NUMERICAL COMPUTATION OF SUCCESSIVE APPROXIMATIONS METHOD AND VARIATIONAL ITERATION METHOD FOR SOLVING KLEIN-GORDON SCHRÖDINGER EQUATION

This paper is devoted to investigating and comparing the Successive Approximations Method (SAM) and Variational Iteration Method (VIM) for solving Klein-Gordon Schrödinger (KGS) Equation. Furthermore, the approximate solutions that obtained by both methods have been represented numerically and graphically.


INTRODUCTION
Nowadays, many authors have their attention toward studying the solution of linear and nonlinear partial differential equations. Because, partial differential equations can be used as a proper tool for describing most of the natural phenomena of engineering and science models. Moreover, a wide range of significant phenomena arising in physics, biology, mathematics, mathematical 4068 SAAD A. MANAA, FADHIL H. EASIF, JOMAA J. MURAD physics and another fields are modeled through partial differential equations. While, finding the exact solutions of partial differential equations has become one of the most significant and challenging problems in engineering and physics, biology, mathematics and other science [1]- [2]. Therefore, numerical methods were applied to overcome such problems. Currently, the numerical methods are in competition with each other to find the best and more accurate approximate solutions [3].

Consider the Klein-Gordon Schrödinger (KGS) Equation [12]
Where , , , are considered as arbitrary constants. While, is the complex nucleon field and is the neutral real meson fields. Moreover, the system (1) has wide-range applications in many fields such as quantum physics and modern physics [12]- [13].
We can separate equation (1) into real and imaginary parts. Therefore, one can obtain a tripled system (a system of three real equations) in the following form:

BASIC IDEA OF SUCCESSIVE APPROXIMATIONS METHOD [3]-[6]:
Consider the following general nonlinear partial differential equation: Subject to the initial condition: For = , ∈ ℕ , is the highest order partial derivative with respect to time . Moreover, the reminder linear term is , the nonlinear operator is and the inhomogeneous source term is The SAM considers the approximate solution of an integral equation as a sequence, which is usually, convergent to an accurate solution. For solving equation (4) by using SAM, we apply −1 [. ], which is: on both sides of equation (4), we get: The solution of (7) by SAM is a sequence as follows: Now, for ( , ) = 0, then SAM introduces the recurrence relation of the form: Therefore, The solution is computed as: The Successive Approximations Method is simple in its principles. While the difficulties appear in proving the convergence of the introduced series [3].

BASIC IDEA OF VARIATIONAL ITERATION METHOD [6]-[11]
Applying VIM on eq. (4) , then we write the correction functionals of equation (4) Where is called a Lagrange multiplier, which will be identified optimally by variational iteration method. Now, Ũ is a restricted variation, which demonstrates that Ũ = 0. Creating the correct functional of (4), that yields: Therefore, its stationary conditions can be find by applying integration by parts on equation (12).
Then, we get the general form of Lagrange multiplier as follows [11]: Substituting equation (12) into equation (13)  Then, the approximate solution of equation (4) is given by:

DERIVATION OF SAM FOR SOLVING KGS-SYSTEM
Applying For all ≥ 0 .
To find the initial approximations, we use equation (5) Where is the closing iteration step.

DERIVATION OF VIM FOR SOLVING KGS-SYSTEM
The variational iteration formula in equation (14) is used to find the iteration formulas of equation Where is the closing iteration step.

APPLICATION WITH NUMERICAL RESULTS (TABLES, FIGURES)
This section will be devoted to find the numerical results (Tables, Figures)

CONCLUSION
In this paper the Klein-Gordon Schrödinger (KGS) system was solved numerically by using Successive Approximations Method and Variational Iteration Method. We took an example of (KGS) equation to find the comparison between our solutions and the exact solution, and we showed that both methods are very accurate and effective in solving (KGS) equation. However, it is clear form Table1 and Fingure1 that the obtained solutions for ( , ) by both methods are equivalent. While, Table2 and Figure2 showed that the obtained solutions for ( , ) by VIM is more accurate than SAM.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.