Solvability of nonlinear Klein-Gordon equation by Laplace Decomposition Method

A wide variety of physically significant problems such as nonlinear Klein-Gordon equation, modeled by linear and nonlinear partial differential equations has been the focus of extensive studies for the last decades. A huge number of research and investigations have been invested in these scientific applications. Several approaches such as the characteristics method, spectral methods and perturbation techniques have been extensively used to examine these problems (Wazwaz, 2009). Solving of nonlinear equations using Adomian decomposition method (ADM) has been done in Wazwaz 2009; 2006; El-Wakil et al. 2006; Adomian 1994;1984;1986; Abassy et al. 2007; 2004; Cherrualt 1990; Lesnic 2006; 2007; Wazwaz 2001; Mohammed and Tarig 2013; 2014 and modified decomposition method (MD) in Mohammed and Tarig 2013; 2014. The aim of this paper is in two folds: firstly, to solve the nonlinear Klein-Gordon equation via LDM, ADM and MD. Secondly, to show these three methods yielded exactly the same result. As we know the nonlinear Klein-Gordon, equation comes from quantum field theory and describes nonlinear wave interaction. The nonlinear Klein-Gordon equation in its standard form is


INTRODUCTION
A wide variety of physically significant problems such as nonlinear Klein-Gordon equation, modeled by linear and nonlinear partial differential equations has been the focus of extensive studies for the last decades.A huge number of research and investigations have been invested in these scientific applications.
Several approaches such as the characteristics method, spectral methods and perturbation techniques have been extensively used to examine these problems (Wazwaz, 2009).Solving of nonlinear equations using Adomian decomposition method (ADM) has been done in Wazwaz 2009;2006;El-Wakil et al. 2006;Adomian 1994;1984;1986;Abassy et al. 2007;2004;Cherrualt 1990;Lesnic 2006;2007;Wazwaz 2001;Mohammed and Tarig 2013;2014 and modified decomposition method (MD) in Mohammed and Tarig 2013;2014.The aim of this paper is in two folds: firstly, to solve the nonlinear Klein-Gordon equation via LDM, ADM and MD.Secondly, to show these three methods yielded exactly the same result.
As we know the nonlinear Klein-Gordon, equation comes from quantum field theory and describes nonlinear wave interaction.The nonlinear Klein-Gordon equation in its standard form is Subject to the initial conditions Where a is a constant,   , h x t is a source term and In this work, the noise terms phenomenon was used (Wazwaz, 2009), which, provides a major advantage in that it demonstrates a fast convergence of the solution.It is important to note that the noise terms phenomenon, may appear only for inhomogeneous partial differential equations; in addition, this phenomenon is applicable to all inhomogeneous PDEs of any order.The noise terms, if existed in the components and will provide, in general, the solution in a closed form with only two successive iterations.

Solution of Nonlinear Klein -Gordon Equation by ADM
The decomposition method will be employed.The nonlinear term     , F u x t will be equated to the infinite series of Adomian polynomials (Adomian, 1994).In an operator form Equation (1) given by, Where , and Applying to both sides of (3) and using the initial conditions to obtain, We obtain the recursive relation: (5) that leads to: This completes the determination of the first few components of the solution.Based on this determination, the solution in a series form is readily obtained.In many cases, a closed form solution obtained conductively.

Example
Given the following nonlinear Klein-Gordon equation: , ,0 1 , ,0 , following the discussion presented above, we find: Canceling the noise terms The modified decomposition method suggests that Operating with on both sides of Equation ( 9) subject to Equation (2) and using the above assumption, we obtain:

Example:
Consider the initial nonlinear Klein-Gordon problem (7) Following the previous discussion, we find: The solution in a series form given by

Solution of Nonlinear Klein -Gordon Equation by LDM
Here the LDM will be implemented to Klein-Gordon equation .To illustrate the method consider the general form of Klien-Gordon Equation (1) subject to the initial condition (2) and applying Laplace transform (denoted throughout this paper by ) on both sides of Equation (1) , yields: Consider the nonlinear Klein-Gordon Equation ( 7), using Equation ( 16) subject to the initial condition, we get

Conclusion
In this paper, we introduced Klein-Gordon equation, and solved it by using ADM, and MD then applied LDM .Clearly, these three methods are very effective, it accelerates the solutions.If we compare it with the other methods, it will be the best.In addition, the LDM may give the exact solutions for nonlinear PDEs.Moreover, the noise terms may appear if the exact solution is a part of the 'theroth component ( .
Solution of Nonlinear Klein -Gordon Equation by MDIn an operator form Equation (1 form, follow immediately. Laplace transform to the Equation (16), then the required recurrence relation is immediately obtained which complete the solution Application 1:

u
and verifying that the remaining non- canceled terms satisfies the equation, the exact solution Application 2: Consider the nonlinear Equation   −   +  2 = 6 3  − 6 3 +  6  6

u
Following the analysis presented above and using the given initial conditions, we obtain the recursive relation in the form and verifying that the remaining non-canceled terms of 0 u satisfies Equation (21), we find that the exact solution is given by  ,  =  3  3