Solving of Klein-Gordon by Two Methods of Numerical Analysis

In this paper, the Decomposion Laplace-Adomian method and He-Laplace method are used to construct the solution of Klein-Gordon equation.


Introduction
In field theory, the description of the free partide for the wave function in quantum physics obeys to Klein-Gordon equation [1].In addition, it also appears in nonlinear optics and plasma physics.
In sum, the Klein-Gordon equation rises in physics in linear and non linear forms.In this paper we examine the Klein-Gordon equation, using the Laplace-Adomian decomposition method and He-Laplace method to get the exact solution.The Klein-Gordon equation is described as: where , α β are constants (spin zero) charged field, ( ) , h x t is a source term and ( ) ( ) , N u x t is a nonlinear function of ( ) , u x t .

The Laplace Transform [2]
Let's note the laplace transform by ( )

U x s u x t u x t t
From (1), we have:

Laplace-Adomian Decomposition Method (LADM) [3]-[6]
Suppose that we need to solve the following equation: subject to initial conditions: E is a Banach space, where is a linear or a nonlinear operator, h E ∈ and u is the unknown function.
Let's suppose that operator F can be decomposed under the following form: where L R + is linear, N nonlinear.Let's suppose that L is inversible to the sense of Adomian with 1 L − as inverse.
From above, by applying the Laplace transform to both sides of Equation (4), we have: From the Equation (7), it follows: and this equation gives So, from the above Equation (9), we can write: We have now ( ) , , We research solution of (4) in the following series expansion form and we consider ( ) where n A are the Adomian polynomials of 0 1 , , , n u u u  and it can be calculated by formula given below.

Remark
In order overcome the short coming, we assume that ( ) Instead of the iteration procedure Equation (17) we suggest the following modification , , 1 The solution through the modified Laplace decomposition method highly depends upon the choice of ( )

He-Laplace Method [7]
We consider a general nonlinear non homogeneous partial differential equation with initial conditions of the form , 0 N represents the general nonlinear differential operateur and ( ) Taking the Laplace transform on both sides of (20), we obtain: Applying the initial conditions given in (22), we Operating the inverse Laplace transform on both sides of (23), we have , , Now, we apply the homotopy perturbation method and the non linear term can be decomposed as for some He's polynomials ( ) n H u that are given by ( ) Sustituding Equation (25) and Equation (26) in Equation ( 24), we get Comparing the coefficients of like powers of p, we have the following approximations: , ,

Illustrative Examples
To demonstrate the applicability of the above-presented method, we have applied it to two linear and two non linear partial differential equations.These examples have been chosen because they have been widely discussed in literature.

Example 1
Consider the following linear Klein-Gordon equation , 0 cos

Application of the LADM
Applying the Laplace transform on both side of Equation (30) with the initial conditions, we have: The inverse Laplace transform give us: We suppose that solution of (30) has the following form: From ( 34) and (33).we have: This result garantee that the following Adomian algorithm is: Consequently,we obtain: So that the solution of (30) is given by which is the exact solution of problem.

Application of the He-Laplace Method
Applying the Laplace transform on both side of Equation ( 30) with the initial conditions, we obtain: By applying inverse Laplace transform, we have: Now applying the homotopy perturbation method, we have: Comparing the coefficient of like powers of p, we have , , , So that, the solution ( ) , u x t is given by: ( ) ( )

Laplace-Adomian Method
Using the Laplace transform, we have , 0 by applying inverse Laplace transformation to Equation (48), we hace ( ) ( ) ( ) Supposing that the solution of (46) has the following form: and Taking (50) and (51) in to (49), we obtain: According to the standard Adomian algorithm (52), we need to chose ( ) then garantee that: So the exact solution of ( 46) is

He-Laplace Method
Using the Laplace transform, we have: Now, we apply the inverse Laplace transformation to Equation (46), we have: Applying the homotopy perturbation method, we have: Comparing the coefficients of the like powers of p, we have: ( ) , 30 ) So that, the exact solution ( ) , u x t is given by: 4. Applications

Problem 1
Consider the following linear Klein-Gordon equation (

Application of the LADM
Using the Laplace transform, we have By appling the inverse Laplace transform, we have: From above equation, we have the following modified Adomian allgorithm: Equation ( 69) give us: , sin sin , 0, and the exact solution of Equation ( 64) is ( )

Problem 2
Consider the following nonlinear Klein-Gordon equation , 0 Application of the LADM Using the Laplace transform from (73), we have: Now, we apply the inverse Laplace transform, we have: Denoting that the solution of (73) has the following form: Taking ( 77) and ( 78) into (76), we have: and we obtain the following Adomian algorithm: As before, we defines the solution ( ) He's polynomials.The first few components of He's polynomials are given by = is decomposed in term of Adomian polynomials