are obtained by MOHAM and more accurate solution as compared to OHAM. The convergence of